50:
6677:
5032:
6669:
107:
3953:) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.
3703:
4292:. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.
3695: â 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for
3133:
2582:) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
4373:
In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a Μ-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume
2577:
The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes
4756:
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories
432:
as the founder of dynamical systems. Poincaré published two now classical monographs, "New
Methods of Celestial Mechanics" (1892â1899) and "Lectures on Celestial Mechanics" (1905â1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and
379:
The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all
3901:
is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the
363:
The systems studied may only be known approximatelyâthe parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of
4478:
In a
Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a
2578:
it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the
4139:
This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If
355:, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.
4645:
it becomes possible to classify the ergodic properties of Ί. In using the
Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Ί gets mapped into an infinite-dimensional linear problem
4716:
deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a
4757:
are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of "backwards-in-time unpredictability" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for
4544:
is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function Ï. This introduces an operator
3017:
443:
developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.
4964:
316:. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a
3833:. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.
2695:
402:
The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for
3961:
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point
3197:
2991:
2563:
940:
358:
For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:
248:
involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.
4854:
4462:
1810:
2751:
372:. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their
1580:
2219:
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called
2479:
4134:
674:
2628:
is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of
7253:
4687:
systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called
4636:
3663:
1423:
1365:
1013:
2414:
1080:
3937:
the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space
1151:
4471:, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the
3559:
2786:
733:
4237:. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of
3461:
1908:
1477:
771:
5096:
3786:
1679:
5438:
3128:{\displaystyle {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}
1976:
1940:
1858:
1834:
4266:
it is derived from) depends on a parameter Ό, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the
831:
1194:
4865:
328:.) To determine the state for all future times requires iterating the relation many timesâeach advancing time a small step. The iteration procedure is referred to as
5801:
4993:
4732:
has been known for years to involve complexâeven chaoticâbehavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The
1725:
1237:
5061:
395:
where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the
5019:
6966:
4741:
3885:
The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a
6796:
391:
The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have
5071:
2199:
is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Ί a (locally defined) evolution function. As such
3334:
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the
520:
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer
3902:
approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.
2646:
336:. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a
4245:
2616:
of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
2569:
are studied. For continuous dynamical systems, the map Ί is understood to be a finite time evolution map and the construction is more complicated.
6676:
6436:
4665:. This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems.
4336:
on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on
2605:
and the invariant measures must be singular with respect to the
Lebesgue measure. A small region of phase space shrinks under time evolution.
380:
possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes.
6227:
6208:
6186:
6162:
6131:
6112:
6080:
6058:
6018:
5999:
5980:
5942:
5923:
5904:
5874:
5853:
5825:
5785:
5762:
5740:
5710:
5649:
5607:
5474:
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gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map
6629:
4472:
2586:
3141:
2935:
2609:
6576:
5462:
2304:
437:, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
6909:
4513:
One of the questions raised by
Boltzmann's work was the possible equality between time averages and space averages, what he called the
2517:
837:
6037:
5961:
4701:
perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the
4277:
is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.
226:
6951:
6639:
4479:
sub-manifold of the phase space. The volume of the energy shell, computed using the
Liouville measure, is preserved under evolution.
4388:
2268:. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (
1769:
385:
7144:
5619:
5548:
5390:
2701:
498:
93:
71:
5382:
3945:. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where
2593:, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic
452:
6826:
4774:
3926:
where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.
1510:
5076:
4765:. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.
4762:
4733:
4021:. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes
225:
of the dynamical system is a function that describes what future states follow from the current state. Often the function is
5241:
Gintautas, V.; et al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics".
4160:
they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form
3716:
2633:
1747:
495:
that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.
171:
6644:
6634:
4483:
2579:
1836:
is the domain for time â there are many choices, usually the reals or the integers, possibly restricted to be non-negative.
434:
4697:
are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the
2427:
6104:
5809:
5591:
5086:
4193:
does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of
3478:
the position vector. The solution to this system can be found by using the superposition principle (linearity). The case
3214:
2929:
The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above
6899:
6991:
6764:
5111:
3311:
3301:
3261:
1865:
4222:
4073:
629:
536:
systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of
6176:
4356:
2051:
229:, that is, for a given time interval only one future state follows from the current state. However, some systems are
6904:
433:
studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the
6971:
6358:
4532:
and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.
7243:
6429:
4559:
3724:
3596:
3579: = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the
3362:) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will
3209:
Many of the concepts in dynamical systems can be extended to infinite-dimensional manifoldsâthose that are locally
2258:
1371:
1313:
948:
38:
2380:
1018:
7019:
5411:
4684:
4281:
3688:
it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
3246:
3236:
1085:
506:
3496:
6723:
6668:
6100:
3934:
3200:
2220:
253:
64:
58:
31:
5091:
2762:
685:
7248:
6884:
6649:
6556:
5813:
4344:
3343:
3329:
774:
502:
381:
186:
or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a
6364:
3404:
6924:
6536:
5416:
5037:
4210:
3251:
2597:
the choice of invariant measure is technically more challenging. The measure needs to be supported on the
621:
206:
178:
by allowing different choices of the space and how time is measured. Time can be measured by integers, by
131:
75:
4329:) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of
1883:
1434:
738:
7074:
6981:
6779:
6606:
6541:
6516:
6422:
6250:
4694:
4658:
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4529:
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3720:
3395:
3271:
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2637:
2511:
The map Ί embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the
2358:
1755:
482:
448:
412:
408:
317:
245:
6836:
3733:
3286:
411:. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of
6352:
4031:
The intersection of the periodic orbit with the
Poincaré section is a fixed point of the Poincaré map
1640:
7084:
6914:
6736:
6581:
5260:
5207:
5066:
4758:
3969:
in the orbit Îł and consider the points in phase space in that neighborhood that are perpendicular to
3874:
3291:
373:
369:
4181:
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of
1957:
1921:
1839:
1815:
7044:
7001:
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6831:
6784:
6769:
6754:
6654:
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6531:
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The
Liouville measure restricted to the energy surface Ω is the basis for the averages computed in
4537:
4352:
3981:
3281:
3242:
3227:
2629:
2374:
2075:
2047:
1600:
325:
321:
313:
167:
6301:
4959:{\displaystyle y(t)={\frac {1}{4}}\left(1-{\frac {t}{2}}+\left|1-{\frac {t}{2}}\right|\right)^{2}}
795:
7222:
7089:
6919:
6806:
6801:
6693:
6571:
6473:
6379:
5795:
5595:
5519:
5334:
5276:
5250:
4657:. An average in time along a trajectory is equivalent to an average in space computed with the
4528:
The ergodic hypothesis turned out not to be the essential property needed for the development of
4514:
4337:
4257:
3580:
3387:
2594:
2590:
2088:
2003:
1167:
525:
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456:
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392:
365:
289:
244:
is described as a "particle or ensemble of particles whose state varies over time and thus obeys
151:
5633:
1746:
More commonly there are two classes of definitions for a dynamical system: one is motivated by
7109:
7094:
7059:
7049:
6946:
6566:
6488:
6223:
6204:
6182:
6158:
6127:
6108:
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6033:
6014:
5995:
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5957:
5938:
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5900:
5892:
5870:
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5821:
5781:
5758:
5736:
5724:
5706:
5645:
5629:
5615:
5603:
5544:
5470:
5446:
5386:
5342:
5314:
5223:
5156:
4550:
4374:
is measured in units of (position) Ă (momentum). The flow takes points of a subset
4348:
3820:
3338:-dimensional Euclidean space, so any point in phase space can be represented by a vector with
2512:
2334:
2251:
2200:
617:
509:
in 1964. One of the implications of the theorem is that if a discrete dynamical system on the
429:
297:
230:
195:
191:
6073:
Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering
4972:
4169:â ÎŁ (multiples of other eigenvalues) occurs in the denominator of the terms for the function
3851:, with a real eigenvalue smaller than one, then the straight lines given by the points along
7202:
7114:
7064:
6961:
6889:
6841:
6718:
6698:
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2224:
1695:
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1240:
1207:
344:
202:
147:
139:
111:
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Some bifurcations can lead to very complicated structures in phase space. For example, the
3870:, is an invariant curve of the map. Points in this straight line run into the fixed point.
7134:
7029:
6956:
6789:
6601:
6591:
6396:
6383:
6266:
6068:
5694:
5536:
5101:
5051:
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4533:
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2613:
2248:
2244:
2192:
2184:
1996:
549:
466:
163:
17:
7124:
7069:
5439:
Dynamical systems on monoids: Toward a general theory of deterministic systems and motion
4517:. The hypothesis states that the length of time a typical trajectory spends in a region
3999:
3572: â 0 the origin is an equilibrium (or singular) point of the flow, that is, if
30:
This article is about the general aspects of dynamical systems. For the study field, see
5264:
5211:
7217:
7184:
7179:
7174:
6976:
6866:
6861:
6759:
6708:
6624:
6498:
5841:
5729:
5699:
5642:
Dynamics Beyond
Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective
5361:
Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"."
5149:
5106:
4368:
3306:
3232:
3206:
This equation is useful when modeling mechanical systems with complicated constraints.
2612:
appear to be the natural choice. They are constructed on the geometrical structure of
2309:
A dynamical system may be defined formally as a measure-preserving transformation of a
1915:
1751:
569:
514:
478:
465:. Birkhoff's most durable result has been his 1931 discovery of what is now called the
404:
183:
175:
159:
6292:. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
461:
388:
are examples of dynamical systems where the possible classes of orbits are understood.
7237:
7212:
7169:
7159:
7154:
7054:
7034:
6894:
6816:
6713:
6521:
6311:
6307:
6289:
6150:
6094:
6090:
5888:
5884:
5659:
4745:
4698:
4503:
2858:
There is no need for higher order derivatives in the equation, nor for the parameter
2848:
2350:
2346:
2310:
2288:
2114:
2039:
2020:
521:
492:
488:
301:
293:
143:
115:
5680:
5663:
5523:
4669:
replace the
Boltzmann factor and they are defined on attractors of chaotic systems.
4214:
and when the eigenvalues are on the unit circle and complex, the dynamics is called
3823:, the origin is a fixed point of the map and the solutions are of the linear system
7164:
7129:
7039:
6996:
6851:
6846:
6445:
6335:
6260:
6254:
6196:
6172:
6146:
5750:
5280:
5176:
4737:
4725:?" or "Does the long-term behavior of the system depend on its initial condition?"
4718:
4689:
4678:
4510:'s derivation of the increase in entropy in a dynamical system of colliding atoms.
4263:
3696:
3210:
3011:
shown above gives a more general form of equations a dynamical system must satisfy
2877:
Depending on the properties of this vector field, the mechanical system is called
2820:
2136:
2043:
416:
285:
281:
273:
256:, which has applications to a wide variety of fields such as mathematics, physics,
218:
214:
106:
6811:
3276:
5134:
Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry
7149:
7139:
7024:
6774:
6596:
6503:
5720:
5406:
5198:
Melby, P.; et al. (2005). "Dynamics of Self-Adjusting Systems With Noise".
5081:
4729:
4713:
4666:
4310:
4267:
3681:
2874:), because these can be eliminated by considering systems of higher dimensions.
2024:
1288:
565:
561:
533:
265:
179:
123:
6337:, SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
5503:
4482:
For systems where the volume is preserved by the flow, Poincaré discovered the
2690:{\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}
7207:
7104:
6551:
5467:
Methods, models, simulations and approaches towards a general theory of change
5443:
Methods, models, simulations and approaches towards a general theory of change
5272:
5027:
3897:) = 0) will remain a singular point under smooth transformations; a
3673:
3256:
1496:
577:
541:
396:
338:
170:. The most general definition unifies several concepts in mathematics such as
6298:. An introduction to dynamical systems from the periodic orbit point of view.
6269:
provides definitions, explanations and resources related to nonlinear science
3910:
A flow in most small patches of the phase space can be made very simple. If
1942:
into the space of diffeomorphisms of the manifold to itself. In other terms,
7119:
7079:
6821:
6483:
6468:
5515:
4722:
4347:
describes how a periodic orbit bifurcates into a torus and the torus into a
3342:
numbers. The analysis of linear systems is possible because they satisfy a
2598:
2284:
2280:
1864:, i.e. locally a Banach space or Euclidean space, or in the discrete case a
557:
529:
510:
352:
269:
261:
6244:
5227:
3922:) â 0, then there is a change of coordinates for a region around
2636:
must be solved before it becomes a dynamic system. For example consider an
517:
of period 3, then it must have periodic points of every other period.
5565:
3702:
3680:
determine the structure of the phase space. From the eigenvalues and the
1918:
of the manifold to itself. So, f is a "smooth" mapping of the time-domain
364:
stability have been introduced in the study of dynamical systems, such as
233:, in that random events also affect the evolution of the state variables.
6856:
6247:
has daily submissions of (non-refereed) manuscripts in dynamical systems.
5636:) has a sub-series on dynamical systems with reviews of current research.
4173:, the non-resonant condition is also known as the small divisor problem.
3316:
3203:
from the set of evolution functions to the field of the complex numbers.
2832:
2789:
2132:
2035:
1861:
573:
537:
277:
187:
155:
6279:
5378:
IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems
2096:
is restricted to the non-negative reals, then the dynamical system is a
6526:
6478:
5897:
Differential Equations, dynamical systems, and an introduction to chaos
2492:
to itself, it is ÎŁ-measurable, and is measure-preserving. The triplet (
2188:
553:
481:, this theorem solved, at least in principle, a fundamental problem of
470:
257:
237:
5560:
Introduction to the Theory of Infinite-Dimensional Dissipative Systems
5219:
2257:, it is often useful to study the continuous extension Ί* of Ί to the
7099:
4280:
Bifurcation theory considers a structure in phase space (typically a
609:
6402:
3192:{\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }
2986:{\displaystyle {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}
491:
made significant advances as well. His first contribution was the
6340:
5255:
4285:
3701:
3691:
The distance between two different initial conditions in the case
2836:
593:
545:
210:
105:
6323:
4241:
in the complex plane, implying that the map is still hyperbolic.
2558:{\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi }
935:{\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}
459:, a result that made him world-famous. In 1927, he published his
6030:
Introduction to Modern Dynamics: Chaos, Networks, Space and Time
3799:
a vector. As in the continuous case, the change of coordinates
3486: = 0 is just a straight line in the direction of
2155:
is restricted to the non-negative integers we call the system a
135:
6418:
6414:
5573:
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
5375:
Rega, Giuseppe (2019). "Tribute to Ali H. Nayfeh (1933â2017)".
4502:
infinitely often. The Poincaré recurrence theorem was used by
4486:: Assume the phase space has a finite Liouville volume and let
4457:{\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}
4355:
describes how a stable periodic orbit goes through a series of
4288:) and studies its behavior as a function of the parameter
3836:
As in the continuous case, the eigenvalues and eigenvectors of
1805:{\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle }
1766:
In the geometrical definition, a dynamical system is the tuple
485:. The ergodic theorem has also had repercussions for dynamics.
6346:
6283:
4197:
are not in the unit circle, the dynamics near the fixed point
2746:{\displaystyle {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}
43:
6408:
4705:) and another of the points that diverge from the orbit (the
27:
Mathematical model of the time dependence of a point in space
6373:, Instituto Superior TĂ©cnico, Technical University of Lisbon
6329:
5973:
Dynamical Systems with Applications using Mathematica 2nd Ed
5954:
Dynamical Systems with Applications using MATLAB 2nd Edition
1963:
1927:
1895:
1845:
1821:
1788:
1778:
1750:
and is geometrical in flavor; and the other is motivated by
6399:, Institute of Computer Science, Czech Academy of Sciences.
4849:{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}
4641:
By studying the spectral properties of the linear operator
1428:
if we take one of the variables as constant. The function
6386:, IMPA, Instituto Nacional de MatemĂĄtica Pura e Applicada.
5755:
Elements of Differentiable Dynamics and Bifurcation Theory
5640:
Christian Bonatti; Lorenzo J. DĂaz; Marcelo Viana (2005).
4728:
The chaotic behavior of complex systems is not the issue.
1575:{\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}
407:
and a more detailed understanding has been worked out for
5296:
Applications of Dynamical Systems in Biology and Medicine
4494:
a subset of the phase space. Then almost every point of
3840:
determine the structure of phase space. For example, if
2082:
is taken to be the reals, the dynamical system is called
6389:
6370:
6332:. Concentrates on the applications of dynamical systems.
6295:
5200:
Chaos: An Interdisciplinary Journal of Nonlinear Science
5778:
Ergodic theory, symbolic dynamics and hyperbolic spaces
6409:
Center for Control, Dynamical Systems, and Computation
6376:
5935:
Dynamical Systems with Applications using Maple 2nd Ed
5916:
Introduction to the modern theory of dynamical systems
5731:
Geometric theory of dynamical systems: an introduction
5151:
Introduction to the Modern Theory of Dynamical Systems
4536:
approached the study of ergodic systems by the use of
4035:. By a translation, the point can be assumed to be at
3398:
function of the position in the phase space, that is,
2839:
acting on the given material point in the phase space
1266:
of the dynamical system: it associates to every point
6308:
Ordinary Differential Equations and Dynamical Systems
6096:
Ordinary Differential Equations and Dynamical Systems
5001:
4975:
4868:
4777:
4761:
differential equations according to the proof of the
4562:
4391:
4076:
3736:
3599:
3499:
3407:
3144:
3020:
2938:
2843:. The change is not a vector in the phase space
2765:
2704:
2649:
2520:
2474:{\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )}
2430:
2383:
2223:. Such systems are useful for modeling, for example,
1960:
1924:
1886:
1842:
1818:
1772:
1698:
1643:
1513:
1437:
1374:
1316:
1210:
1170:
1088:
1021:
951:
840:
798:
741:
688:
632:
312:
The concept of a dynamical system has its origins in
7193:
7010:
6937:
6875:
6745:
6732:
6684:
6615:
6459:
6452:
6263:. Models of bifurcation and chaos by Elmer G. Wiens
6124:
Introduction to Applied Dynamical Systems and Chaos
4995:and is not Lipschitz continuous at its ending time
4721:in the long term, and if so, what are the possible
4248:theorem gives the behavior near an elliptic point.
6355:, Ecole Polytechnique Fédérale de Lausanne (EPFL).
5728:
5698:
5463:Reversible dynamics and the directionality of time
5148:
5013:
4987:
4958:
4848:
4630:
4456:
4128:
3780:
3657:
3553:
3455:
3191:
3127:
2985:
2780:
2745:
2689:
2557:
2473:
2408:
1970:
1934:
1902:
1852:
1828:
1804:
1739:must be defined for all time for every element of
1719:
1673:
1574:
1471:
1417:
1359:
1231:
1188:
1145:
1074:
1007:
934:
825:
765:
727:
668:
7254:Mathematical and quantitative methods (economics)
5564:online version of first edition on the EMIS site
5508:1985 24th IEEE Conference on Decision and Control
5097:Conley's fundamental theorem of dynamical systems
4129:{\displaystyle h^{-1}\circ F\circ h(x)=J\cdot x.}
3998:), of the orbit. The flow now defines a map, the
2585:Some systems have a natural measure, such as the
2329:is a monoid (usually the non-negative integers),
2207:represents the "space" lattice, while the one in
669:{\displaystyle \Phi :U\subseteq (T\times X)\to X}
6365:Systems Analysis, Modelling and Prediction Group
6282:. Nils Berglund's lecture notes for a course at
6220:Does God Play Dice? The New Mathematics of Chaos
5992:Dynamical Systems with Applications using Python
5772:Tim Bedford, Michael Keane and Caroline Series,
4039: = 0. The Taylor series of the map is
386:systems that have two numbers describing a state
376:changes with the different notions of stability.
37:"Dynamical" redirects here. For other uses, see
6257:â peer reviewed and written by invited experts.
5062:Dynamic approach to second language development
4744:arose with just second-degree polynomials; the
4382:) and invariance of the phase space means that
2481:. Combining the above, a map Ί is said to be a
252:The study of dynamical systems is the focus of
5818:Dynamicsâthe geometry of behavior, 2nd edition
5689:Introductory texts with a unique perspective:
4683:Simple nonlinear dynamical systems, including
3213:âin which case the differential equations are
280:. Dynamical systems are a fundamental part of
6430:
5668:Bulletin of the American Mathematical Society
4631:{\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}
4295:The bifurcations of a hyperbolic fixed point
3658:{\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}
2504:), Ί), for such a Ί, is then defined to be a
1418:{\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}
1360:{\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}
1008:{\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)}
8:
5800:: CS1 maint: multiple names: authors list (
5469:, pp. 161â171, Singapore: World Scientific.
5465:". In Minati G., Abram M., Pessa E. (eds.),
5445:, pp. 173â185, Singapore: World Scientific.
5441:". In Minati G., Abram M., Pessa E. (eds.),
3706:Linear vector fields and a few trajectories.
2409:{\displaystyle \Phi ^{-1}\sigma \in \Sigma }
1799:
1773:
1569:
1527:
1140:
1104:
1075:{\displaystyle \ t_{2}\in I(\Phi (t_{1},x))}
201:At any given time, a dynamical system has a
168:the number of fish each springtime in a lake
5846:Chaos. An introduction to dynamical systems
5701:Mathematical methods of classical mechanics
5072:Infinite compositions of analytic functions
4490:be a phase space volume-preserving map and
4185:and the degree of smoothness required from
1146:{\displaystyle I(x):=\{t\in T:(t,x)\in U\}}
6742:
6456:
6437:
6423:
6415:
6411:, University of California, Santa Barbara.
6201:Mathematics and the Unexpected (Paperback)
3554:{\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}
3007:Some formal manipulation of the system of
2377:if and only if, for every Ï in ÎŁ, one has
1274:a unique image, depending on the variable
6304:. Tutorial on learning dynamical systems.
5914:Anatole Katok; Boris Hasselblatt (1996).
5679:
5254:
5155:. Cambridge: Cambridge University Press.
5000:
4974:
4950:
4930:
4906:
4884:
4867:
4827:
4826:
4816:
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4806:
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4415:
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4075:
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3741:
3735:
3646:
3633:
3617:
3604:
3598:
3533:
3517:
3504:
3498:
3409:
3408:
3406:
3184:
3174:
3157:
3155:
3146:
3145:
3143:
3105:
3100:
3070:
3069:
3050:
3036:
3022:
3021:
3019:
2974:
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2939:
2937:
2767:
2766:
2764:
2737:
2732:
2716:
2711:
2705:
2703:
2679:
2665:
2651:
2650:
2648:
2525:
2519:
2441:
2429:
2388:
2382:
1962:
1961:
1959:
1926:
1925:
1923:
1894:
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1885:
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1820:
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1817:
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1442:
1436:
1379:
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1321:
1315:
1209:
1169:
1087:
1054:
1029:
1020:
984:
971:
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957:
952:
950:
914:
901:
870:
851:
839:
797:
757:
743:
740:
704:
690:
687:
631:
94:Learn how and when to remove this message
6405:, Polytechnical University of Catalonia.
3889:of the vector field (a point where
2781:{\displaystyle {\dot {\boldsymbol {x}}}}
728:{\displaystyle \mathrm {proj} _{2}(U)=X}
428:Many people regard French mathematician
57:This article includes a list of general
6280:Geometrical theory of dynamical systems
5840:Kathleen T. Alligood, Tim D. Sauer and
5124:
3101:
3051:
3037:
3024:
2970:
2940:
2769:
2733:
2706:
2680:
2666:
2653:
544:that are common in daily life, such as
205:representing a point in an appropriate
5793:
5626:Encyclopaedia of Mathematical Sciences
4313:of the first derivative of the system
2608:For hyperbolic dynamical systems, the
2231:Compactification of a dynamical system
2203:are dynamical systems. The lattice in
1950:) is a diffeomorphism, for every time
154:that describe the swinging of a clock
6403:UPC Dynamical Systems Group Barcelona
5994:. Springer International Publishing.
5956:. Springer International Publishing.
5341:(Fourth ed.). Berlin: Springer.
5339:Economic Dynamics: Methods and Models
4859:Admits the finite duration solution:
4673:Nonlinear dynamical systems and chaos
3456:{\displaystyle {\dot {x}}=v(x)=Ax+b,}
2483:measure-preserving transformation of
2143:is taken to be the integers, it is a
164:random motion of particles in the air
7:
6286:at the advanced undergraduate level.
5504:"Finite Time Differential Equations"
5294:Jackson, T.; Radunskaya, A. (2015).
4284:, a periodic orbit, or an invariant
198:space-time structure defined on it.
6577:Measure-preserving dynamical system
5147:Katok, A.; Hasselblatt, B. (1995).
3723:dynamical system has the form of a
3147:
3071:
2305:Measure-preserving dynamical system
1903:{\displaystyle t\in {\mathcal {T}}}
1472:{\displaystyle \Phi _{x}:I(x)\to X}
766:{\displaystyle \mathrm {proj} _{2}}
397:transition to turbulence of a fluid
5664:"Differentiable dynamical systems"
5586:Works providing a broad coverage:
5461:Mazzola C. and Giunti M. (2012), "
5437:Giunti M. and Mazzola C. (2012), "
4601:
4430:
4422:
4419:
4416:
4399:
4396:
4393:
3914:is a point where the vector field
3601:
3501:
3087:
2956:
2552:
2540:
2534:
2522:
2438:
2403:
2385:
2070:; if not, the dynamical system is
1644:
1530:
1439:
1397:
1376:
1339:
1318:
1044:
891:
860:
841:
799:
753:
750:
747:
744:
700:
697:
694:
691:
633:
63:it lacks sufficient corresponding
25:
7145:Oleksandr Mykolayovych Sharkovsky
6324:Dynamical Systems Group Groningen
6267:Sci.Nonlinear FAQ 2.0 (Sept 2003)
6251:Encyclopedia of dynamical systems
4655:equilibrium statistical mechanics
4262:When the evolution map Ί (or the
4063:can only be expected to simplify
3781:{\displaystyle x_{n+1}=Ax_{n}+b,}
2620:Construction of dynamical systems
2279:In compact dynamical systems the
2235:Given a global dynamical system (
499:Oleksandr Mykolaiovych Sharkovsky
209:. This state is often given by a
6675:
6667:
6341:Center for Dynamics and Geometry
5539:(2006). "Fundamental concepts".
5311:Advanced Engineering Mathematics
5030:
4742:FermiâPastaâUlamâTsingou problem
3875:other discrete dynamical systems
3185:
2801:is a finite dimensional manifold
2573:Relation to geometric definition
1674:{\displaystyle \Phi (t,x)\in S.}
1164:In particular, in the case that
1082:, where we have defined the set
48:
6353:Laboratory of Nonlinear Systems
6326:, IWI, University of Groningen.
6203:. University Of Chicago Press.
5681:10.1090/S0002-9904-1967-11798-1
5541:Ordinary Differential Equations
5077:List of dynamical system topics
3302:Quadratic map simulation system
3068:
3064:
2634:ordinary differential equations
2215:Multidimensional generalization
2211:represents the "time" lattice.
2056:differentiable dynamical system
1748:ordinary differential equations
221:in a geometrical manifold. The
172:ordinary differential equations
6910:RabinovichâFabrikant equations
6050:Chaos and time-series analysis
6047:Julien Clinton Sprott (2003).
6011:Differential Dynamical Systems
4878:
4872:
4837:
4831:
4817:
4809:
4803:
4797:
4622:
4619:
4613:
4597:
4588:
4582:
4579:
4563:
4448:
4445:
4439:
4426:
4409:
4403:
4233:. The hyperbolic case is also
4108:
4102:
4059:), so a change of coordinates
3819:from the equation. In the new
3623:
3610:
3523:
3510:
3432:
3426:
3215:partial differential equations
3181:
3170:
3158:
3111:
3090:
3065:
3055:
3041:
2996:The dynamical system is then (
2980:
2959:
2950:
2944:
2712:
2684:
2670:
2468:
2462:
2453:
2434:
2299:Measure theoretical definition
1971:{\displaystyle {\mathcal {T}}}
1935:{\displaystyle {\mathcal {T}}}
1853:{\displaystyle {\mathcal {M}}}
1829:{\displaystyle {\mathcal {T}}}
1708:
1702:
1659:
1647:
1566:
1560:
1545:
1533:
1463:
1460:
1454:
1412:
1400:
1391:
1385:
1354:
1342:
1333:
1327:
1220:
1214:
1131:
1119:
1098:
1092:
1069:
1066:
1047:
1041:
1002:
996:
926:
894:
885:
882:
863:
844:
814:
802:
716:
710:
660:
657:
645:
1:
6347:Control and Dynamical Systems
6274:Online books or lecture notes
6105:American Mathematical Society
5087:People in systems and control
4246:KolmogorovâArnoldâMoser (KAM)
2831:and represents the change of
2614:stable and unstable manifolds
2139:, and Ί is a function. When
588:In the most general sense, a
6390:Nonlinear Dynamics Workgroup
6359:Center for Dynamical Systems
6296:Chaos: classical and quantum
5112:Principle of maximum caliber
4752:Solutions of finite duration
4357:period-doubling bifurcations
4309:can be characterized by the
4221:In the hyperbolic case, the
3807: + (1 â
3262:Complex quadratic polynomial
2630:classical mechanical systems
2062:is locally diffeomorphic to
1735:. That is, the flow through
826:{\displaystyle \Phi (0,x)=x}
6645:Poincaré recurrence theorem
6178:Chaos: Making a New Science
6053:. Oxford University Press.
6032:. Oxford University Press.
5780:. Oxford University Press.
5543:. Berlin: Springer Verlag.
3235:is an example of a chaotic
2610:SinaiâRuelleâBowen measures
2601:, but attractors have zero
2135:locally diffeomorphic to a
2052:continuously differentiable
1189:{\displaystyle U=T\times X}
435:Poincaré recurrence theorem
160:the flow of water in a pipe
114:arises in the study of the
7270:
6640:PoincarĂ©âBendixson theorem
6302:Learning Dynamical Systems
5866:Discrete Dynamical Systems
5489:Discrete Dynamical Systems
4768:As example, the equation:
4734:PomeauâManneville scenario
4676:
4366:
4353:Feigenbaum period-doubling
4255:
3725:matrix difference equation
3327:
2420:if and only if, for every
2302:
2259:one-point compactification
2066:, the dynamical system is
1992:real-time dynamical system
1239:and thus that Ί defines a
507:discrete dynamical systems
469:. Combining insights from
39:Dynamical (disambiguation)
36:
29:
18:Nonlinear dynamical system
6992:Swinging Atwood's machine
6665:
6635:KrylovâBogolyubov theorem
6512:
6371:Non-Linear Dynamics Group
5614:(available as a reprint:
5412:Franklin Institute Awards
5273:10.1007/s10955-007-9444-4
4013:, for points starting in
3312:Swinging Atwood's machine
2580:KrylovâBogolyubov theorem
2110:discrete dynamical system
2104:Discrete dynamical system
2074:. This does not assume a
455:", a special case of the
6900:LotkaâVolterra equations
6724:Synchronization of chaos
6527:axiom A dynamical system
6122:Stephen Wiggins (2003).
5600:Foundations of mechanics
5502:Vardia T. Haimo (1985).
5309:Kreyszig, Erwin (2011).
5132:Strogatz, S. H. (2001).
3941:the dynamical system is
3474:a vector of numbers and
3324:Linear dynamical systems
2847:, but is instead in the
2221:multidimensional systems
2191:or a higher-dimensional
1684:Thus, in particular, if
382:Linear dynamical systems
254:dynamical systems theory
194:, without the need of a
32:Dynamical systems theory
6885:Double scroll attractor
6650:Stable manifold theorem
6557:False nearest neighbors
6028:David D. Nolte (2015).
5516:10.1109/CDC.1985.268832
4988:{\displaystyle t\geq 2}
4763:Picard-Lindelof theorem
4659:Boltzmann factor exp(âÎČ
4223:HartmanâGrobman theorem
4156:are the eigenvalues of
3980:). These points are a
3583:: for an initial point
3581:exponential of a matrix
3344:superposition principle
3330:Linear dynamical system
2640:such as the following:
2054:we say the system is a
612:, written additively,
150:. Examples include the
130:is a system in which a
78:more precise citations.
6925:Van der Pol oscillator
6905:MackeyâGlass equations
6537:Box-counting dimension
6367:, University of Oxford
6361:, University of Bremen
5990:Stephen Lynch (2018).
5971:Stephen Lynch (2017).
5952:Stephen Lynch (2014).
5933:Stephen Lynch (2010).
5571:Temam, Roger (1997) .
5510:. pp. 1729â1733.
5417:The Franklin Institute
5365:193.3 (1990): 137â163.
5038:Systems science portal
5015:
4989:
4960:
4850:
4632:
4458:
4351:. In another example,
4345:RuelleâTakens scenario
4270:until a special value
4130:
4017:and returning to
3782:
3707:
3659:
3555:
3457:
3252:Bouncing ball dynamics
3193:
3129:
3009:differential equations
2987:
2792:of the material point
2782:
2747:
2691:
2559:
2488:, if it is a map from
2475:
2410:
2349:, meaning that ÎŁ is a
1972:
1936:
1904:
1854:
1830:
1806:
1762:Geometrical definition
1721:
1720:{\displaystyle I(x)=T}
1675:
1576:
1473:
1419:
1361:
1233:
1232:{\displaystyle I(x)=T}
1190:
1147:
1076:
1009:
936:
827:
767:
729:
670:
453:Last Geometric Theorem
334:integrating the system
246:differential equations
119:
7075:Svetlana Jitomirskaya
6982:Multiscroll attractor
6827:Interval exchange map
6780:Dyadic transformation
6765:Complex quadratic map
6607:Topological conjugacy
6542:Correlation dimension
6517:Anosov diffeomorphism
6245:Arxiv preprint server
5602:. BenjaminâCummings.
5092:Sharkovskii's theorem
5016:
4990:
4961:
4851:
4748:is piecewise linear.
4633:
4530:statistical mechanics
4469:Hamiltonian formalism
4459:
4131:
3931:rectification theorem
3847:is an eigenvector of
3783:
3705:
3660:
3556:
3458:
3390:, the vector field v(
3272:Dyadic transformation
3194:
3130:
2988:
2835:induced by the known
2783:
2748:
2692:
2638:initial value problem
2560:
2476:
2416:. A map Ί is said to
2411:
1988:real dynamical system
1982:Real dynamical system
1973:
1937:
1905:
1872:is an evolution rule
1855:
1831:
1807:
1722:
1676:
1577:
1474:
1420:
1362:
1296:, while the variable
1234:
1191:
1148:
1077:
1010:
937:
828:
768:
730:
671:
483:statistical mechanics
449:George David Birkhoff
413:statistical mechanics
351:Before the advent of
318:differential equation
118:, a dynamical system.
109:
7085:Edward Norton Lorenz
6218:Ian Stewart (1997).
6155:Celestial Encounters
6009:James Meiss (2007).
5487:Galor, Oded (2010).
5067:Feedback passivation
5014:{\displaystyle t=2.}
4999:
4973:
4866:
4775:
4759:Lipschitz continuous
4560:
4389:
4074:
3957:Near periodic orbits
3933:says that away from
3873:There are also many
3734:
3597:
3497:
3482: â 0 with
3405:
3292:List of chaotic maps
3142:
3018:
2936:
2763:
2702:
2647:
2518:
2428:
2418:preserve the measure
2381:
2076:symplectic structure
2072:infinite-dimensional
1958:
1922:
1884:
1840:
1816:
1770:
1696:
1641:
1603:of the flow through
1595:. The orbit through
1511:
1435:
1372:
1314:
1208:
1168:
1086:
1019:
949:
838:
796:
739:
686:
630:
503:Sharkovsky's theorem
370:structural stability
7045:Mitchell Feigenbaum
6987:Population dynamics
6972:HĂ©nonâHeiles system
6832:Irrational rotation
6785:Dynamical billiards
6770:Coupled map lattice
6630:Liouville's theorem
6562:Hausdorff dimension
6547:Conservative system
6532:Bifurcation diagram
6310:. Lecture notes by
5863:Oded Galor (2011).
5848:. Springer Verlag.
5814:Christopher D. Shaw
5735:. Springer-Verlag.
5705:. Springer-Verlag.
5537:Arnold, Vladimir I.
5335:Gandolfo, Giancarlo
5265:2008JSP...130..617G
5212:2005Chaos..15c3902M
5047:Behavioral modeling
4538:functional analysis
4302:of a system family
4235:structurally stable
4177:Conjugation results
4067:to its linear part
3282:Irrational rotation
2595:dissipative systems
2591:Hamiltonian systems
2058:. If the manifold
2048:continuous function
1756:measure theoretical
1611:of the state space
1280:evolution parameter
451:proved Poincaré's "
322:difference equation
314:Newtonian mechanics
300:processes, and the
152:mathematical models
7223:Santa Fe Institute
7090:Aleksandr Lyapunov
6920:Three-body problem
6807:Gingerbreadman map
6694:Bifurcation theory
6572:Lyapunov stability
6395:2015-01-21 at the
6382:2017-06-02 at the
6261:Nonlinear Dynamics
6075:. Addison Wesley.
6069:Steven H. Strogatz
5899:. Academic Press.
5820:. Addison-Wesley.
5757:. Academic Press.
5596:Jerrold E. Marsden
5575:. Springer Verlag.
5407:"Ali Hasan Nayfeh"
5313:. Hoboken: Wiley.
5052:Cognitive modeling
5011:
4985:
4956:
4846:
4695:Hyperbolic systems
4628:
4515:ergodic hypothesis
4484:recurrence theorem
4454:
4378:into the points Ί(
4338:Bifurcation theory
4258:Bifurcation theory
4252:Bifurcation theory
4126:
3778:
3708:
3655:
3551:
3453:
3189:
3125:
2983:
2778:
2743:
2687:
2632:. But a system of
2565:for every integer
2555:
2471:
2406:
2357:and Ό is a finite
2169:cellular automaton
2163:Cellular automaton
2068:finite-dimensional
1968:
1932:
1900:
1850:
1826:
1802:
1717:
1671:
1572:
1469:
1415:
1357:
1264:evolution function
1229:
1196:we have for every
1186:
1143:
1072:
1005:
932:
823:
763:
725:
666:
526:nonlinear dynamics
505:on the periods of
475:ergodic hypothesis
457:three-body problem
441:Aleksandr Lyapunov
409:hyperbolic systems
393:bifurcation points
366:Lyapunov stability
330:solving the system
290:bifurcation theory
120:
7244:Dynamical systems
7231:
7230:
7095:BenoĂźt Mandelbrot
7060:Martin Gutzwiller
7050:Peter Grassberger
6933:
6932:
6915:Rössler attractor
6663:
6662:
6567:Invariant measure
6489:Lyapunov exponent
6377:Dynamical Systems
6290:Dynamical systems
6229:978-0-14-025602-4
6210:978-0-226-19990-0
6188:978-0-14-009250-9
6164:978-0-691-02743-2
6141:Popularizations:
6133:978-0-387-00177-7
6114:978-0-8218-8328-0
6082:978-0-201-54344-5
6060:978-0-19-850839-7
6020:978-0-89871-635-1
6001:978-3-319-78145-7
5982:978-3-319-61485-4
5944:978-0-8176-4389-8
5925:978-0-521-57557-7
5906:978-0-12-349703-1
5893:Robert L. Devaney
5876:978-3-642-07185-0
5855:978-0-387-94677-1
5827:978-0-201-56716-8
5787:978-0-19-853390-0
5764:978-0-12-601710-6
5742:978-0-387-90668-3
5725:Welington de Melo
5712:978-0-387-96890-2
5651:978-3-540-22066-4
5609:978-0-8053-0102-1
5475:978-981-4383-32-5
5451:978-981-4383-32-5
5419:. 4 February 2014
5348:978-3-642-13503-3
5320:978-0-470-64613-7
5220:10.1063/1.1953147
5179:. Springer Nature
5162:978-0-521-34187-5
4969:that is zero for
4938:
4914:
4892:
4821:
4795:
4707:unstable manifold
4551:transfer operator
4540:. An observable
4473:Liouville measure
4349:strange attractor
4147:, ...,
3821:coordinate system
3815:removes the term
3417:
3030:
2775:
2659:
2626:evolution in time
2587:Liouville measure
2347:probability space
2252:topological space
2201:cellular automata
1024:
584:Formal definition
462:Dynamical Systems
298:self-organization
116:Lorenz oscillator
104:
103:
96:
16:(Redirected from
7261:
7203:Butterfly effect
7115:Itamar Procaccia
7065:Brosl Hasslacher
6962:Elastic pendulum
6890:Duffing equation
6837:KaplanâYorke map
6755:Arnold's cat map
6743:
6719:Stability theory
6704:Dynamical system
6699:Control of chaos
6679:
6671:
6655:Takens's theorem
6587:Poincaré section
6457:
6439:
6432:
6425:
6416:
6233:
6214:
6192:
6168:
6137:
6118:
6086:
6064:
6043:
6024:
6005:
5986:
5967:
5948:
5929:
5910:
5885:Morris W. Hirsch
5880:
5859:
5831:
5810:Ralph H. Abraham
5805:
5799:
5791:
5768:
5746:
5734:
5716:
5704:
5685:
5683:
5655:
5613:
5576:
5563:
5558:Chueshov, I. D.
5554:
5528:
5527:
5499:
5493:
5492:
5484:
5478:
5459:
5453:
5435:
5429:
5428:
5426:
5424:
5403:
5397:
5396:
5385:. pp. 1â2.
5372:
5366:
5359:
5353:
5352:
5331:
5325:
5324:
5306:
5300:
5299:
5291:
5285:
5284:
5258:
5238:
5232:
5231:
5195:
5189:
5188:
5186:
5184:
5173:
5167:
5166:
5154:
5144:
5138:
5137:
5129:
5057:Complex dynamics
5040:
5035:
5034:
5033:
5020:
5018:
5017:
5012:
4994:
4992:
4991:
4986:
4965:
4963:
4962:
4957:
4955:
4954:
4949:
4945:
4944:
4940:
4939:
4931:
4915:
4907:
4893:
4885:
4855:
4853:
4852:
4847:
4822:
4820:
4812:
4807:
4796:
4793:
4785:
4685:piecewise linear
4637:
4635:
4634:
4629:
4612:
4611:
4575:
4574:
4463:
4461:
4460:
4455:
4438:
4437:
4425:
4402:
4135:
4133:
4132:
4127:
4089:
4088:
3982:Poincaré section
3787:
3785:
3784:
3779:
3768:
3767:
3752:
3751:
3697:chaotic behavior
3664:
3662:
3661:
3656:
3651:
3650:
3641:
3640:
3622:
3621:
3609:
3608:
3560:
3558:
3557:
3552:
3538:
3537:
3522:
3521:
3509:
3508:
3462:
3460:
3459:
3454:
3419:
3418:
3410:
3287:KaplanâYorke map
3237:piecewise linear
3228:Arnold's cat map
3198:
3196:
3195:
3190:
3188:
3180:
3179:
3178:
3173:
3151:
3150:
3134:
3132:
3131:
3126:
3118:
3114:
3110:
3109:
3104:
3075:
3074:
3054:
3040:
3032:
3031:
3023:
2992:
2990:
2989:
2984:
2979:
2978:
2973:
2943:
2787:
2785:
2784:
2779:
2777:
2776:
2768:
2752:
2750:
2749:
2744:
2742:
2741:
2736:
2727:
2726:
2715:
2709:
2696:
2694:
2693:
2688:
2683:
2669:
2661:
2660:
2652:
2603:Lebesgue measure
2564:
2562:
2561:
2556:
2530:
2529:
2506:dynamical system
2480:
2478:
2477:
2472:
2449:
2448:
2415:
2413:
2412:
2407:
2396:
2395:
2293:simply connected
2283:of any orbit is
2225:image processing
2117:dynamical system
1999:dynamical system
1977:
1975:
1974:
1969:
1967:
1966:
1941:
1939:
1938:
1933:
1931:
1930:
1909:
1907:
1906:
1901:
1899:
1898:
1859:
1857:
1856:
1851:
1849:
1848:
1835:
1833:
1832:
1827:
1825:
1824:
1811:
1809:
1808:
1803:
1792:
1791:
1782:
1781:
1726:
1724:
1723:
1718:
1680:
1678:
1677:
1672:
1581:
1579:
1578:
1573:
1523:
1522:
1478:
1476:
1475:
1470:
1447:
1446:
1424:
1422:
1421:
1416:
1384:
1383:
1366:
1364:
1363:
1358:
1326:
1325:
1307:We often write
1262:) is called the
1238:
1236:
1235:
1230:
1195:
1193:
1192:
1187:
1152:
1150:
1149:
1144:
1081:
1079:
1078:
1073:
1059:
1058:
1034:
1033:
1022:
1014:
1012:
1011:
1006:
989:
988:
976:
975:
962:
961:
941:
939:
938:
933:
919:
918:
906:
905:
875:
874:
856:
855:
832:
830:
829:
824:
772:
770:
769:
764:
762:
761:
756:
734:
732:
731:
726:
709:
708:
703:
675:
673:
672:
667:
590:dynamical system
242:dynamical system
148:parametric curve
138:dependence of a
128:dynamical system
112:Lorenz attractor
99:
92:
88:
85:
79:
74:this article by
65:inline citations
52:
51:
44:
21:
7269:
7268:
7264:
7263:
7262:
7260:
7259:
7258:
7234:
7233:
7232:
7227:
7195:
7189:
7135:Caroline Series
7030:Mary Cartwright
7012:
7006:
6957:Double pendulum
6939:
6929:
6878:
6871:
6797:Exponential map
6748:
6734:
6728:
6686:
6680:
6673:
6659:
6625:Ergodic theorem
6618:
6611:
6602:Stable manifold
6592:Recurrence plot
6508:
6462:
6448:
6443:
6397:Wayback Machine
6384:Wayback Machine
6318:Research groups
6241:
6236:
6230:
6217:
6211:
6195:
6189:
6171:
6165:
6145:
6134:
6121:
6115:
6089:
6083:
6067:
6061:
6046:
6040:
6027:
6021:
6008:
6002:
5989:
5983:
5970:
5964:
5951:
5945:
5932:
5926:
5913:
5907:
5883:
5877:
5862:
5856:
5839:
5828:
5808:
5792:
5788:
5771:
5765:
5749:
5743:
5719:
5713:
5693:
5658:
5652:
5639:
5610:
5590:
5583:
5581:Further reading
5570:
5557:
5551:
5535:
5532:
5531:
5501:
5500:
5496:
5486:
5485:
5481:
5460:
5456:
5436:
5432:
5422:
5420:
5405:
5404:
5400:
5393:
5374:
5373:
5369:
5363:Physics Reports
5360:
5356:
5349:
5333:
5332:
5328:
5321:
5308:
5307:
5303:
5293:
5292:
5288:
5240:
5239:
5235:
5197:
5196:
5192:
5182:
5180:
5175:
5174:
5170:
5163:
5146:
5145:
5141:
5131:
5130:
5126:
5121:
5116:
5102:System dynamics
5036:
5031:
5029:
5026:
4997:
4996:
4971:
4970:
4923:
4919:
4899:
4895:
4894:
4864:
4863:
4778:
4773:
4772:
4754:
4712:This branch of
4703:stable manifold
4681:
4675:
4646:involving
4600:
4566:
4558:
4557:
4429:
4387:
4386:
4371:
4365:
4363:Ergodic systems
4334:
4328:
4321:
4307:
4301:
4276:
4260:
4254:
4203:
4179:
4168:
4155:
4146:
4077:
4072:
4071:
4055: + O(
4027:
3997:
3979:
3968:
3959:
3935:singular points
3908:
3883:
3861:
3846:
3832:
3759:
3737:
3732:
3731:
3713:
3642:
3629:
3613:
3600:
3595:
3594:
3589:
3578:
3529:
3513:
3500:
3495:
3494:
3403:
3402:
3384:
3332:
3326:
3321:
3267:Double pendulum
3247:outer billiards
3223:
3156:
3140:
3139:
3099:
3080:
3076:
3016:
3015:
2968:
2934:
2933:
2788:represents the
2761:
2760:
2731:
2710:
2700:
2699:
2645:
2644:
2624:The concept of
2622:
2575:
2521:
2516:
2515:
2437:
2426:
2425:
2384:
2379:
2378:
2365:, Σ). A map Ί:
2313:, the triplet (
2307:
2301:
2245:locally compact
2233:
2217:
2165:
2106:
1997:continuous time
1984:
1956:
1955:
1920:
1919:
1882:
1881:
1838:
1837:
1814:
1813:
1768:
1767:
1764:
1694:
1693:
1639:
1638:
1514:
1509:
1508:
1438:
1433:
1432:
1375:
1370:
1369:
1317:
1312:
1311:
1304:of the system.
1254:The function Ί(
1206:
1205:
1166:
1165:
1084:
1083:
1050:
1025:
1017:
1016:
980:
967:
953:
947:
946:
910:
897:
866:
847:
836:
835:
794:
793:
742:
737:
736:
689:
684:
683:
628:
627:
616:is a non-empty
586:
493:Smale horseshoe
467:ergodic theorem
426:
405:ergodic systems
310:
184:complex numbers
146:, such as in a
100:
89:
83:
80:
70:Please help to
69:
53:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7267:
7265:
7257:
7256:
7251:
7249:Systems theory
7246:
7236:
7235:
7229:
7228:
7226:
7225:
7220:
7218:Predictability
7215:
7210:
7205:
7199:
7197:
7191:
7190:
7188:
7187:
7185:Lai-Sang Young
7182:
7180:James A. Yorke
7177:
7175:Amie Wilkinson
7172:
7167:
7162:
7157:
7152:
7147:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7110:Henri Poincaré
7107:
7102:
7097:
7092:
7087:
7082:
7077:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7016:
7014:
7008:
7007:
7005:
7004:
6999:
6994:
6989:
6984:
6979:
6977:Kicked rotator
6974:
6969:
6964:
6959:
6954:
6949:
6947:Chua's circuit
6943:
6941:
6935:
6934:
6931:
6930:
6928:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6881:
6879:
6876:
6873:
6872:
6870:
6869:
6867:Zaslavskii map
6864:
6862:Tinkerbell map
6859:
6854:
6849:
6844:
6839:
6834:
6829:
6824:
6819:
6814:
6809:
6804:
6799:
6794:
6793:
6792:
6782:
6777:
6772:
6767:
6762:
6757:
6751:
6749:
6746:
6740:
6730:
6729:
6727:
6726:
6721:
6716:
6711:
6709:Ergodic theory
6706:
6701:
6696:
6690:
6688:
6682:
6681:
6666:
6664:
6661:
6660:
6658:
6657:
6652:
6647:
6642:
6637:
6632:
6627:
6621:
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6616:
6613:
6612:
6610:
6609:
6604:
6599:
6594:
6589:
6584:
6579:
6574:
6569:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6524:
6519:
6513:
6510:
6509:
6507:
6506:
6501:
6499:Periodic point
6496:
6491:
6486:
6481:
6476:
6471:
6465:
6463:
6460:
6454:
6450:
6449:
6444:
6442:
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6434:
6427:
6419:
6413:
6412:
6406:
6400:
6387:
6374:
6368:
6362:
6356:
6350:
6344:
6338:
6333:
6327:
6320:
6319:
6315:
6314:
6305:
6299:
6293:
6287:
6276:
6275:
6271:
6270:
6264:
6258:
6248:
6240:
6239:External links
6237:
6235:
6234:
6228:
6215:
6209:
6193:
6187:
6169:
6163:
6139:
6138:
6132:
6119:
6113:
6091:Teschl, Gerald
6087:
6081:
6065:
6059:
6044:
6039:978-0199657032
6038:
6025:
6019:
6006:
6000:
5987:
5981:
5968:
5963:978-3319068190
5962:
5949:
5943:
5930:
5924:
5911:
5905:
5881:
5875:
5860:
5854:
5842:James A. Yorke
5833:
5832:
5826:
5806:
5786:
5769:
5763:
5747:
5741:
5717:
5711:
5687:
5686:
5674:(6): 747â817.
5656:
5650:
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5555:
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5190:
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5117:
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5114:
5109:
5107:Systems theory
5104:
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4978:
4967:
4966:
4953:
4948:
4943:
4937:
4934:
4929:
4926:
4922:
4918:
4913:
4910:
4905:
4902:
4898:
4891:
4888:
4883:
4880:
4877:
4874:
4871:
4857:
4856:
4845:
4842:
4839:
4836:
4833:
4830:
4825:
4819:
4815:
4811:
4805:
4802:
4799:
4791:
4788:
4784:
4781:
4753:
4750:
4699:tangent spaces
4677:Main article:
4674:
4671:
4639:
4638:
4627:
4624:
4621:
4618:
4615:
4610:
4607:
4603:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4573:
4569:
4565:
4465:
4464:
4453:
4450:
4447:
4444:
4441:
4436:
4432:
4428:
4424:
4421:
4418:
4414:
4411:
4408:
4405:
4401:
4398:
4395:
4369:Ergodic theory
4367:Main article:
4364:
4361:
4332:
4326:
4317:
4305:
4299:
4274:
4256:Main article:
4253:
4250:
4201:
4178:
4175:
4164:
4151:
4144:
4137:
4136:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4087:
4084:
4080:
4047:) =
4025:
3995:
3977:
3966:
3958:
3955:
3907:
3904:
3899:periodic orbit
3887:singular point
3882:
3881:Local dynamics
3879:
3859:
3844:
3830:
3789:
3788:
3777:
3774:
3771:
3766:
3762:
3758:
3755:
3750:
3747:
3744:
3740:
3712:
3709:
3666:
3665:
3654:
3649:
3645:
3639:
3636:
3632:
3628:
3625:
3620:
3616:
3612:
3607:
3603:
3587:
3576:
3562:
3561:
3550:
3547:
3544:
3541:
3536:
3532:
3528:
3525:
3520:
3516:
3512:
3507:
3503:
3464:
3463:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3416:
3413:
3383:
3380:
3370:) +
3328:Main article:
3325:
3322:
3320:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3240:
3230:
3224:
3222:
3219:
3187:
3183:
3177:
3172:
3169:
3166:
3163:
3160:
3154:
3149:
3136:
3135:
3124:
3121:
3117:
3113:
3108:
3103:
3098:
3095:
3092:
3089:
3086:
3083:
3079:
3073:
3067:
3063:
3060:
3057:
3053:
3049:
3046:
3043:
3039:
3035:
3029:
3026:
2994:
2993:
2982:
2977:
2972:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2942:
2927:
2926:
2922:) = 0 for all
2905:
2856:
2855:
2802:
2796:
2774:
2771:
2754:
2753:
2740:
2735:
2730:
2725:
2722:
2719:
2714:
2708:
2697:
2686:
2682:
2678:
2675:
2672:
2668:
2664:
2658:
2655:
2621:
2618:
2574:
2571:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2528:
2524:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2447:
2444:
2440:
2436:
2433:
2424:in ÎŁ, one has
2405:
2402:
2399:
2394:
2391:
2387:
2373:is said to be
2303:Main article:
2300:
2297:
2232:
2229:
2216:
2213:
2164:
2161:
2105:
2102:
1983:
1980:
1965:
1954:in the domain
1929:
1916:diffeomorphism
1897:
1892:
1889:
1847:
1823:
1801:
1798:
1795:
1790:
1785:
1780:
1775:
1763:
1760:
1752:ergodic theory
1716:
1713:
1710:
1707:
1704:
1701:
1682:
1681:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1585:is called the
1583:
1582:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1521:
1517:
1494:is called the
1482:is called the
1480:
1479:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1445:
1441:
1426:
1425:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1382:
1378:
1367:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1324:
1320:
1300:represents an
1228:
1225:
1222:
1219:
1216:
1213:
1185:
1182:
1179:
1176:
1173:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1071:
1068:
1065:
1062:
1057:
1053:
1049:
1046:
1043:
1040:
1037:
1032:
1028:
1004:
1001:
998:
995:
992:
987:
983:
979:
974:
970:
965:
960:
956:
943:
942:
931:
928:
925:
922:
917:
913:
909:
904:
900:
896:
893:
890:
887:
884:
881:
878:
873:
869:
865:
862:
859:
854:
850:
846:
843:
833:
822:
819:
816:
813:
810:
807:
804:
801:
779:
778:
775:projection map
760:
755:
752:
749:
746:
724:
721:
718:
715:
712:
707:
702:
699:
696:
693:
677:
676:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
585:
582:
570:rocket engines
515:periodic point
479:measure theory
430:Henri Poincaré
425:
422:
421:
420:
400:
389:
377:
309:
306:
223:evolution rule
176:ergodic theory
134:describes the
102:
101:
56:
54:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7266:
7255:
7252:
7250:
7247:
7245:
7242:
7241:
7239:
7224:
7221:
7219:
7216:
7214:
7213:Edge of chaos
7211:
7209:
7206:
7204:
7201:
7200:
7198:
7192:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7170:Marcelo Viana
7168:
7166:
7163:
7161:
7160:Audrey Terras
7158:
7156:
7155:Floris Takens
7153:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7055:Celso Grebogi
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7035:Chen Guanrong
7033:
7031:
7028:
7026:
7023:
7021:
7020:Michael Berry
7018:
7017:
7015:
7009:
7003:
7000:
6998:
6995:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6948:
6945:
6944:
6942:
6936:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6895:Lorenz system
6893:
6891:
6888:
6886:
6883:
6882:
6880:
6874:
6868:
6865:
6863:
6860:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6842:Langton's ant
6840:
6838:
6835:
6833:
6830:
6828:
6825:
6823:
6820:
6818:
6817:Horseshoe map
6815:
6813:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6791:
6788:
6787:
6786:
6783:
6781:
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6761:
6758:
6756:
6753:
6752:
6750:
6744:
6741:
6738:
6731:
6725:
6722:
6720:
6717:
6715:
6714:Quantum chaos
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6695:
6692:
6691:
6689:
6683:
6678:
6674:
6670:
6656:
6653:
6651:
6648:
6646:
6643:
6641:
6638:
6636:
6633:
6631:
6628:
6626:
6623:
6622:
6620:
6614:
6608:
6605:
6603:
6600:
6598:
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6523:
6522:Arnold tongue
6520:
6518:
6515:
6514:
6511:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6466:
6464:
6458:
6455:
6451:
6447:
6440:
6435:
6433:
6428:
6426:
6421:
6420:
6417:
6410:
6407:
6404:
6401:
6398:
6394:
6391:
6388:
6385:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6343:, Penn State.
6342:
6339:
6336:
6334:
6331:
6328:
6325:
6322:
6321:
6317:
6316:
6313:
6312:Gerald Teschl
6309:
6306:
6303:
6300:
6297:
6294:
6291:
6288:
6285:
6281:
6278:
6277:
6273:
6272:
6268:
6265:
6262:
6259:
6256:
6252:
6249:
6246:
6243:
6242:
6238:
6231:
6225:
6221:
6216:
6212:
6206:
6202:
6198:
6194:
6190:
6184:
6180:
6179:
6174:
6170:
6166:
6160:
6157:. Princeton.
6156:
6152:
6151:Philip Holmes
6148:
6144:
6143:
6142:
6135:
6129:
6125:
6120:
6116:
6110:
6106:
6102:
6098:
6097:
6092:
6088:
6084:
6078:
6074:
6070:
6066:
6062:
6056:
6052:
6049:
6045:
6041:
6035:
6031:
6026:
6022:
6016:
6012:
6007:
6003:
5997:
5993:
5988:
5984:
5978:
5974:
5969:
5965:
5959:
5955:
5950:
5946:
5940:
5936:
5931:
5927:
5921:
5918:. Cambridge.
5917:
5912:
5908:
5902:
5898:
5894:
5890:
5889:Stephen Smale
5886:
5882:
5878:
5872:
5868:
5865:
5861:
5857:
5851:
5847:
5843:
5838:
5837:
5836:
5829:
5823:
5819:
5815:
5811:
5807:
5803:
5797:
5789:
5783:
5779:
5775:
5770:
5766:
5760:
5756:
5752:
5748:
5744:
5738:
5733:
5732:
5726:
5722:
5718:
5714:
5708:
5703:
5702:
5696:
5692:
5691:
5690:
5682:
5677:
5673:
5669:
5665:
5661:
5660:Stephen Smale
5657:
5653:
5647:
5643:
5638:
5635:
5631:
5627:
5624:
5621:
5620:0-201-40840-6
5617:
5611:
5605:
5601:
5597:
5593:
5592:Ralph Abraham
5589:
5588:
5587:
5580:
5574:
5569:
5566:
5561:
5556:
5552:
5550:3-540-34563-9
5546:
5542:
5538:
5534:
5533:
5525:
5521:
5517:
5513:
5509:
5505:
5498:
5495:
5490:
5483:
5480:
5476:
5472:
5468:
5464:
5458:
5455:
5452:
5448:
5444:
5440:
5434:
5431:
5418:
5414:
5413:
5408:
5402:
5399:
5394:
5392:9783030236922
5388:
5384:
5380:
5379:
5371:
5368:
5364:
5358:
5355:
5350:
5344:
5340:
5336:
5330:
5327:
5322:
5316:
5312:
5305:
5302:
5297:
5290:
5287:
5282:
5278:
5274:
5270:
5266:
5262:
5257:
5252:
5248:
5244:
5243:J. Stat. Phys
5237:
5234:
5229:
5225:
5221:
5217:
5213:
5209:
5206:(3): 033902.
5205:
5201:
5194:
5191:
5178:
5172:
5169:
5164:
5158:
5153:
5152:
5143:
5140:
5135:
5128:
5125:
5118:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5044:
5039:
5028:
5023:
5021:
5008:
5005:
5002:
4982:
4979:
4976:
4951:
4946:
4941:
4935:
4932:
4927:
4924:
4920:
4916:
4911:
4908:
4903:
4900:
4896:
4889:
4886:
4881:
4875:
4869:
4862:
4861:
4860:
4843:
4840:
4834:
4828:
4823:
4813:
4800:
4789:
4786:
4782:
4779:
4771:
4770:
4769:
4766:
4764:
4760:
4751:
4749:
4747:
4746:horseshoe map
4743:
4739:
4735:
4731:
4726:
4724:
4720:
4715:
4710:
4708:
4704:
4700:
4696:
4692:
4691:
4686:
4680:
4672:
4670:
4668:
4664:
4662:
4656:
4651:
4649:
4644:
4625:
4616:
4608:
4605:
4594:
4591:
4585:
4576:
4571:
4567:
4556:
4555:
4554:
4552:
4548:
4543:
4539:
4535:
4531:
4526:
4524:
4520:
4516:
4511:
4509:
4506:to object to
4505:
4501:
4497:
4493:
4489:
4485:
4480:
4476:
4474:
4470:
4451:
4442:
4434:
4412:
4406:
4385:
4384:
4383:
4381:
4377:
4370:
4362:
4360:
4358:
4354:
4350:
4346:
4341:
4339:
4335:
4325:
4320:
4316:
4312:
4308:
4298:
4293:
4291:
4287:
4283:
4278:
4273:
4269:
4265:
4259:
4251:
4249:
4247:
4242:
4240:
4236:
4232:
4229: ·
4228:
4224:
4219:
4217:
4213:
4212:
4207:
4200:
4196:
4192:
4188:
4184:
4176:
4174:
4172:
4167:
4163:
4159:
4154:
4150:
4143:
4123:
4120:
4117:
4114:
4111:
4105:
4099:
4096:
4093:
4090:
4085:
4082:
4078:
4070:
4069:
4068:
4066:
4062:
4058:
4054:
4051: ·
4050:
4046:
4042:
4038:
4034:
4029:
4024:
4020:
4016:
4012:
4009: â
4008:
4005: :
4004:
4001:
3994:
3990:
3986:
3983:
3976:
3972:
3965:
3956:
3954:
3952:
3948:
3944:
3940:
3936:
3932:
3927:
3925:
3921:
3917:
3913:
3906:Rectification
3905:
3903:
3900:
3896:
3892:
3888:
3880:
3878:
3876:
3871:
3869:
3866: â
3865:
3858:
3854:
3850:
3843:
3839:
3834:
3829:
3826:
3822:
3818:
3814:
3810:
3806:
3803: â
3802:
3798:
3795:a matrix and
3794:
3775:
3772:
3769:
3764:
3760:
3756:
3753:
3748:
3745:
3742:
3738:
3730:
3729:
3728:
3726:
3722:
3718:
3717:discrete-time
3710:
3704:
3700:
3698:
3694:
3689:
3687:
3683:
3679:
3675:
3671:
3652:
3647:
3643:
3637:
3634:
3630:
3626:
3618:
3614:
3605:
3593:
3592:
3591:
3586:
3582:
3575:
3571:
3567:
3548:
3545:
3542:
3539:
3534:
3530:
3526:
3518:
3514:
3505:
3493:
3492:
3491:
3489:
3485:
3481:
3477:
3473:
3469:
3450:
3447:
3444:
3441:
3438:
3435:
3429:
3423:
3420:
3414:
3411:
3401:
3400:
3399:
3397:
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3389:
3381:
3379:
3377:
3373:
3369:
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3337:
3331:
3323:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3297:Lorenz system
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3244:
3241:
3238:
3234:
3231:
3229:
3226:
3225:
3220:
3218:
3216:
3212:
3211:Banach spaces
3207:
3204:
3202:
3175:
3167:
3164:
3161:
3152:
3122:
3119:
3115:
3106:
3096:
3093:
3084:
3081:
3077:
3061:
3058:
3047:
3044:
3033:
3027:
3014:
3013:
3012:
3010:
3005:
3003:
2999:
2975:
2965:
2962:
2953:
2947:
2932:
2931:
2930:
2925:
2921:
2917:
2913:
2909:
2906:
2903:
2899:
2895:
2891:
2887:
2883:
2880:
2879:
2878:
2875:
2873:
2869:
2865:
2861:
2853:
2850:
2849:tangent space
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2803:
2800:
2797:
2795:
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2772:
2759:
2758:
2757:
2738:
2728:
2723:
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2717:
2698:
2676:
2673:
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2656:
2643:
2642:
2641:
2639:
2635:
2631:
2627:
2619:
2617:
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2611:
2606:
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2600:
2596:
2592:
2588:
2583:
2581:
2572:
2570:
2568:
2549:
2546:
2543:
2537:
2531:
2526:
2514:
2509:
2507:
2503:
2499:
2495:
2491:
2487:
2486:
2465:
2459:
2456:
2450:
2445:
2442:
2431:
2423:
2419:
2400:
2397:
2392:
2389:
2376:
2372:
2368:
2364:
2360:
2356:
2352:
2351:sigma-algebra
2348:
2344:
2340:
2336:
2332:
2328:
2325:), Ί). Here,
2324:
2320:
2316:
2312:
2311:measure space
2306:
2298:
2296:
2294:
2290:
2286:
2282:
2277:
2275:
2271:
2267:
2263:
2260:
2256:
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2250:
2246:
2242:
2238:
2230:
2228:
2226:
2222:
2214:
2212:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2162:
2160:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2116:
2115:discrete-time
2111:
2103:
2101:
2099:
2095:
2091:
2090:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2040:diffeomorphic
2037:
2033:
2029:
2026:
2022:
2021:open interval
2018:
2014:
2010:
2006:
2005:
2000:
1998:
1993:
1989:
1981:
1979:
1953:
1949:
1945:
1917:
1913:
1890:
1887:
1879:
1876: â
1875:
1871:
1867:
1863:
1796:
1793:
1783:
1761:
1759:
1757:
1753:
1749:
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1711:
1705:
1699:
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1668:
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1656:
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1637:
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1626:
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1598:
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1589:
1563:
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1551:
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1536:
1524:
1519:
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1506:
1505:
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1499:
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1493:
1489:
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1466:
1457:
1451:
1448:
1443:
1431:
1430:
1429:
1409:
1406:
1403:
1394:
1388:
1380:
1368:
1351:
1348:
1345:
1336:
1330:
1322:
1310:
1309:
1308:
1305:
1303:
1302:initial state
1299:
1295:
1291:
1290:
1285:
1281:
1278:, called the
1277:
1273:
1269:
1265:
1261:
1257:
1252:
1250:
1246:
1242:
1241:monoid action
1226:
1223:
1217:
1211:
1203:
1199:
1183:
1180:
1177:
1174:
1171:
1162:
1160:
1156:
1137:
1134:
1128:
1125:
1122:
1116:
1113:
1110:
1107:
1101:
1095:
1089:
1063:
1060:
1055:
1051:
1038:
1035:
1030:
1026:
999:
993:
990:
985:
981:
977:
972:
968:
963:
958:
954:
929:
923:
920:
915:
911:
907:
902:
898:
888:
879:
876:
871:
867:
857:
852:
848:
834:
820:
817:
811:
808:
805:
792:
791:
790:
788:
784:
776:
758:
722:
719:
713:
705:
682:
681:
680:
663:
654:
651:
648:
642:
639:
636:
626:
625:
624:
623:
619:
615:
611:
607:
603:
599:
595:
591:
583:
581:
579:
575:
571:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
522:Ali H. Nayfeh
518:
516:
512:
508:
504:
500:
496:
494:
490:
489:Stephen Smale
486:
484:
480:
476:
472:
468:
464:
463:
458:
454:
450:
445:
442:
438:
436:
431:
423:
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414:
410:
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401:
398:
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390:
387:
383:
378:
375:
371:
367:
362:
361:
360:
356:
354:
349:
347:
346:
341:
340:
335:
331:
327:
323:
319:
315:
307:
305:
303:
302:edge of chaos
299:
295:
294:self-assembly
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
250:
247:
243:
239:
234:
232:
228:
227:deterministic
224:
220:
216:
212:
208:
204:
199:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
149:
145:
144:ambient space
141:
137:
133:
129:
125:
117:
113:
108:
98:
95:
87:
84:February 2022
77:
73:
67:
66:
60:
55:
46:
45:
40:
33:
19:
7165:Mary Tsingou
7130:David Ruelle
7125:Otto Rössler
7070:Michel HĂ©non
7040:Leon O. Chua
6997:Tilt-A-Whirl
6967:FPUT problem
6852:Standard map
6847:Logistic map
6703:
6672:
6446:Chaos theory
6255:Scholarpedia
6219:
6200:
6197:Ivar Ekeland
6177:
6173:James Gleick
6154:
6147:Florin Diacu
6140:
6126:. Springer.
6123:
6095:
6072:
6051:
6048:
6029:
6010:
5991:
5975:. Springer.
5972:
5953:
5937:. Springer.
5934:
5915:
5896:
5869:. Springer.
5867:
5864:
5845:
5834:
5817:
5777:
5773:
5754:
5751:David Ruelle
5730:
5700:
5695:V. I. Arnold
5688:
5671:
5667:
5644:. Springer.
5641:
5625:
5599:
5585:
5572:
5559:
5540:
5507:
5497:
5488:
5482:
5466:
5457:
5442:
5433:
5421:. Retrieved
5410:
5401:
5377:
5370:
5362:
5357:
5338:
5329:
5310:
5304:
5295:
5289:
5246:
5242:
5236:
5203:
5199:
5193:
5181:. Retrieved
5171:
5150:
5142:
5133:
5127:
4968:
4858:
4767:
4755:
4738:logistic map
4727:
4719:steady state
4711:
4706:
4702:
4688:
4682:
4679:Chaos theory
4667:SRB measures
4660:
4652:
4647:
4642:
4640:
4546:
4541:
4527:
4522:
4518:
4512:
4499:
4495:
4491:
4487:
4481:
4477:
4466:
4379:
4375:
4372:
4342:
4330:
4323:
4318:
4314:
4303:
4296:
4294:
4289:
4279:
4271:
4264:vector field
4261:
4243:
4238:
4234:
4230:
4226:
4220:
4215:
4209:
4205:
4198:
4194:
4190:
4186:
4182:
4180:
4170:
4165:
4161:
4157:
4152:
4148:
4141:
4138:
4064:
4060:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4030:
4022:
4018:
4014:
4010:
4006:
4002:
4000:Poincaré map
3992:
3988:
3984:
3974:
3970:
3963:
3960:
3950:
3946:
3942:
3938:
3930:
3928:
3923:
3919:
3915:
3911:
3909:
3898:
3894:
3890:
3886:
3884:
3872:
3867:
3863:
3856:
3852:
3848:
3841:
3837:
3835:
3827:
3824:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3790:
3714:
3692:
3690:
3685:
3682:eigenvectors
3677:
3669:
3667:
3584:
3573:
3569:
3568:is zero and
3565:
3563:
3487:
3483:
3479:
3475:
3471:
3467:
3465:
3391:
3385:
3375:
3371:
3367:
3363:
3359:
3355:
3351:
3347:
3339:
3335:
3333:
3208:
3205:
3137:
3006:
3001:
2997:
2995:
2928:
2923:
2919:
2915:
2911:
2907:
2901:
2897:
2893:
2889:
2885:
2881:
2876:
2871:
2867:
2863:
2859:
2857:
2851:
2844:
2840:
2828:
2824:
2821:vector field
2816:
2812:
2808:
2804:
2798:
2793:
2755:
2625:
2623:
2607:
2584:
2576:
2566:
2510:
2505:
2501:
2497:
2493:
2489:
2484:
2482:
2421:
2417:
2375:ÎŁ-measurable
2370:
2366:
2362:
2354:
2342:
2338:
2330:
2326:
2322:
2318:
2314:
2308:
2278:
2273:
2269:
2265:
2261:
2254:
2240:
2236:
2234:
2218:
2208:
2204:
2196:
2193:integer grid
2187:such as the
2180:
2176:
2172:
2171:is a tuple (
2168:
2166:
2157:semi-cascade
2156:
2152:
2148:
2144:
2140:
2137:Banach space
2128:
2127:, Ί), where
2124:
2120:
2119:is a tuple (
2113:
2109:
2107:
2097:
2093:
2087:
2083:
2079:
2071:
2067:
2063:
2059:
2055:
2044:Banach space
2031:
2027:
2025:real numbers
2016:
2012:
2008:
2007:is a tuple (
2002:
1995:
1991:
1987:
1985:
1951:
1947:
1943:
1911:
1910:) such that
1877:
1873:
1869:
1765:
1745:
1740:
1736:
1732:
1728:
1689:
1685:
1683:
1632:
1628:
1624:
1620:
1616:
1615:is called Ί-
1612:
1608:
1604:
1596:
1592:
1586:
1584:
1501:
1495:
1487:
1483:
1481:
1427:
1306:
1301:
1297:
1293:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1255:
1253:
1248:
1244:
1201:
1197:
1163:
1158:
1154:
944:
786:
782:
781:and for any
780:
678:
613:
605:
601:
597:
589:
587:
519:
497:
487:
460:
446:
439:
427:
357:
350:
343:
337:
333:
329:
311:
286:logistic map
282:chaos theory
251:
241:
235:
222:
215:real numbers
200:
190:or simply a
127:
121:
90:
81:
62:
7150:Nina Snaith
7140:Yakov Sinai
7025:Rufus Bowen
6775:Duffing map
6760:Baker's map
6685:Theoretical
6597:SRB measure
6504:Phase space
6474:Bifurcation
6330:Chaos @ UMD
6222:. Penguin.
6181:. Penguin.
5721:Jacob Palis
5491:. Springer.
5298:. Springer.
5183:17 February
5082:Oscillation
4730:Meteorology
4714:mathematics
4498:returns to
4311:eigenvalues
4282:fixed point
4268:phase space
3674:eigenvalues
3307:Rössler map
3233:Baker's map
2908:homogeneous
2179:, Ί), with
2046:, and Ί a
1758:in flavor.
1619:if for all
1607:. A subset
1294:state space
1289:phase space
1270:in the set
773:is the 2nd
620:and Ί is a
604:, Ί) where
566:jet engines
562:skyscrapers
534:engineering
374:equivalence
266:engineering
207:state space
124:mathematics
76:introducing
7238:Categories
7208:Complexity
7105:Edward Ott
6952:Convection
6877:Continuous
6552:Ergodicity
6349:, Caltech.
6253:A part of
6101:Providence
5835:Textbooks
5249:(3): 617.
5136:. Perseus.
5119:References
4723:attractors
4525:)/vol(Ω).
4211:hyperbolic
4208:is called
3943:integrable
3470:a matrix,
3257:Circle map
3201:functional
2882:autonomous
2243:, Ί) on a
2050:. If Ί is
2015:, Ί) with
1504:. The set
1497:trajectory
1286:is called
578:spacecraft
542:structures
530:mechanical
501:developed
339:trajectory
326:time scale
288:dynamics,
231:stochastic
59:references
7120:Mary Rees
7080:Bryna Kra
7013:theorists
6822:Ikeda map
6812:HĂ©non map
6802:Gauss map
6484:Limit set
6469:Attractor
5796:cite book
5634:0938-0396
5423:25 August
5337:(2009) .
5256:0705.0311
4980:≥
4928:−
4904:−
4790:−
4606:−
4602:Φ
4508:Boltzmann
4431:Φ
4118:⋅
4097:∘
4091:∘
4083:−
3672:= 0, the
3602:Φ
3502:Φ
3415:˙
3277:HĂ©non map
3243:Billiards
3182:→
3165:×
3088:Φ
3066:⇔
3034:−
3028:˙
2957:Φ
2773:˙
2657:˙
2599:attractor
2553:Φ
2550:∘
2547:⋯
2544:∘
2541:Φ
2538:∘
2535:Φ
2523:Φ
2466:σ
2460:μ
2451:σ
2443:−
2439:Φ
2432:μ
2404:Σ
2401:∈
2398:σ
2390:−
2386:Φ
2285:non-empty
2281:limit set
2249:Hausdorff
2098:semi-flow
2092:; and if
1891:∈
1800:⟩
1774:⟨
1690:invariant
1663:∈
1645:Φ
1617:invariant
1555:∈
1531:Φ
1525:≡
1516:γ
1464:→
1440:Φ
1398:Φ
1395:≡
1377:Φ
1340:Φ
1337:≡
1319:Φ
1181:×
1135:∈
1111:∈
1045:Φ
1036:∈
991:∈
892:Φ
861:Φ
842:Φ
800:Φ
661:→
652:×
643:⊆
634:Φ
558:buildings
511:real line
447:In 1913,
353:computers
324:or other
304:concept.
270:economics
262:chemistry
7196:articles
6938:Physical
6857:Tent map
6747:Discrete
6687:branches
6617:Theorems
6453:Concepts
6393:Archived
6380:Archived
6199:(1990).
6175:(1988).
6153:(1996).
6093:(2012).
6071:(1994).
6013:. SIAM.
5895:(2003).
5844:(2000).
5816:(1992).
5776:(1991).
5753:(1989).
5727:(1982).
5697:(1982).
5662:(1967).
5598:(1978).
5524:45426376
5383:Springer
5228:16252993
5177:"Nature"
5024:See also
4783:′
4740:and the
4216:elliptic
3394:) is an
3317:Tent map
3221:Examples
2833:velocity
2790:velocity
2513:iterates
2189:integers
2133:manifold
2038:locally
2036:manifold
1862:manifold
1727:for all
1627:and all
1591:through
1500:through
1490:and its
1486:through
1153:for any
622:function
574:aircraft
538:machines
524:applied
308:Overview
278:medicine
217:or by a
188:manifold
156:pendulum
132:function
7194:Related
7002:Weather
6940:systems
6733:Chaotic
6479:Fractal
5281:8677631
5261:Bibcode
5208:Bibcode
4736:of the
4534:Koopman
4521:is vol(
4504:Zermelo
4467:In the
3991:,
3862:, with
2884:, when
2756:where
2359:measure
2345:) is a
2337:, and (
2289:compact
2276:, Ί*).
2185:lattice
2145:cascade
2078:. When
2023:in the
1754:and is
1599:is the
735:(where
554:bridges
473:on the
471:physics
424:History
415:and of
274:history
258:biology
238:physics
72:improve
7100:Hee Oh
6735:maps (
6582:Mixing
6226:
6207:
6185:
6161:
6130:
6111:
6079:
6057:
6036:
6017:
5998:
5979:
5960:
5941:
5922:
5903:
5873:
5852:
5824:
5784:
5761:
5739:
5709:
5648:
5632:
5618:
5606:
5547:
5522:
5473:
5449:
5389:
5345:
5317:
5279:
5226:
5159:
4549:, the
4189:. As
3855:
3721:affine
3396:affine
3386:For a
3354:) and
3138:where
3004:, Ί).
2837:forces
2084:global
1880:(with
1023:
610:monoid
550:cranes
513:has a
292:, the
276:, and
219:vector
196:smooth
166:, and
162:, the
142:in an
61:, but
7011:Chaos
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