39:
6590:
4945:
6582:
96:
3942:) = 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.
3692:
4281:. 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.
3684: 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
3122:
2571:) 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.
4362:
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
2566:
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
421:
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
368:
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
4731:
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions cannot be analytical
3890:
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
352:
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
4467:
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
2567:
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
4128:
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
344:, 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.
4634:
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
4699:
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
4683:
are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the
4672:
Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called
4533:
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
3006:
432:
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.
4931:
305:. 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
3822:. 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.
2684:
391:
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
3950:
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
3186:
2980:
2552:
929:
347:
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:
237:
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.
4821:
4451:
1799:
2740:
361:. 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
1569:
2208:
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
2468:
4123:
663:
2617:
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
7166:
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3652:
1412:
1354:
1002:
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1069:
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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
1140:
4460:, 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
3548:
2775:
722:
4226:. 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
3450:
1897:
1466:
760:
5009:
3775:
1668:
5351:
3117:{\displaystyle {\dot {\boldsymbol {x}}}-{\boldsymbol {v}}(t,{\boldsymbol {x}})=0\qquad \Leftrightarrow \qquad {\mathfrak {G}}\left(t,\Phi (t,{\boldsymbol {x}}_{0})\right)=0}
1965:
1929:
1847:
1823:
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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
820:
1183:
4832:
4732:
functions on the whole real line, and because they are non-Lipschitz functions at their ending time, they are not unique solutions of
Lipschitz differential equations.
317:.) 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
5714:
4715:
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
1714:
1226:
4974:
384:
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
6879:
3874:
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
6709:
380:
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
4984:
2188:
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
3323:
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
509:
In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer
3891:
approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.
2635:
325:. 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
4234:
2605:
of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
2558:
are studied. For continuous dynamical systems, the map Ί is understood to be a finite time evolution map and the construction is more complicated.
6589:
6349:
4654:. This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems.
4325:
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
2594:
and the invariant measures must be singular with respect to the
Lebesgue measure. A small region of phase space shrinks under time evolution.
369:
possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes.
6140:
6121:
6099:
6075:
6044:
6025:
5993:
5971:
5931:
5912:
5893:
5855:
5836:
5817:
5787:
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5738:
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5675:
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5562:
5520:
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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
6542:
4461:
2575:
60:
3130:
2924:
2598:
6489:
5375:
2293:
426:, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.
6822:
4502:
One of the questions raised by
Boltzmann's work was the possible equality between time averages and space averages, what he called the
2506:
826:
5950:
5874:
4266:
is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.
215:
6864:
6552:
4468:
sub-manifold of the phase space. The volume of the energy shell, computed using the
Liouville measure, is preserved under evolution.
4377:
2257:. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (
1758:
374:
7057:
5532:
5461:
5303:
2690:
487:
82:
5295:
3934:. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where
2582:, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic
441:
6739:
4741:
3915:
where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.
1499:
4989:
4010:. 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
214:
of the dynamical system is a function that describes what future states follow from the current state. Often the function is
5154:
Gintautas, V.; et al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics".
4149:
they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form
3705:
2622:
1736:
484:
that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.
160:
6557:
6547:
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is the domain for time â there are many choices, usually the reals or the integers, possibly restricted to be non-negative.
423:
2416:
6017:
5722:
5504:
4999:
4182:
does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of
3467:
the position vector. The solution to this system can be found by using the superposition principle (linearity). The case
3203:
2918:
The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above
6812:
6904:
6677:
5024:
3300:
3290:
3250:
1854:
4211:
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618:
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systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of
6089:
4345:
2040:
218:, that is, for a given time interval only one future state follows from the current state. However, some systems are
6817:
422:
studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the
53:
47:
6884:
6271:
4521:
and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems.
7156:
6342:
4548:
3713:
3585:
3568: = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the
3351:) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will
3198:
Many of the concepts in dynamical systems can be extended to infinite-dimensional manifoldsâthose that are locally
2247:
1360:
1302:
937:
27:
2369:
1007:
6932:
5324:
4270:
3677:
it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
3235:
3225:
1074:
495:
3485:
64:
6636:
6581:
6013:
3923:
3189:
2209:
242:
20:
5004:
2751:
674:
7161:
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6562:
6469:
5726:
4333:
3332:
3318:
763:
491:
370:
175:
or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a
6277:
3393:
6837:
6449:
5329:
4950:
4199:
3240:
2586:
the choice of invariant measure is technically more challenging. The measure needs to be supported on the
610:
195:
167:
by allowing different choices of the space and how time is measured. Time can be measured by integers, by
120:
4318:) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of
1872:
1423:
727:
6987:
6894:
6692:
6519:
6454:
6429:
6335:
6163:
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4647:
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The map Ί embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the
2347:
1744:
471:
437:
401:
397:
306:
234:
6749:
3722:
3275:
400:. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of
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The intersection of the periodic orbit with the
Poincaré section is a fixed point of the Poincaré map
1629:
6997:
6827:
6649:
6494:
5173:
5120:
4979:
3958:
in the orbit Îł and consider the points in phase space in that neighborhood that are perpendicular to
3863:
3280:
362:
358:
4170:
The results on the existence of a solution to the conjugation equation depend on the eigenvalues of
1946:
1910:
1828:
1804:
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6697:
6682:
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6567:
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The
Liouville measure restricted to the energy surface Ω is the basis for the averages computed in
4526:
4341:
3970:
3270:
3231:
3216:
2618:
2363:
2064:
2036:
1589:
314:
310:
302:
156:
6214:
4926:{\displaystyle y(x)={\frac {1}{4}}\left(1-{\frac {x}{2}}+\left|1-{\frac {x}{2}}\right|\right)^{2}}
784:
7135:
7002:
6832:
6719:
6714:
6606:
6484:
6386:
6292:
5708:
5508:
5432:
5247:
5189:
5163:
4646:. An average in time along a trajectory is equivalent to an average in space computed with the
4517:
The ergodic hypothesis turned out not to be the essential property needed for the development of
4503:
4326:
4246:
3569:
3376:
2583:
2579:
2077:
1992:
1156:
514:
463:
445:
429:
381:
354:
278:
233:
is described as a "particle or ensemble of particles whose state varies over time and thus obeys
140:
5546:
1735:
More commonly there are two classes of definitions for a dynamical system: one is motivated by
7022:
7007:
6972:
6962:
6859:
6479:
6401:
6136:
6117:
6095:
6071:
6040:
6021:
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5927:
5908:
5889:
5870:
5851:
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5805:
5783:
5762:
5734:
5694:
5671:
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5637:
5619:
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5528:
5516:
5457:
5383:
5359:
5299:
5255:
5227:
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is measured in units of (position) Ă (momentum). The flow takes points of a subset
4337:
3809:
3327:-dimensional Euclidean space, so any point in phase space can be represented by a vector with
2501:
2323:
2240:
2189:
606:
498:
in 1964. One of the implications of the theorem is that if a discrete dynamical system on the
418:
286:
219:
184:
180:
5986:
Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering
4158:â ÎŁ (multiples of other eigenvalues) occurs in the denominator of the terms for the function
3840:, with a real eigenvalue smaller than one, then the straight lines given by the points along
7115:
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6977:
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6802:
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333:
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128:
100:
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Some bifurcations can lead to very complicated structures in phase space. For example, the
3859:, is an invariant curve of the map. Points in this straight line run into the fixed point.
7047:
6942:
6869:
6702:
6514:
6504:
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2602:
2237:
2233:
2181:
2173:
1985:
538:
455:
152:
7037:
6982:
5352:
Dynamical systems on monoids: Toward a general theory of deterministic systems and motion
4506:. The hypothesis states that the length of time a typical trajectory spends in a region
3988:
3561: â 0 the origin is an equilibrium (or singular) point of the flow, that is, if
19:
This article is about the general aspects of dynamical systems. For the study field, see
5177:
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6779:
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6672:
6621:
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5555:
Dynamics Beyond
Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective
5274:
Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"."
5062:
5019:
4357:
3295:
3221:
3195:
This equation is useful when modeling mechanical systems with complicated constraints.
2601:
appear to be the natural choice. They are constructed on the geometrical structure of
2298:
A dynamical system may be defined formally as a measure-preserving transformation of a
1904:
1740:
558:
503:
467:
454:. Birkhoff's most durable result has been his 1931 discovery of what is now called the
393:
172:
164:
148:
6205:. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
450:
377:
are examples of dynamical systems where the possible classes of orbits are understood.
7150:
7125:
7082:
7072:
7067:
6967:
6947:
6807:
6729:
6626:
6434:
6224:
6220:
6202:
6063:
6007:
6003:
5801:
5797:
5572:
4720:
4492:
2847:
There is no need for higher order derivatives in the equation, nor for the parameter
2837:
2339:
2335:
2299:
2277:
2103:
2028:
2009:
510:
481:
477:
290:
282:
132:
104:
5593:
5576:
5436:
4658:
replace the Boltzmann factor and they are defined on attractors of chaotic systems.
4203:
and when the eigenvalues are on the unit circle and complex, the dynamics is called
3812:, the origin is a fixed point of the map and the solutions are of the linear system
7077:
7042:
6952:
6909:
6764:
6759:
6358:
6248:
6173:
6167:
6109:
6085:
6059:
5663:
5193:
5089:
4716:
4708:?" or "Does the long-term behavior of the system depend on its initial condition?"
4701:
4675:
4667:
4499:'s derivation of the increase in entropy in a dynamical system of colliding atoms.
4252:
3685:
3199:
3000:
shown above gives a more general form of equations a dynamical system must satisfy
2866:
Depending on the properties of this vector field, the mechanical system is called
2809:
2125:
2032:
405:
274:
270:
262:
245:, which has applications to a wide variety of fields such as mathematics, physics,
207:
203:
95:
6724:
3265:
5047:
Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry
7062:
7052:
6937:
6687:
6509:
6416:
5633:
5319:
5111:
Melby, P.; et al. (2005). "Dynamics of Self-Adjusting Systems With Noise".
4994:
4712:
4696:
4655:
4299:
4256:
3670:
2863:), because these can be eliminated by considering systems of higher dimensions.
2013:
1277:
554:
550:
522:
254:
168:
112:
6250:, SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
5416:
4471:
For systems where the volume is preserved by the flow, Poincaré discovered the
2679:{\displaystyle {\dot {\boldsymbol {x}}}={\boldsymbol {v}}(t,{\boldsymbol {x}})}
7120:
7017:
6464:
5380:
Methods, models, simulations and approaches towards a general theory of change
5356:
Methods, models, simulations and approaches towards a general theory of change
5185:
4940:
3886:) = 0) will remain a singular point under smooth transformations; a
3662:
3245:
1485:
566:
530:
385:
327:
159:. The most general definition unifies several concepts in mathematics such as
6211:. An introduction to dynamical systems from the periodic orbit point of view.
6182:
provides definitions, explanations and resources related to nonlinear science
3899:
A flow in most small patches of the phase space can be made very simple. If
1931:
into the space of diffeomorphisms of the manifold to itself. In other terms,
7032:
6992:
6734:
6396:
6381:
5428:
4705:
4336:
describes how a periodic orbit bifurcates into a torus and the torus into a
3331:
numbers. The analysis of linear systems is possible because they satisfy a
2587:
2273:
2269:
1853:, i.e. locally a Banach space or Euclidean space, or in the discrete case a
546:
518:
499:
341:
258:
250:
6157:
5140:
3911:) â 0, then there is a change of coordinates for a region around
2625:
must be solved before it becomes a dynamic system. For example consider an
506:
of period 3, then it must have periodic points of every other period.
5478:
3691:
3669:
determine the structure of the phase space. From the eigenvalues and the
1907:
of the manifold to itself. So, f is a "smooth" mapping of the time-domain
353:
stability have been introduced in the study of dynamical systems, such as
222:, in that random events also affect the evolution of the state variables.
6769:
6160:
has daily submissions of (non-refereed) manuscripts in dynamical systems.
5549:) has a sub-series on dynamical systems with reviews of current research.
4162:, the non-resonant condition is also known as the small divisor problem.
3305:
3192:
from the set of evolution functions to the field of the complex numbers.
2821:
2778:
2121:
2024:
1850:
562:
526:
266:
176:
144:
6192:
5291:
IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems
2085:
is restricted to the non-negative reals, then the dynamical system is a
6439:
6391:
5810:
Differential Equations, dynamical systems, and an introduction to chaos
2481:
to itself, it is ÎŁ-measurable, and is measure-preserving. The triplet (
2177:
542:
470:, this theorem solved, at least in principle, a fundamental problem of
459:
246:
226:
5473:
Introduction to the Theory of Infinite-Dimensional Dissipative Systems
5132:
2246:, it is often useful to study the continuous extension Ί* of Ί to the
7012:
4269:
Bifurcation theory considers a structure in phase space (typically a
598:
6315:
3181:{\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} }
2975:{\displaystyle {\boldsymbol {x}}(t)=\Phi (t,{\boldsymbol {x}}_{0})}
480:
made significant advances as well. His first contribution was the
6253:
5168:
4274:
3690:
3680:
The distance between two different initial conditions in the case
2825:
582:
534:
199:
94:
6236:
4230:
in the complex plane, implying that the map is still hyperbolic.
2547:{\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi }
924:{\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}
448:, a result that made him world-famous. In 1927, he published his
5943:
Introduction to Modern Dynamics: Chaos, Networks, Space and Time
3788:
a vector. As in the continuous case, the change of coordinates
3475: = 0 is just a straight line in the direction of
2144:
is restricted to the non-negative integers we call the system a
124:
6331:
6327:
5486:
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
5288:
Rega, Giuseppe (2019). "Tribute to Ali H. Nayfeh (1933â2017)".
4491:
infinitely often. The Poincaré recurrence theorem was used by
4475:: Assume the phase space has a finite Liouville volume and let
4446:{\displaystyle \mathrm {vol} (A)=\mathrm {vol} (\Phi ^{t}(A)).}
4344:
describes how a stable periodic orbit goes through a series of
4277:) and studies its behavior as a function of the parameter
3825:
As in the continuous case, the eigenvalues and eigenvectors of
1794:{\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle }
1755:
In the geometrical definition, a dynamical system is the tuple
474:. The ergodic theorem has also had repercussions for dynamics.
6259:
6196:
4186:
are not in the unit circle, the dynamics near the fixed point
2735:{\displaystyle {\boldsymbol {x}}|_{t=0}={\boldsymbol {x}}_{0}}
32:
6321:
4688:) and another of the points that diverge from the orbit (the
16:
Mathematical model of the time dependence of a point in space
6286:, Instituto Superior TĂ©cnico, Technical University of Lisbon
6242:
5886:
Dynamical Systems with Applications using Mathematica 2nd Ed
5867:
Dynamical Systems with Applications using MATLAB 2nd Edition
1952:
1916:
1884:
1834:
1810:
1777:
1767:
1739:
and is geometrical in flavor; and the other is motivated by
6312:, Institute of Computer Science, Czech Academy of Sciences.
4816:{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}
4630:
By studying the spectral properties of the linear operator
1417:
if we take one of the variables as constant. The function
6299:, IMPA, Instituto Nacional de MatemĂĄtica Pura e Applicada.
5668:
Elements of Differentiable Dynamics and Bifurcation Theory
5553:
Christian Bonatti; Lorenzo J. DĂaz; Marcelo Viana (2005).
4711:
The chaotic behavior of complex systems is not the issue.
1564:{\displaystyle \gamma _{x}\equiv \{\Phi (t,x):t\in I(x)\}}
396:
and a more detailed understanding has been worked out for
5209:
Applications of Dynamical Systems in Biology and Medicine
4483:
a subset of the phase space. Then almost every point of
3829:
determine the structure of phase space. For example, if
2071:
is taken to be the reals, the dynamical system is called
6302:
6283:
6245:. Concentrates on the applications of dynamical systems.
6208:
5113:
Chaos: An Interdisciplinary Journal of Nonlinear Science
5691:
Ergodic theory, symbolic dynamics and hyperbolic spaces
6322:
Center for Control, Dynamical Systems, and Computation
6289:
5848:
Dynamical Systems with Applications using Maple 2nd Ed
5829:
Introduction to the modern theory of dynamical systems
5644:
Geometric theory of dynamical systems: an introduction
5064:
Introduction to the Modern Theory of Dynamical Systems
4525:
approached the study of ergodic systems by the use of
4024:. By a translation, the point can be assumed to be at
3387:
function of the position in the phase space, that is,
2828:
acting on the given material point in the phase space
1255:
of the dynamical system: it associates to every point
6221:
Ordinary Differential Equations and Dynamical Systems
6009:
Ordinary Differential Equations and Dynamical Systems
4835:
4744:
4551:
4380:
4065:
3725:
3588:
3488:
3396:
3133:
3009:
2927:
2832:. The change is not a vector in the phase space
2754:
2693:
2638:
2509:
2463:{\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )}
2419:
2372:
2212:. Such systems are useful for modeling, for example,
1949:
1913:
1875:
1831:
1807:
1761:
1687:
1632:
1502:
1426:
1363:
1305:
1199:
1159:
1077:
1010:
940:
829:
787:
730:
677:
621:
301:
The concept of a dynamical system has its origins in
7106:
6923:
6850:
6788:
6658:
6645:
6597:
6528:
6372:
6365:
6176:. Models of bifurcation and chaos by Elmer G. Wiens
6037:
Introduction to Applied Dynamical Systems and Chaos
4704:in the long term, and if so, what are the possible
4237:theorem gives the behavior near an elliptic point.
6268:, Ecole Polytechnique Fédérale de Lausanne (EPFL).
5641:
5611:
5376:Reversible dynamics and the directionality of time
5061:
4925:
4815:
4619:
4445:
4117:
3769:
3646:
3542:
3444:
3180:
3116:
2974:
2769:
2734:
2678:
2546:
2462:
2397:
1959:
1923:
1891:
1841:
1817:
1793:
1728:must be defined for all time for every element of
1708:
1662:
1563:
1460:
1406:
1348:
1220:
1177:
1134:
1063:
996:
923:
814:
754:
716:
657:
7167:Mathematical and quantitative methods (economics)
5477:online version of first edition on the EMIS site
5421:1985 24th IEEE Conference on Decision and Control
5010:Conley's fundamental theorem of dynamical systems
4118:{\displaystyle h^{-1}\circ F\circ h(x)=J\cdot x.}
3987:), of the orbit. The flow now defines a map, the
2574:Some systems have a natural measure, such as the
2318:is a monoid (usually the non-negative integers),
2196:represents the "space" lattice, while the one in
658:{\displaystyle \Phi :U\subseteq (T\times X)\to X}
6278:Systems Analysis, Modelling and Prediction Group
6195:. Nils Berglund's lecture notes for a course at
6133:Does God Play Dice? The New Mathematics of Chaos
5905:Dynamical Systems with Applications using Python
5685:Tim Bedford, Michael Keane and Caroline Series,
4028: = 0. The Taylor series of the map is
375:systems that have two numbers describing a state
365:changes with the different notions of stability.
26:"Dynamical" redirects here. For other uses, see
6170:â peer reviewed and written by invited experts.
4975:Dynamic approach to second language development
4371:) and invariance of the phase space means that
2470:. Combining the above, a map Ί is said to be a
241:The study of dynamical systems is the focus of
5731:Dynamicsâthe geometry of behavior, 2nd edition
5602:Introductory texts with a unique perspective:
3202:âin which case the differential equations are
269:. Dynamical systems are a fundamental part of
6343:
5581:Bulletin of the American Mathematical Society
4620:{\displaystyle (U^{t}a)(x)=a(\Phi ^{-t}(x)).}
4284:The bifurcations of a hyperbolic fixed point
3647:{\displaystyle \Phi ^{t}(x_{0})=e^{tA}x_{0}.}
2493:), Ί), for such a Ί, is then defined to be a
1407:{\displaystyle \Phi ^{t}(x)\equiv \Phi (t,x)}
1349:{\displaystyle \Phi _{x}(t)\equiv \Phi (t,x)}
997:{\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)}
8:
5713:: CS1 maint: multiple names: authors list (
5382:, pp. 161â171, Singapore: World Scientific.
5378:". In Minati G., Abram M., Pessa E. (eds.),
5358:, pp. 173â185, Singapore: World Scientific.
5354:". In Minati G., Abram M., Pessa E. (eds.),
3695:Linear vector fields and a few trajectories.
2398:{\displaystyle \Phi ^{-1}\sigma \in \Sigma }
1788:
1762:
1558:
1516:
1129:
1093:
1064:{\displaystyle \ t_{2}\in I(\Phi (t_{1},x))}
190:At any given time, a dynamical system has a
157:the number of fish each springtime in a lake
5759:Chaos. An introduction to dynamical systems
5614:Mathematical methods of classical mechanics
4985:Infinite compositions of analytic functions
4479:be a phase space volume-preserving map and
4174:and the degree of smoothness required from
1135:{\displaystyle I(x):=\{t\in T:(t,x)\in U\}}
6655:
6369:
6350:
6336:
6328:
6324:, University of California, Santa Barbara.
6114:Mathematics and the Unexpected (Paperback)
3543:{\displaystyle \Phi ^{t}(x_{1})=x_{1}+bt.}
2996:Some formal manipulation of the system of
2366:if and only if, for every Ï in ÎŁ, one has
1263:a unique image, depending on the variable
6217:. Tutorial on learning dynamical systems.
5827:Anatole Katok; Boris Hasselblatt (1996).
5592:
5167:
5068:. Cambridge: Cambridge University Press.
4917:
4897:
4873:
4851:
4834:
4794:
4793:
4783:
4775:
4773:
4759:
4743:
4593:
4559:
4550:
4422:
4404:
4381:
4379:
4070:
4064:
3752:
3730:
3724:
3635:
3622:
3606:
3593:
3587:
3522:
3506:
3493:
3487:
3398:
3397:
3395:
3173:
3163:
3146:
3144:
3135:
3134:
3132:
3094:
3089:
3059:
3058:
3039:
3025:
3011:
3010:
3008:
2963:
2958:
2928:
2926:
2756:
2755:
2753:
2726:
2721:
2705:
2700:
2694:
2692:
2668:
2654:
2640:
2639:
2637:
2514:
2508:
2430:
2418:
2377:
2371:
1951:
1950:
1948:
1915:
1914:
1912:
1883:
1882:
1874:
1833:
1832:
1830:
1809:
1808:
1806:
1776:
1775:
1766:
1765:
1760:
1686:
1631:
1507:
1501:
1431:
1425:
1368:
1362:
1310:
1304:
1198:
1158:
1076:
1043:
1018:
1009:
973:
960:
955:
946:
941:
939:
903:
890:
859:
840:
828:
786:
746:
732:
729:
693:
679:
676:
620:
83:Learn how and when to remove this message
6318:, Polytechnical University of Catalonia.
4719:is only a second-degree polynomial; the
3878:of the vector field (a point where
2770:{\displaystyle {\dot {\boldsymbol {x}}}}
717:{\displaystyle \mathrm {proj} _{2}(U)=X}
417:Many people regard French mathematician
46:This article includes a list of general
6193:Geometrical theory of dynamical systems
5753:Kathleen T. Alligood, Tim D. Sauer and
5037:
3090:
3040:
3026:
3013:
2959:
2929:
2758:
2722:
2695:
2669:
2655:
2642:
533:that are common in daily life, such as
194:representing a point in an appropriate
5706:
5539:Encyclopaedia of Mathematical Sciences
4302:of the first derivative of the system
2597:For hyperbolic dynamical systems, the
2220:Compactification of a dynamical system
2192:are dynamical systems. The lattice in
1939:) is a diffeomorphism, for every time
143:that describe the swinging of a clock
6316:UPC Dynamical Systems Group Barcelona
5907:. Springer International Publishing.
5869:. Springer International Publishing.
5254:(Fourth ed.). Berlin: Springer.
5252:Economic Dynamics: Methods and Models
4826:Admits the finite duration solution:
4662:Nonlinear dynamical systems and chaos
3445:{\displaystyle {\dot {x}}=v(x)=Ax+b,}
2472:measure-preserving transformation of
2132:is taken to be the integers, it is a
153:random motion of particles in the air
7:
6199:at the advanced undergraduate level.
5417:"Finite Time Differential Equations"
5207:Jackson, T.; Radunskaya, A. (2015).
4273:, a periodic orbit, or an invariant
187:space-time structure defined on it.
6490:Measure-preserving dynamical system
5060:Katok, A.; Hasselblatt, B. (1995).
3712:dynamical system has the form of a
3136:
3060:
2294:Measure-preserving dynamical system
1892:{\displaystyle t\in {\mathcal {T}}}
1461:{\displaystyle \Phi _{x}:I(x)\to X}
755:{\displaystyle \mathrm {proj} _{2}}
386:transition to turbulence of a fluid
5577:"Differentiable dynamical systems"
5499:Works providing a broad coverage:
5374:Mazzola C. and Giunti M. (2012), "
5350:Giunti M. and Mazzola C. (2012), "
4590:
4419:
4411:
4408:
4405:
4388:
4385:
4382:
3903:is a point where the vector field
3590:
3490:
3076:
2945:
2541:
2529:
2523:
2511:
2427:
2392:
2374:
2059:; if not, the dynamical system is
1633:
1519:
1428:
1386:
1365:
1328:
1307:
1033:
880:
849:
830:
788:
742:
739:
736:
733:
689:
686:
683:
680:
622:
52:it lacks sufficient corresponding
14:
7058:Oleksandr Mykolayovych Sharkovsky
6237:Dynamical Systems Group Groningen
6180:Sci.Nonlinear FAQ 2.0 (Sept 2003)
6164:Encyclopedia of dynamical systems
4644:equilibrium statistical mechanics
4251:When the evolution map Ί (or the
4052:can only be expected to simplify
3770:{\displaystyle x_{n+1}=Ax_{n}+b,}
2609:Construction of dynamical systems
2268:In compact dynamical systems the
2224:Given a global dynamical system (
488:Oleksandr Mykolaiovych Sharkovsky
198:. This state is often given by a
6588:
6580:
6254:Center for Dynamics and Geometry
5452:(2006). "Fundamental concepts".
5224:Advanced Engineering Mathematics
4943:
3864:other discrete dynamical systems
3174:
2790:is a finite dimensional manifold
2562:Relation to geometric definition
1663:{\displaystyle \Phi (t,x)\in S.}
1153:In particular, in the case that
1071:, where we have defined the set
37:
6266:Laboratory of Nonlinear Systems
6239:, IWI, University of Groningen.
6116:. University Of Chicago Press.
5594:10.1090/S0002-9904-1967-11798-1
5454:Ordinary Differential Equations
4990:List of dynamical system topics
3291:Quadratic map simulation system
3057:
3053:
2623:ordinary differential equations
2204:Multidimensional generalization
2200:represents the "time" lattice.
2045:differentiable dynamical system
1737:ordinary differential equations
210:in a geometrical manifold. The
161:ordinary differential equations
6823:RabinovichâFabrikant equations
5963:Chaos and time-series analysis
5960:Julien Clinton Sprott (2003).
5924:Differential Dynamical Systems
4845:
4839:
4804:
4798:
4784:
4776:
4770:
4764:
4611:
4608:
4602:
4586:
4577:
4571:
4568:
4552:
4437:
4434:
4428:
4415:
4398:
4392:
4222:. The hyperbolic case is also
4097:
4091:
4048:), so a change of coordinates
3808:from the equation. In the new
3612:
3599:
3512:
3499:
3421:
3415:
3204:partial differential equations
3170:
3159:
3147:
3100:
3079:
3054:
3044:
3030:
2985:The dynamical system is then (
2969:
2948:
2939:
2933:
2701:
2673:
2659:
2457:
2451:
2442:
2423:
2288:Measure theoretical definition
1960:{\displaystyle {\mathcal {T}}}
1924:{\displaystyle {\mathcal {T}}}
1842:{\displaystyle {\mathcal {M}}}
1818:{\displaystyle {\mathcal {T}}}
1697:
1691:
1648:
1636:
1555:
1549:
1534:
1522:
1452:
1449:
1443:
1401:
1389:
1380:
1374:
1343:
1331:
1322:
1316:
1209:
1203:
1120:
1108:
1087:
1081:
1058:
1055:
1036:
1030:
991:
985:
915:
883:
874:
871:
852:
833:
803:
791:
705:
699:
649:
646:
634:
1:
6260:Control and Dynamical Systems
6187:Online books or lecture notes
6018:American Mathematical Society
5000:People in systems and control
4235:KolmogorovâArnoldâMoser (KAM)
2820:and represents the change of
2603:stable and unstable manifolds
2128:, and Ί is a function. When
577:In the most general sense, a
6303:Nonlinear Dynamics Workgroup
6272:Center for Dynamical Systems
6209:Chaos: classical and quantum
5025:Principle of maximum caliber
4727:Solutions of Finite Duration
4346:period-doubling bifurcations
4298:can be characterized by the
4210:In the hyperbolic case, the
3796: + (1 â
3251:Complex quadratic polynomial
2619:classical mechanical systems
2051:is locally diffeomorphic to
1724:. That is, the flow through
815:{\displaystyle \Phi (0,x)=x}
6558:Poincaré recurrence theorem
6091:Chaos: Making a New Science
5966:. Oxford University Press.
5945:. Oxford University Press.
5693:. Oxford University Press.
5456:. Berlin: Springer Verlag.
3224:is an example of a chaotic
2599:SinaiâRuelleâBowen measures
2590:, but attractors have zero
2124:locally diffeomorphic to a
2041:continuously differentiable
1178:{\displaystyle U=T\times X}
424:Poincaré recurrence theorem
149:the flow of water in a pipe
103:arises in the study of the
7185:
6553:PoincarĂ©âBendixson theorem
6215:Learning Dynamical Systems
5779:Discrete Dynamical Systems
5402:Discrete Dynamical Systems
4735:As example, the equation:
4665:
4355:
4342:Feigenbaum period-doubling
4244:
3714:matrix difference equation
3316:
2409:if and only if, for every
2291:
2248:one-point compactification
2055:, the dynamical system is
1981:real-time dynamical system
1228:and thus that Ί defines a
496:discrete dynamical systems
458:. Combining insights from
28:Dynamical (disambiguation)
25:
18:
6905:Swinging Atwood's machine
6578:
6548:KrylovâBogolyubov theorem
6425:
6284:Non-Linear Dynamics Group
5527:(available as a reprint:
5325:Franklin Institute Awards
5186:10.1007/s10955-007-9444-4
4002:, for points starting in
3301:Swinging Atwood's machine
2569:KrylovâBogolyubov theorem
2099:discrete dynamical system
2093:Discrete dynamical system
2063:. This does not assume a
444:", a special case of the
6813:LotkaâVolterra equations
6637:Synchronization of chaos
6440:axiom A dynamical system
6035:Stephen Wiggins (2003).
5513:Foundations of mechanics
5415:Vardia T. Haimo (1985).
5222:Kreyszig, Erwin (2011).
5045:Strogatz, S. H. (2001).
3930:the dynamical system is
3463:a vector of numbers and
3313:Linear dynamical systems
2836:, but is instead in the
2210:multidimensional systems
2180:or a higher-dimensional
1673:Thus, in particular, if
371:Linear dynamical systems
243:dynamical systems theory
183:, without the need of a
21:Dynamical systems theory
6798:Double scroll attractor
6563:Stable manifold theorem
6470:False nearest neighbors
5941:David D. Nolte (2015).
5429:10.1109/CDC.1985.268832
4648:Boltzmann factor exp(âÎČ
4212:HartmanâGrobman theorem
4145:are the eigenvalues of
3969:). These points are a
3572:: for an initial point
3570:exponential of a matrix
3333:superposition principle
3319:Linear dynamical system
2629:such as the following:
2043:we say the system is a
601:, written additively,
139:. Examples include the
119:is a system in which a
67:more precise citations.
6838:Van der Pol oscillator
6818:MackeyâGlass equations
6450:Box-counting dimension
6280:, University of Oxford
6274:, University of Bremen
5903:Stephen Lynch (2018).
5884:Stephen Lynch (2017).
5865:Stephen Lynch (2014).
5846:Stephen Lynch (2010).
5484:Temam, Roger (1997) .
5423:. pp. 1729â1733.
5330:The Franklin Institute
5278:193.3 (1990): 137â163.
4951:Systems science portal
4927:
4817:
4621:
4447:
4340:. In another example,
4334:RuelleâTakens scenario
4259:until a special value
4119:
4006:and returning to
3771:
3696:
3648:
3544:
3446:
3241:Bouncing ball dynamics
3182:
3118:
2998:differential equations
2976:
2781:of the material point
2771:
2736:
2680:
2548:
2477:, if it is a map from
2464:
2399:
2338:, meaning that ÎŁ is a
1961:
1925:
1893:
1843:
1819:
1795:
1751:Geometrical definition
1710:
1709:{\displaystyle I(x)=T}
1664:
1565:
1462:
1408:
1350:
1222:
1221:{\displaystyle I(x)=T}
1179:
1136:
1065:
998:
925:
816:
756:
718:
659:
442:Last Geometric Theorem
323:integrating the system
235:differential equations
108:
6988:Svetlana Jitomirskaya
6895:Multiscroll attractor
6740:Interval exchange map
6693:Dyadic transformation
6678:Complex quadratic map
6520:Topological conjugacy
6455:Correlation dimension
6430:Anosov diffeomorphism
6158:Arxiv preprint server
5515:. BenjaminâCummings.
5005:Sharkovskii's theorem
4928:
4818:
4723:is piecewise linear.
4622:
4519:statistical mechanics
4458:Hamiltonian formalism
4448:
4120:
3920:rectification theorem
3836:is an eigenvector of
3772:
3694:
3649:
3545:
3447:
3379:, the vector field v(
3261:Dyadic transformation
3183:
3119:
2977:
2824:induced by the known
2772:
2737:
2681:
2627:initial value problem
2549:
2465:
2405:. A map Ί is said to
2400:
1977:real dynamical system
1971:Real dynamical system
1962:
1926:
1894:
1861:is an evolution rule
1844:
1820:
1796:
1711:
1665:
1566:
1463:
1409:
1351:
1285:, while the variable
1223:
1180:
1137:
1066:
999:
926:
817:
757:
719:
660:
472:statistical mechanics
438:George David Birkhoff
402:statistical mechanics
340:Before the advent of
307:differential equation
107:, a dynamical system.
98:
6998:Edward Norton Lorenz
6131:Ian Stewart (1997).
6068:Celestial Encounters
5922:James Meiss (2007).
5400:Galor, Oded (2010).
4980:Feedback passivation
4833:
4742:
4549:
4378:
4063:
3946:Near periodic orbits
3922:says that away from
3862:There are also many
3723:
3586:
3486:
3471: â 0 with
3394:
3281:List of chaotic maps
3131:
3007:
2925:
2752:
2691:
2636:
2507:
2417:
2407:preserve the measure
2370:
2065:symplectic structure
2061:infinite-dimensional
1947:
1911:
1873:
1829:
1805:
1759:
1685:
1630:
1592:of the flow through
1584:. The orbit through
1500:
1424:
1361:
1303:
1197:
1157:
1075:
1008:
938:
827:
785:
728:
675:
619:
492:Sharkovsky's theorem
359:structural stability
6958:Mitchell Feigenbaum
6900:Population dynamics
6885:HĂ©nonâHeiles system
6745:Irrational rotation
6698:Dynamical billiards
6683:Coupled map lattice
6543:Liouville's theorem
6475:Hausdorff dimension
6460:Conservative system
6445:Bifurcation diagram
6223:. Lecture notes by
5776:Oded Galor (2011).
5761:. Springer Verlag.
5727:Christopher D. Shaw
5648:. Springer-Verlag.
5618:. Springer-Verlag.
5450:Arnold, Vladimir I.
5248:Gandolfo, Giancarlo
5178:2008JSP...130..617G
5125:2005Chaos..15c3902M
4960:Behavioral modeling
4527:functional analysis
4291:of a system family
4224:structurally stable
4166:Conjugation results
4056:to its linear part
3271:Irrational rotation
2584:dissipative systems
2580:Hamiltonian systems
2047:. If the manifold
2037:continuous function
1745:measure theoretical
1600:of the state space
1269:evolution parameter
440:proved Poincaré's "
311:difference equation
303:Newtonian mechanics
289:processes, and the
141:mathematical models
7136:Santa Fe Institute
7003:Aleksandr Lyapunov
6833:Three-body problem
6720:Gingerbreadman map
6607:Bifurcation theory
6485:Lyapunov stability
6308:2015-01-21 at the
6295:2017-06-02 at the
6174:Nonlinear Dynamics
5988:. Addison Wesley.
5982:Steven H. Strogatz
5812:. Academic Press.
5733:. Addison-Wesley.
5670:. Academic Press.
5509:Jerrold E. Marsden
5488:. Springer Verlag.
5320:"Ali Hasan Nayfeh"
5226:. Hoboken: Wiley.
4965:Cognitive modeling
4923:
4813:
4681:Hyperbolic systems
4617:
4504:ergodic hypothesis
4473:recurrence theorem
4443:
4367:into the points Ί(
4327:Bifurcation theory
4247:Bifurcation theory
4241:Bifurcation theory
4115:
3767:
3697:
3644:
3540:
3442:
3178:
3114:
2972:
2767:
2732:
2676:
2621:. But a system of
2554:for every integer
2544:
2460:
2395:
2346:and Ό is a finite
2158:cellular automaton
2152:Cellular automaton
2057:finite-dimensional
1957:
1921:
1889:
1839:
1815:
1791:
1706:
1660:
1561:
1458:
1404:
1346:
1253:evolution function
1218:
1185:we have for every
1175:
1132:
1061:
994:
921:
812:
752:
714:
655:
515:nonlinear dynamics
494:on the periods of
464:ergodic hypothesis
446:three-body problem
430:Aleksandr Lyapunov
398:hyperbolic systems
382:bifurcation points
355:Lyapunov stability
319:solving the system
279:bifurcation theory
109:
7157:Dynamical systems
7144:
7143:
7008:BenoĂźt Mandelbrot
6973:Martin Gutzwiller
6963:Peter Grassberger
6846:
6845:
6828:Rössler attractor
6576:
6575:
6480:Invariant measure
6402:Lyapunov exponent
6290:Dynamical Systems
6203:Dynamical systems
6142:978-0-14-025602-4
6123:978-0-226-19990-0
6101:978-0-14-009250-9
6077:978-0-691-02743-2
6054:Popularizations:
6046:978-0-387-00177-7
6027:978-0-8218-8328-0
5995:978-0-201-54344-5
5973:978-0-19-850839-7
5933:978-0-89871-635-1
5914:978-3-319-78145-7
5895:978-3-319-61485-4
5857:978-0-8176-4389-8
5838:978-0-521-57557-7
5819:978-0-12-349703-1
5806:Robert L. Devaney
5789:978-3-642-07185-0
5768:978-0-387-94677-1
5740:978-0-201-56716-8
5700:978-0-19-853390-0
5677:978-0-12-601710-6
5655:978-0-387-90668-3
5638:Welington de Melo
5625:978-0-387-96890-2
5564:978-3-540-22066-4
5522:978-0-8053-0102-1
5388:978-981-4383-32-5
5364:978-981-4383-32-5
5332:. 4 February 2014
5261:978-3-642-13503-3
5233:978-0-470-64613-7
5133:10.1063/1.1953147
5092:. Springer Nature
5075:978-0-521-34187-5
4905:
4881:
4859:
4788:
4762:
4690:unstable manifold
4540:transfer operator
4529:. An observable
4462:Liouville measure
4338:strange attractor
4136:, ...,
3810:coordinate system
3804:removes the term
3406:
3019:
2764:
2648:
2615:evolution in time
2576:Liouville measure
2336:probability space
2241:topological space
2190:cellular automata
1013:
573:Formal definition
451:Dynamical Systems
287:self-organization
105:Lorenz oscillator
93:
92:
85:
7174:
7116:Butterfly effect
7028:Itamar Procaccia
6978:Brosl Hasslacher
6875:Elastic pendulum
6803:Duffing equation
6750:KaplanâYorke map
6668:Arnold's cat map
6656:
6632:Stability theory
6617:Dynamical system
6612:Control of chaos
6592:
6584:
6568:Takens's theorem
6500:Poincaré section
6370:
6352:
6345:
6338:
6329:
6146:
6127:
6105:
6081:
6050:
6031:
5999:
5977:
5956:
5937:
5918:
5899:
5880:
5861:
5842:
5823:
5798:Morris W. Hirsch
5793:
5772:
5744:
5723:Ralph H. Abraham
5718:
5712:
5704:
5681:
5659:
5647:
5629:
5617:
5598:
5596:
5568:
5526:
5489:
5476:
5471:Chueshov, I. D.
5467:
5441:
5440:
5412:
5406:
5405:
5397:
5391:
5372:
5366:
5348:
5342:
5341:
5339:
5337:
5316:
5310:
5309:
5298:. pp. 1â2.
5285:
5279:
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5204:
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5145:
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5102:
5101:
5099:
5097:
5086:
5080:
5079:
5067:
5057:
5051:
5050:
5042:
4970:Complex dynamics
4953:
4948:
4947:
4946:
4932:
4930:
4929:
4924:
4922:
4921:
4916:
4912:
4911:
4907:
4906:
4898:
4882:
4874:
4860:
4852:
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4789:
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4779:
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4618:
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4452:
4450:
4449:
4444:
4427:
4426:
4414:
4391:
4124:
4122:
4121:
4116:
4078:
4077:
3971:Poincaré section
3776:
3774:
3773:
3768:
3757:
3756:
3741:
3740:
3686:chaotic behavior
3653:
3651:
3650:
3645:
3640:
3639:
3630:
3629:
3611:
3610:
3598:
3597:
3549:
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3541:
3527:
3526:
3511:
3510:
3498:
3497:
3451:
3449:
3448:
3443:
3408:
3407:
3399:
3276:KaplanâYorke map
3226:piecewise linear
3217:Arnold's cat map
3187:
3185:
3184:
3179:
3177:
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2716:
2715:
2704:
2698:
2685:
2683:
2682:
2677:
2672:
2658:
2650:
2649:
2641:
2592:Lebesgue measure
2553:
2551:
2550:
2545:
2519:
2518:
2495:dynamical system
2469:
2467:
2466:
2461:
2438:
2437:
2404:
2402:
2401:
2396:
2385:
2384:
2282:simply connected
2272:of any orbit is
2214:image processing
2106:dynamical system
1988:dynamical system
1966:
1964:
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1568:
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1562:
1512:
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1413:
1411:
1410:
1405:
1373:
1372:
1355:
1353:
1352:
1347:
1315:
1314:
1296:We often write
1251:) is called the
1227:
1225:
1224:
1219:
1184:
1182:
1181:
1176:
1141:
1139:
1138:
1133:
1070:
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1001:
1000:
995:
978:
977:
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930:
928:
927:
922:
908:
907:
895:
894:
864:
863:
845:
844:
821:
819:
818:
813:
761:
759:
758:
753:
751:
750:
745:
723:
721:
720:
715:
698:
697:
692:
664:
662:
661:
656:
579:dynamical system
231:dynamical system
137:parametric curve
127:dependence of a
117:dynamical system
101:Lorenz attractor
88:
81:
77:
74:
68:
63:this article by
54:inline citations
41:
40:
33:
7184:
7183:
7177:
7176:
7175:
7173:
7172:
7171:
7147:
7146:
7145:
7140:
7108:
7102:
7048:Caroline Series
6943:Mary Cartwright
6925:
6919:
6870:Double pendulum
6852:
6842:
6791:
6784:
6710:Exponential map
6661:
6647:
6641:
6599:
6593:
6586:
6572:
6538:Ergodic theorem
6531:
6524:
6515:Stable manifold
6505:Recurrence plot
6421:
6375:
6361:
6356:
6310:Wayback Machine
6297:Wayback Machine
6231:Research groups
6154:
6149:
6143:
6130:
6124:
6108:
6102:
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6078:
6058:
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5684:
5678:
5662:
5656:
5632:
5626:
5606:
5571:
5565:
5552:
5523:
5503:
5496:
5494:Further reading
5483:
5470:
5464:
5448:
5445:
5444:
5414:
5413:
5409:
5399:
5398:
5394:
5373:
5369:
5349:
5345:
5335:
5333:
5318:
5317:
5313:
5306:
5287:
5286:
5282:
5276:Physics Reports
5273:
5269:
5262:
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5241:
5234:
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5216:
5206:
5205:
5201:
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5109:
5105:
5095:
5093:
5088:
5087:
5083:
5076:
5059:
5058:
5054:
5044:
5043:
5039:
5034:
5029:
5015:System dynamics
4949:
4944:
4942:
4939:
4890:
4886:
4866:
4862:
4861:
4831:
4830:
4745:
4740:
4739:
4729:
4695:This branch of
4686:stable manifold
4670:
4664:
4635:involving
4589:
4555:
4547:
4546:
4418:
4376:
4375:
4360:
4354:
4352:Ergodic systems
4323:
4317:
4310:
4296:
4290:
4265:
4249:
4243:
4192:
4168:
4157:
4144:
4135:
4066:
4061:
4060:
4044: + O(
4016:
3986:
3968:
3957:
3948:
3924:singular points
3897:
3872:
3850:
3835:
3821:
3748:
3726:
3721:
3720:
3702:
3631:
3618:
3602:
3589:
3584:
3583:
3578:
3567:
3518:
3502:
3489:
3484:
3483:
3392:
3391:
3373:
3321:
3315:
3310:
3256:Double pendulum
3236:outer billiards
3212:
3145:
3129:
3128:
3088:
3069:
3065:
3005:
3004:
2957:
2923:
2922:
2777:represents the
2750:
2749:
2720:
2699:
2689:
2688:
2634:
2633:
2613:The concept of
2611:
2564:
2510:
2505:
2504:
2426:
2415:
2414:
2373:
2368:
2367:
2354:, Σ). A map Ί:
2302:, the triplet (
2296:
2290:
2234:locally compact
2222:
2206:
2154:
2095:
1986:continuous time
1973:
1945:
1944:
1909:
1908:
1871:
1870:
1827:
1826:
1803:
1802:
1757:
1756:
1753:
1683:
1682:
1628:
1627:
1503:
1498:
1497:
1427:
1422:
1421:
1364:
1359:
1358:
1306:
1301:
1300:
1293:of the system.
1243:The function Ί(
1195:
1194:
1155:
1154:
1073:
1072:
1039:
1014:
1006:
1005:
969:
956:
942:
936:
935:
899:
886:
855:
836:
825:
824:
783:
782:
731:
726:
725:
678:
673:
672:
617:
616:
605:is a non-empty
575:
482:Smale horseshoe
456:ergodic theorem
415:
394:ergodic systems
299:
173:complex numbers
135:, such as in a
89:
78:
72:
69:
59:Please help to
58:
42:
38:
31:
24:
17:
12:
11:
5:
7182:
7181:
7178:
7170:
7169:
7164:
7162:Systems theory
7159:
7149:
7148:
7142:
7141:
7139:
7138:
7133:
7131:Predictability
7128:
7123:
7118:
7112:
7110:
7104:
7103:
7101:
7100:
7098:Lai-Sang Young
7095:
7093:James A. Yorke
7090:
7088:Amie Wilkinson
7085:
7080:
7075:
7070:
7065:
7060:
7055:
7050:
7045:
7040:
7035:
7030:
7025:
7023:Henri Poincaré
7020:
7015:
7010:
7005:
7000:
6995:
6990:
6985:
6980:
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6965:
6960:
6955:
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6935:
6929:
6927:
6921:
6920:
6918:
6917:
6912:
6907:
6902:
6897:
6892:
6890:Kicked rotator
6887:
6882:
6877:
6872:
6867:
6862:
6860:Chua's circuit
6856:
6854:
6848:
6847:
6844:
6843:
6841:
6840:
6835:
6830:
6825:
6820:
6815:
6810:
6805:
6800:
6794:
6792:
6789:
6786:
6785:
6783:
6782:
6780:Zaslavskii map
6777:
6775:Tinkerbell map
6772:
6767:
6762:
6757:
6752:
6747:
6742:
6737:
6732:
6727:
6722:
6717:
6712:
6707:
6706:
6705:
6695:
6690:
6685:
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6670:
6664:
6662:
6659:
6653:
6643:
6642:
6640:
6639:
6634:
6629:
6624:
6622:Ergodic theory
6619:
6614:
6609:
6603:
6601:
6595:
6594:
6579:
6577:
6574:
6573:
6571:
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6452:
6447:
6442:
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6426:
6423:
6422:
6420:
6419:
6414:
6412:Periodic point
6409:
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6189:
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6184:
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6177:
6171:
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6153:
6152:External links
6150:
6148:
6147:
6141:
6128:
6122:
6106:
6100:
6082:
6076:
6052:
6051:
6045:
6032:
6026:
6004:Teschl, Gerald
6000:
5994:
5978:
5972:
5957:
5952:978-0199657032
5951:
5938:
5932:
5919:
5913:
5900:
5894:
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5876:978-3319068190
5875:
5862:
5856:
5843:
5837:
5824:
5818:
5794:
5788:
5773:
5767:
5755:James A. Yorke
5746:
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5587:(6): 747â817.
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5020:Systems theory
5017:
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4772:
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4666:Main article:
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4403:
4400:
4397:
4394:
4390:
4387:
4384:
4358:Ergodic theory
4356:Main article:
4353:
4350:
4321:
4315:
4306:
4294:
4288:
4263:
4245:Main article:
4242:
4239:
4190:
4167:
4164:
4153:
4140:
4133:
4126:
4125:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4076:
4073:
4069:
4036:) =
4014:
3984:
3966:
3955:
3947:
3944:
3896:
3893:
3888:periodic orbit
3876:singular point
3871:
3870:Local dynamics
3868:
3848:
3833:
3819:
3778:
3777:
3766:
3763:
3760:
3755:
3751:
3747:
3744:
3739:
3736:
3733:
3729:
3701:
3698:
3655:
3654:
3643:
3638:
3634:
3628:
3625:
3621:
3617:
3614:
3609:
3605:
3601:
3596:
3592:
3576:
3565:
3551:
3550:
3539:
3536:
3533:
3530:
3525:
3521:
3517:
3514:
3509:
3505:
3501:
3496:
3492:
3453:
3452:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3405:
3402:
3372:
3369:
3359:) +
3317:Main article:
3314:
3311:
3309:
3308:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3258:
3253:
3248:
3243:
3238:
3229:
3219:
3213:
3211:
3208:
3176:
3172:
3166:
3161:
3158:
3155:
3152:
3149:
3143:
3138:
3125:
3124:
3113:
3110:
3106:
3102:
3097:
3092:
3087:
3084:
3081:
3078:
3075:
3072:
3068:
3062:
3056:
3052:
3049:
3046:
3042:
3038:
3035:
3032:
3028:
3024:
3018:
3015:
2983:
2982:
2971:
2966:
2961:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2931:
2916:
2915:
2911:) = 0 for all
2894:
2845:
2844:
2791:
2785:
2763:
2760:
2743:
2742:
2729:
2724:
2719:
2714:
2711:
2708:
2703:
2697:
2686:
2675:
2671:
2667:
2664:
2661:
2657:
2653:
2647:
2644:
2610:
2607:
2563:
2560:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2517:
2513:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2436:
2433:
2429:
2425:
2422:
2413:in ÎŁ, one has
2394:
2391:
2388:
2383:
2380:
2376:
2362:is said to be
2292:Main article:
2289:
2286:
2221:
2218:
2205:
2202:
2153:
2150:
2094:
2091:
1972:
1969:
1954:
1943:in the domain
1918:
1905:diffeomorphism
1886:
1881:
1878:
1836:
1812:
1790:
1787:
1784:
1779:
1774:
1769:
1764:
1752:
1749:
1741:ergodic theory
1705:
1702:
1699:
1696:
1693:
1690:
1671:
1670:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1574:is called the
1572:
1571:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1510:
1506:
1483:is called the
1471:is called the
1469:
1468:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1434:
1430:
1415:
1414:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1371:
1367:
1356:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1313:
1309:
1289:represents an
1217:
1214:
1211:
1208:
1205:
1202:
1174:
1171:
1168:
1165:
1162:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1060:
1057:
1054:
1051:
1046:
1042:
1038:
1035:
1032:
1029:
1026:
1021:
1017:
993:
990:
987:
984:
981:
976:
972:
968:
963:
959:
954:
949:
945:
932:
931:
920:
917:
914:
911:
906:
902:
898:
893:
889:
885:
882:
879:
876:
873:
870:
867:
862:
858:
854:
851:
848:
843:
839:
835:
832:
822:
811:
808:
805:
802:
799:
796:
793:
790:
768:
767:
764:projection map
749:
744:
741:
738:
735:
713:
710:
707:
704:
701:
696:
691:
688:
685:
682:
666:
665:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
574:
571:
559:rocket engines
504:periodic point
468:measure theory
419:Henri Poincaré
414:
411:
410:
409:
389:
378:
366:
298:
295:
212:evolution rule
165:ergodic theory
123:describes the
91:
90:
45:
43:
36:
15:
13:
10:
9:
6:
4:
3:
2:
7180:
7179:
7168:
7165:
7163:
7160:
7158:
7155:
7154:
7152:
7137:
7134:
7132:
7129:
7127:
7126:Edge of chaos
7124:
7122:
7119:
7117:
7114:
7113:
7111:
7105:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7083:Marcelo Viana
7081:
7079:
7076:
7074:
7073:Audrey Terras
7071:
7069:
7068:Floris Takens
7066:
7064:
7061:
7059:
7056:
7054:
7051:
7049:
7046:
7044:
7041:
7039:
7036:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6996:
6994:
6991:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6968:Celso Grebogi
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6948:Chen Guanrong
6946:
6944:
6941:
6939:
6936:
6934:
6933:Michael Berry
6931:
6930:
6928:
6922:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6857:
6855:
6849:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6808:Lorenz system
6806:
6804:
6801:
6799:
6796:
6795:
6793:
6787:
6781:
6778:
6776:
6773:
6771:
6768:
6766:
6763:
6761:
6758:
6756:
6755:Langton's ant
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6730:Horseshoe map
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6711:
6708:
6704:
6701:
6700:
6699:
6696:
6694:
6691:
6689:
6686:
6684:
6681:
6679:
6676:
6674:
6671:
6669:
6666:
6665:
6663:
6657:
6654:
6651:
6644:
6638:
6635:
6633:
6630:
6628:
6627:Quantum chaos
6625:
6623:
6620:
6618:
6615:
6613:
6610:
6608:
6605:
6604:
6602:
6596:
6591:
6587:
6583:
6569:
6566:
6564:
6561:
6559:
6556:
6554:
6551:
6549:
6546:
6544:
6541:
6539:
6536:
6535:
6533:
6527:
6521:
6518:
6516:
6513:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6468:
6466:
6463:
6461:
6458:
6456:
6453:
6451:
6448:
6446:
6443:
6441:
6438:
6436:
6435:Arnold tongue
6433:
6431:
6428:
6427:
6424:
6418:
6415:
6413:
6410:
6408:
6405:
6403:
6400:
6398:
6395:
6393:
6390:
6388:
6385:
6383:
6380:
6379:
6377:
6371:
6368:
6364:
6360:
6353:
6348:
6346:
6341:
6339:
6334:
6333:
6330:
6323:
6320:
6317:
6314:
6311:
6307:
6304:
6301:
6298:
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6256:, Penn State.
6255:
6252:
6249:
6247:
6244:
6241:
6238:
6235:
6234:
6230:
6229:
6226:
6225:Gerald Teschl
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6194:
6191:
6190:
6186:
6185:
6181:
6178:
6175:
6172:
6169:
6165:
6162:
6159:
6156:
6155:
6151:
6144:
6138:
6134:
6129:
6125:
6119:
6115:
6111:
6107:
6103:
6097:
6093:
6092:
6087:
6083:
6079:
6073:
6070:. Princeton.
6069:
6065:
6064:Philip Holmes
6061:
6057:
6056:
6055:
6048:
6042:
6038:
6033:
6029:
6023:
6019:
6015:
6011:
6010:
6005:
6001:
5997:
5991:
5987:
5983:
5979:
5975:
5969:
5965:
5962:
5958:
5954:
5948:
5944:
5939:
5935:
5929:
5925:
5920:
5916:
5910:
5906:
5901:
5897:
5891:
5887:
5882:
5878:
5872:
5868:
5863:
5859:
5853:
5849:
5844:
5840:
5834:
5831:. Cambridge.
5830:
5825:
5821:
5815:
5811:
5807:
5803:
5802:Stephen Smale
5799:
5795:
5791:
5785:
5781:
5778:
5774:
5770:
5764:
5760:
5756:
5751:
5750:
5749:
5742:
5736:
5732:
5728:
5724:
5720:
5716:
5710:
5702:
5696:
5692:
5688:
5683:
5679:
5673:
5669:
5665:
5661:
5657:
5651:
5646:
5645:
5639:
5635:
5631:
5627:
5621:
5616:
5615:
5609:
5605:
5604:
5603:
5595:
5590:
5586:
5582:
5578:
5574:
5573:Stephen Smale
5570:
5566:
5560:
5556:
5551:
5548:
5544:
5540:
5537:
5534:
5533:0-201-40840-6
5530:
5524:
5518:
5514:
5510:
5506:
5505:Ralph Abraham
5502:
5501:
5500:
5493:
5487:
5482:
5479:
5474:
5469:
5465:
5463:3-540-34563-9
5459:
5455:
5451:
5447:
5446:
5438:
5434:
5430:
5426:
5422:
5418:
5411:
5408:
5403:
5396:
5393:
5389:
5385:
5381:
5377:
5371:
5368:
5365:
5361:
5357:
5353:
5347:
5344:
5331:
5327:
5326:
5321:
5315:
5312:
5307:
5305:9783030236922
5301:
5297:
5293:
5292:
5284:
5281:
5277:
5271:
5268:
5263:
5257:
5253:
5249:
5243:
5240:
5235:
5229:
5225:
5218:
5215:
5210:
5203:
5200:
5195:
5191:
5187:
5183:
5179:
5175:
5170:
5165:
5161:
5157:
5156:J. Stat. Phys
5150:
5147:
5142:
5138:
5134:
5130:
5126:
5122:
5119:(3): 033902.
5118:
5114:
5107:
5104:
5091:
5085:
5082:
5077:
5071:
5066:
5065:
5056:
5053:
5048:
5041:
5038:
5031:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4957:
4952:
4941:
4936:
4918:
4913:
4908:
4902:
4899:
4894:
4891:
4887:
4883:
4878:
4875:
4870:
4867:
4863:
4856:
4853:
4848:
4842:
4836:
4829:
4828:
4827:
4810:
4807:
4801:
4795:
4790:
4780:
4767:
4756:
4753:
4749:
4746:
4738:
4737:
4736:
4733:
4726:
4724:
4722:
4721:horseshoe map
4718:
4714:
4709:
4707:
4703:
4698:
4693:
4691:
4687:
4682:
4678:
4677:
4669:
4661:
4659:
4657:
4653:
4651:
4645:
4640:
4638:
4633:
4614:
4605:
4597:
4594:
4583:
4580:
4574:
4565:
4560:
4556:
4545:
4544:
4543:
4541:
4537:
4532:
4528:
4524:
4520:
4515:
4513:
4509:
4505:
4500:
4498:
4495:to object to
4494:
4490:
4486:
4482:
4478:
4474:
4469:
4465:
4463:
4459:
4440:
4431:
4423:
4401:
4395:
4374:
4373:
4372:
4370:
4366:
4359:
4351:
4349:
4347:
4343:
4339:
4335:
4330:
4328:
4324:
4314:
4309:
4305:
4301:
4297:
4287:
4282:
4280:
4276:
4272:
4267:
4262:
4258:
4254:
4248:
4240:
4238:
4236:
4231:
4229:
4225:
4221:
4218: ·
4217:
4213:
4208:
4206:
4202:
4201:
4196:
4189:
4185:
4181:
4177:
4173:
4165:
4163:
4161:
4156:
4152:
4148:
4143:
4139:
4132:
4112:
4109:
4106:
4103:
4100:
4094:
4088:
4085:
4082:
4079:
4074:
4071:
4067:
4059:
4058:
4057:
4055:
4051:
4047:
4043:
4040: ·
4039:
4035:
4031:
4027:
4023:
4018:
4013:
4009:
4005:
4001:
3998: â
3997:
3994: :
3993:
3990:
3983:
3979:
3975:
3972:
3965:
3961:
3954:
3945:
3943:
3941:
3937:
3933:
3929:
3925:
3921:
3916:
3914:
3910:
3906:
3902:
3895:Rectification
3894:
3892:
3889:
3885:
3881:
3877:
3869:
3867:
3865:
3860:
3858:
3855: â
3854:
3847:
3843:
3839:
3832:
3828:
3823:
3818:
3815:
3811:
3807:
3803:
3799:
3795:
3792: â
3791:
3787:
3784:a matrix and
3783:
3764:
3761:
3758:
3753:
3749:
3745:
3742:
3737:
3734:
3731:
3727:
3719:
3718:
3717:
3715:
3711:
3707:
3706:discrete-time
3699:
3693:
3689:
3687:
3683:
3678:
3676:
3672:
3668:
3664:
3660:
3641:
3636:
3632:
3626:
3623:
3619:
3615:
3607:
3603:
3594:
3582:
3581:
3580:
3575:
3571:
3564:
3560:
3556:
3537:
3534:
3531:
3528:
3523:
3519:
3515:
3507:
3503:
3494:
3482:
3481:
3480:
3478:
3474:
3470:
3466:
3462:
3458:
3439:
3436:
3433:
3430:
3427:
3424:
3418:
3412:
3409:
3403:
3400:
3390:
3389:
3388:
3386:
3382:
3378:
3370:
3368:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3338:
3334:
3330:
3326:
3320:
3312:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3286:Lorenz system
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3249:
3247:
3244:
3242:
3239:
3237:
3233:
3230:
3227:
3223:
3220:
3218:
3215:
3214:
3209:
3207:
3205:
3201:
3200:Banach spaces
3196:
3193:
3191:
3164:
3156:
3153:
3150:
3141:
3111:
3108:
3104:
3095:
3085:
3082:
3073:
3070:
3066:
3050:
3047:
3036:
3033:
3022:
3016:
3003:
3002:
3001:
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2838:tangent space
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2349:
2345:
2341:
2340:sigma-algebra
2337:
2333:
2329:
2325:
2321:
2317:
2314:), Ί). Here,
2313:
2309:
2305:
2301:
2300:measure space
2295:
2287:
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2105:
2104:discrete-time
2100:
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2070:
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2058:
2054:
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2046:
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2034:
2030:
2029:diffeomorphic
2026:
2022:
2018:
2015:
2011:
2010:open interval
2007:
2003:
1999:
1995:
1994:
1989:
1987:
1982:
1978:
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1299:
1298:
1297:
1294:
1292:
1291:initial state
1288:
1284:
1280:
1279:
1274:
1270:
1267:, called the
1266:
1262:
1258:
1254:
1250:
1246:
1241:
1239:
1235:
1231:
1230:monoid action
1215:
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560:
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544:
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536:
532:
528:
524:
520:
516:
512:
511:Ali H. Nayfeh
507:
505:
501:
497:
493:
489:
485:
483:
479:
478:Stephen Smale
475:
473:
469:
465:
461:
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320:
316:
312:
308:
304:
296:
294:
292:
291:edge of chaos
288:
284:
283:self-assembly
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
239:
236:
232:
228:
223:
221:
217:
216:deterministic
213:
209:
205:
201:
197:
193:
188:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
133:ambient space
130:
126:
122:
118:
114:
106:
102:
97:
87:
84:
76:
73:February 2022
66:
62:
56:
55:
49:
44:
35:
34:
29:
22:
7078:Mary Tsingou
7043:David Ruelle
7038:Otto Rössler
6983:Michel HĂ©non
6953:Leon O. Chua
6910:Tilt-A-Whirl
6880:FPUT problem
6765:Standard map
6760:Logistic map
6616:
6585:
6359:Chaos theory
6168:Scholarpedia
6132:
6113:
6110:Ivar Ekeland
6090:
6086:James Gleick
6067:
6060:Florin Diacu
6053:
6039:. Springer.
6036:
6008:
5985:
5964:
5961:
5942:
5923:
5904:
5888:. Springer.
5885:
5866:
5850:. Springer.
5847:
5828:
5809:
5782:. Springer.
5780:
5777:
5758:
5747:
5730:
5690:
5686:
5667:
5664:David Ruelle
5643:
5613:
5608:V. I. Arnold
5601:
5584:
5580:
5557:. Springer.
5554:
5538:
5512:
5498:
5485:
5472:
5453:
5420:
5410:
5401:
5395:
5379:
5370:
5355:
5346:
5334:. Retrieved
5323:
5314:
5290:
5283:
5275:
5270:
5251:
5242:
5223:
5217:
5208:
5202:
5159:
5155:
5149:
5116:
5112:
5106:
5094:. Retrieved
5084:
5063:
5055:
5046:
5040:
4825:
4734:
4730:
4717:logistic map
4710:
4702:steady state
4694:
4689:
4685:
4674:
4671:
4668:Chaos theory
4656:SRB measures
4649:
4641:
4636:
4631:
4629:
4535:
4530:
4516:
4511:
4507:
4501:
4488:
4484:
4480:
4476:
4470:
4466:
4455:
4368:
4364:
4361:
4331:
4319:
4312:
4307:
4303:
4292:
4285:
4283:
4278:
4268:
4260:
4253:vector field
4250:
4232:
4227:
4223:
4219:
4215:
4209:
4204:
4198:
4194:
4187:
4183:
4179:
4175:
4171:
4169:
4159:
4154:
4150:
4146:
4141:
4137:
4130:
4127:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4021:
4019:
4011:
4007:
4003:
3999:
3995:
3991:
3989:Poincaré map
3981:
3977:
3973:
3963:
3959:
3952:
3949:
3939:
3935:
3931:
3927:
3919:
3917:
3912:
3908:
3904:
3900:
3898:
3887:
3883:
3879:
3875:
3873:
3861:
3856:
3852:
3845:
3841:
3837:
3830:
3826:
3824:
3816:
3813:
3805:
3801:
3797:
3793:
3789:
3785:
3781:
3779:
3703:
3681:
3679:
3674:
3671:eigenvectors
3666:
3658:
3656:
3573:
3562:
3558:
3557:is zero and
3554:
3552:
3476:
3472:
3468:
3464:
3460:
3456:
3454:
3380:
3374:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3328:
3324:
3322:
3197:
3194:
3126:
2995:
2990:
2986:
2984:
2917:
2912:
2908:
2904:
2900:
2896:
2890:
2886:
2882:
2878:
2874:
2870:
2865:
2860:
2856:
2852:
2848:
2846:
2840:
2833:
2829:
2817:
2813:
2810:vector field
2805:
2801:
2797:
2793:
2787:
2782:
2744:
2614:
2612:
2596:
2573:
2565:
2555:
2499:
2494:
2490:
2486:
2482:
2478:
2473:
2471:
2410:
2406:
2364:ÎŁ-measurable
2359:
2355:
2351:
2343:
2331:
2327:
2319:
2315:
2311:
2307:
2303:
2297:
2267:
2262:
2258:
2254:
2250:
2243:
2229:
2225:
2223:
2207:
2197:
2193:
2185:
2182:integer grid
2176:such as the
2169:
2165:
2161:
2160:is a tuple (
2157:
2155:
2146:semi-cascade
2145:
2141:
2137:
2133:
2129:
2126:Banach space
2117:
2116:, Ί), where
2113:
2109:
2108:is a tuple (
2102:
2098:
2096:
2086:
2082:
2076:
2072:
2068:
2060:
2056:
2052:
2048:
2044:
2033:Banach space
2020:
2016:
2014:real numbers
2005:
2001:
1997:
1996:is a tuple (
1991:
1984:
1980:
1976:
1974:
1940:
1936:
1932:
1900:
1899:) such that
1866:
1862:
1858:
1754:
1734:
1729:
1725:
1721:
1717:
1678:
1674:
1672:
1621:
1617:
1613:
1609:
1605:
1604:is called Ί-
1601:
1597:
1593:
1585:
1581:
1575:
1573:
1490:
1484:
1476:
1472:
1470:
1416:
1295:
1290:
1286:
1282:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1242:
1237:
1233:
1190:
1186:
1152:
1147:
1143:
933:
775:
771:
770:and for any
769:
667:
602:
594:
590:
586:
578:
576:
508:
486:
476:
449:
435:
428:
416:
346:
339:
332:
326:
322:
318:
300:
275:logistic map
271:chaos theory
240:
230:
224:
211:
204:real numbers
189:
179:or simply a
116:
110:
79:
70:
51:
7063:Nina Snaith
7053:Yakov Sinai
6938:Rufus Bowen
6688:Duffing map
6673:Baker's map
6598:Theoretical
6510:SRB measure
6417:Phase space
6387:Bifurcation
6243:Chaos @ UMD
6135:. Penguin.
6094:. Penguin.
5634:Jacob Palis
5404:. Springer.
5211:. Springer.
5096:17 February
4995:Oscillation
4713:Meteorology
4697:mathematics
4487:returns to
4300:eigenvalues
4271:fixed point
4257:phase space
3663:eigenvalues
3296:Rössler map
3222:Baker's map
2897:homogeneous
2168:, Ί), with
2035:, and Ί a
1747:in flavor.
1608:if for all
1596:. A subset
1283:state space
1278:phase space
1259:in the set
762:is the 2nd
609:and Ί is a
593:, Ί) where
555:jet engines
551:skyscrapers
523:engineering
363:equivalence
255:engineering
196:state space
113:mathematics
65:introducing
7151:Categories
7121:Complexity
7018:Edward Ott
6865:Convection
6790:Continuous
6465:Ergodicity
6262:, Caltech.
6166:A part of
6014:Providence
5748:Textbooks
5162:(3): 617.
5049:. Perseus.
5032:References
4706:attractors
4514:)/vol(Ω).
4200:hyperbolic
4197:is called
3932:integrable
3459:a matrix,
3246:Circle map
3190:functional
2871:autonomous
2232:, Ί) on a
2039:. If Ί is
2004:, Ί) with
1493:. The set
1486:trajectory
1275:is called
567:spacecraft
531:structures
519:mechanical
490:developed
328:trajectory
315:time scale
277:dynamics,
220:stochastic
48:references
7033:Mary Rees
6993:Bryna Kra
6926:theorists
6735:Ikeda map
6725:HĂ©non map
6715:Gauss map
6397:Limit set
6382:Attractor
5709:cite book
5547:0938-0396
5336:25 August
5250:(2009) .
5169:0705.0311
4895:−
4871:−
4757:−
4595:−
4591:Φ
4497:Boltzmann
4420:Φ
4107:⋅
4086:∘
4080:∘
4072:−
3661:= 0, the
3591:Φ
3491:Φ
3404:˙
3266:HĂ©non map
3232:Billiards
3171:→
3154:×
3077:Φ
3055:⇔
3023:−
3017:˙
2946:Φ
2762:˙
2646:˙
2588:attractor
2542:Φ
2539:∘
2536:⋯
2533:∘
2530:Φ
2527:∘
2524:Φ
2512:Φ
2455:σ
2449:μ
2440:σ
2432:−
2428:Φ
2421:μ
2393:Σ
2390:∈
2387:σ
2379:−
2375:Φ
2274:non-empty
2270:limit set
2238:Hausdorff
2087:semi-flow
2081:; and if
1880:∈
1789:⟩
1763:⟨
1679:invariant
1652:∈
1634:Φ
1606:invariant
1544:∈
1520:Φ
1514:≡
1505:γ
1453:→
1429:Φ
1387:Φ
1384:≡
1366:Φ
1329:Φ
1326:≡
1308:Φ
1170:×
1124:∈
1100:∈
1034:Φ
1025:∈
980:∈
881:Φ
850:Φ
831:Φ
789:Φ
650:→
641:×
632:⊆
623:Φ
547:buildings
500:real line
436:In 1913,
342:computers
313:or other
293:concept.
259:economics
251:chemistry
7109:articles
6851:Physical
6770:Tent map
6660:Discrete
6600:branches
6530:Theorems
6366:Concepts
6306:Archived
6293:Archived
6112:(1990).
6088:(1988).
6066:(1996).
6006:(2012).
5984:(1994).
5926:. SIAM.
5808:(2003).
5757:(2000).
5729:(1992).
5689:(1991).
5666:(1989).
5640:(1982).
5610:(1982).
5575:(1967).
5511:(1978).
5437:45426376
5296:Springer
5141:16252993
5090:"Nature"
4937:See also
4750:′
4205:elliptic
3383:) is an
3306:Tent map
3210:Examples
2822:velocity
2779:velocity
2502:iterates
2178:integers
2122:manifold
2027:locally
2025:manifold
1851:manifold
1716:for all
1616:and all
1580:through
1489:through
1479:and its
1475:through
1142:for any
611:function
563:aircraft
527:machines
513:applied
297:Overview
267:medicine
206:or by a
177:manifold
145:pendulum
121:function
7107:Related
6915:Weather
6853:systems
6646:Chaotic
6392:Fractal
5194:8677631
5174:Bibcode
5121:Bibcode
4523:Koopman
4510:is vol(
4493:Zermelo
4456:In the
3980:,
3851:, with
2873:, when
2745:where
2348:measure
2334:) is a
2326:, and (
2278:compact
2265:, Ί*).
2174:lattice
2134:cascade
2067:. When
2012:in the
1743:and is
1588:is the
724:(where
543:bridges
462:on the
460:physics
413:History
404:and of
263:history
247:biology
227:physics
61:improve
7013:Hee Oh
6648:maps (
6495:Mixing
6139:
6120:
6098:
6074:
6043:
6024:
5992:
5970:
5949:
5930:
5911:
5892:
5873:
5854:
5835:
5816:
5786:
5765:
5737:
5697:
5674:
5652:
5622:
5561:
5545:
5531:
5519:
5460:
5435:
5386:
5362:
5302:
5258:
5230:
5192:
5139:
5072:
4538:, the
4178:. As
3844:
3710:affine
3385:affine
3375:For a
3343:) and
3127:where
2993:, Ί).
2826:forces
2073:global
1869:(with
1012:
599:monoid
539:cranes
502:has a
281:, the
265:, and
208:vector
185:smooth
155:, and
151:, the
131:in an
50:, but
6924:Chaos
6703:outer
6407:Orbit
5433:S2CID
5190:S2CID
5164:arXiv
4676:chaos
4275:torus
3780:with
3657:When
3553:When
3455:with
3371:Flows
3335:: if
3188:is a
2899:when
2808:is a
2489:, ÎŁ,
2330:, ÎŁ,
2322:is a
2310:, ÎŁ,
2140:. If
2136:or a
2120:is a
2075:or a
2031:to a
1990:, or
1903:is a
1855:graph
1849:is a
1677:is Ί-
1590:image
1577:orbit
1481:graph
1193:that
668:with
597:is a
583:tuple
581:is a
535:ships
466:with
406:chaos
334:orbit
200:tuple
192:state
129:point
6650:list
6374:Core
6137:ISBN
6118:ISBN
6096:ISBN
6072:ISBN
6062:and
6041:ISBN
6022:ISBN
5990:ISBN
5968:ISBN
5947:ISBN
5928:ISBN
5909:ISBN
5890:ISBN
5871:ISBN
5852:ISBN
5833:ISBN
5814:ISBN
5804:and
5784:ISBN
5763:ISBN
5735:ISBN
5725:and
5715:link
5695:ISBN
5687:eds.
5672:ISBN
5650:ISBN
5636:and
5620:ISBN
5559:ISBN
5543:ISSN
5529:ISBN
5517:ISBN
5507:and
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