772:
published in
Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 . For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanician
110:
43:
4979:
4588:
776:
working at the Kazan
Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct
7421:
Malkin I.G. Theory of
Stability of Motion, Moscow 1952 (Gostekhizdat) Chap II para 4 (Russian) Engl. transl, Language Service Bureau, Washington AEC -tr-3352; originally On stability under constantly acting disturbances Prikl Mat 1944, vol. 8 no.3 241-245 (Russian); Amer. Math. Soc. transl. no.
771:
at
Kharkov University in 1892. A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially
4740:
2858:
7089:
Chetaev, N. G. On stable trajectories of dynamics, Kazan Univ Sci Notes, vol.4 no.1 1936; The
Stability of Motion, Originally published in Russian in 1946 by ОГИЗ. Гос. изд-во технико-теорет. лит., Москва-Ленинград.Translated by Morton Nadler, Oxford, 1961, 200
4064:
7193:
4358:
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3124:. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.
4198:
2448:. The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a
1708:
4223:. It has been shown that near to a point of equilibrium which is Lyapunov stable the system remains stable under small disturbances. For larger input disturbances the study of such systems is the subject of
2360:
6222:
4974:{\displaystyle {\dot {V}}=x_{1}{\dot {x}}_{1}+x_{2}{\dot {x}}_{2}=x_{1}x_{2}-x_{1}x_{2}+\varepsilon {\frac {x_{2}^{4}}{3}}-\varepsilon {x_{2}^{2}}=\varepsilon {\frac {x_{2}^{4}}{3}}-\varepsilon {x_{2}^{2}}.}
4725:
953:
788:
which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature. More recently the concept of the
2695:
3119:
of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the
1490:
1026:
2563:
5541:
1553:
3847:
895:
747:
guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of
Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as
4441:
3592:
3112:
is locally stable.) An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability. Global asymptotic stability (GAS) follows similarly.
1878:
6048:
1991:
6852:
5189:
Instead, Barbalat's lemma allows for
Lyapunov-like analysis of these non-autonomous systems. The lemma is motivated by the following observations. Assuming f is a function of time only:
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1607:
1420:
1253:
7650:
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2017:
4255:
This example shows a system where a
Lyapunov function can be used to prove Lyapunov stability but cannot show asymptotic stability. Consider the following equation, based on the
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5235:
2928:
4646:
3937:
3730:
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5608:
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5643:
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1173:
1050:
981:
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2206:
1927:
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systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.
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2689:
1762:
1734:
1279:
7003:
3001:
2601:
7913:
5910:
5123:
5096:
1097:
729:
702:
675:
648:
617:
590:
563:
6494:
6148:
5710:
5573:
5394:
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2049:
8040:
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if it is both attractive and stable. (There are examples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using
1070:
4583:{\displaystyle {\begin{aligned}&{\dot {x}}_{1}=x_{2},\\&{\dot {x}}_{2}=-x_{1}+\varepsilon \left({\frac {x_{2}^{3}}{3}}-{x_{2}}\right).\end{aligned}}}
7643:
3192:
7700:
5133:
It may be difficult to find a
Lyapunov function with a negative definite derivative as required by the Lyapunov stability criterion, however a function
4139:
64:
51:
7813:
7467:
B. Farkas et al., Variations on Barbălat's Lemma, Amer. Math. Monthly (2016) 128, no. 8, 825-830, DOI: 10.4169/amer.math.monthly.123.8.825, p. 826.
7458:
B. Farkas et al., Variations on Barbălat's Lemma, Amer. Math. Monthly (2016) 128, no. 8, 825-830, DOI: 10.4169/amer.math.monthly.123.8.825, p. 827.
7248:
Smith, M. J.; Wisten, M. B. (1995). "A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium".
505:
4231:. For systems with inputs, one must quantify the effect of inputs on the stability of the system. The main two approaches to this analysis are
7857:
7636:
7080:, (A. T. Fuller trans.) Taylor & Francis, London 1992. Included is a biography by Smirnov and an extensive bibliography of Lyapunov's work.
1810:
Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
534:. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of
1813:
Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate
7751:
7605:
7580:
7518:
7490:
7340:
7305:
8035:
7365:
2208:. However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations". Define
1612:
277:
2255:
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6323:
3115:
It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the
346:
6173:
4654:
2853:{\displaystyle {\dot {V}}(x)={\frac {d}{dt}}V(x)=\sum _{i=1}^{n}{\frac {\partial V}{\partial x_{i}}}f_{i}(x)=\nabla V\cdot f(x)\leq 0}
904:
7746:
7710:
82:
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5183:
3879:
This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices
1425:
986:
7808:
7720:
7449:
I. Barbălat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl. 4 (1959) 267–270, p. 269.
2525:
7776:
2107:
806:
764:
8091:
7847:
777:
successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.
7690:
7123:; Bertram, J. F (1960). "Control System Analysis and Design Via the "Second Method" of Lyapunov: I—Continuous-Time Systems".
5469:
1502:
167:
7862:
5186:
can be applied to prove asymptotic stability, but this theorem is not applicable when the dynamics are a function of time.
3795:
3127:
Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a
815:
784:
period when the so-called "Second Method of
Lyapunov" (see below) was found to be applicable to the stability of aerospace
8167:
7705:
7695:
7572:
773:
7029:
3553:
7715:
7685:
2445:
1744:
Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance
7039:
7024:
1816:
7950:
7945:
7820:
7781:
498:
431:
797:. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.
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1940:
8172:
8162:
4731:
956:
7990:
6792:
8157:
8126:
8096:
8071:
2114:(i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.
1284:
5295:
3393:
7985:
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6548:
6331:
4240:
3694:
2452:
which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system
1558:
1371:
1204:
752:
426:
341:
7995:
6391:
4369:
7965:
7955:
7852:
7366:"The inadequacy of the method of characteristic exponents for the study of nonlinear differential equations"
4076:
3882:
3735:
297:
5948:
793:(related to Lyapunov's First Method of discussing stability) has received wide interest in connection with
781:
4256:
4059:{\displaystyle {{\textbf {x}}_{t+1}}=A_{i_{t}}{\textbf {x}}_{t},\quad A_{i_{t}}\in \{A_{1},\dots ,A_{m}\}}
3786:
3544:
1996:
491:
214:
7120:
3652:
8111:
8106:
8000:
7924:
7903:
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7756:
7680:
7659:
7034:
5799:
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2889:
743:
523:
399:
234:
142:
31:
7622:
4596:
3700:
5877:
5578:
30:
This article is about asymptotic stability of nonlinear systems. For stability of linear systems, see
8005:
7929:
7898:
7324:(In-)Stability of Differential Inclusions: Notions, Equivalences, and Lyapunov-like Characterizations
7202:
6921:
5613:
5007:
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748:
272:
229:
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147:
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7044:
6604:
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527:
314:
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2211:
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1178:
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7066:
5770:
535:
262:
6757:
6708:
5741:
5156:
5045:
4353:{\displaystyle {\ddot {y}}+y-\varepsilon \left({\frac {{\dot {y}}^{3}}{3}}-{\dot {y}}\right)=0.}
2125:
1102:
6253:
6053:
2633:
1790:
1767:
7601:
7576:
7514:
7486:
7336:
7301:
7230:
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3128:
2965:
2054:
790:
302:
239:
118:
2933:
2863:
2668:
1747:
1713:
1258:
751:, which concerns the behavior of different but "nearby" solutions to differential equations.
8121:
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2100:
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472:
421:
186:
130:
5101:
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1075:
707:
680:
653:
626:
595:
568:
541:
8081:
8010:
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6124:
5686:
5549:
5370:
5240:
3380:
2092:
2025:
785:
477:
382:
292:
267:
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2606:
2499:
2369:
7206:
8076:
7883:
6901:
6881:
6861:
6737:
6688:
6668:
6648:
6450:
6153:
5136:
4232:
4224:
3873:
3855:
3629:
3601:
2111:
1055:
449:
365:
359:
282:
207:
201:
196:
7591:
7225:
7188:
8151:
7562:
7558:
7350:
7283:
4236:
3140:
2366:
This is no longer an autonomous system, but it has a guaranteed equilibrium point at
2122:
Instead of considering stability only near an equilibrium point (a constant solution
454:
287:
244:
17:
7408:
7269:
3361:{\displaystyle \forall \epsilon >0\ \exists \delta >0\ \forall y\in X\ \left.}
109:
8131:
8066:
7980:
3852:
is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of
3155:
794:
172:
162:
157:
7072:(In Russian), Doctoral dissertation, Univ. Kharkov 1892 English translations: (1)
27:
Property of a dynamical system where solutions near an equilibrium point remain so
2164:), one can formulate similar definitions of stability near an arbitrary solution
7888:
336:
5767:
is uniformly continuous (a sufficient condition for uniform continuity is that
4193:{\displaystyle {\dot {\textbf {x}}}={\textbf {f}}({\textbf {x}},{\textbf {u}})}
8116:
7786:
7618:
7332:
7297:
3623:
416:
372:
331:
7541:
5182:
that is only negative semi-definite may be available. In autonomous systems,
538:. In simple terms, if the solutions that start out near an equilibrium point
7878:
7322:
7287:
3121:
7234:
7215:
7918:
7628:
6117:
In the following form the Lemma is true also in the vector valued case:
5125:
axis. The equilibrium is Lyapunov stable but not asymptotically stable.
2392:
whose stability is equivalent to the stability of the original solution
7528:
Parks, P. C. (1992). "A. M. Lyapunov's stability theory—100 years on".
7261:
7172:
Parks, P. C. (1962). "Liapunov's method in automatic control theory".
7136:
6315:
The following example is taken from page 125 of Slotine and Li's book
5610:) implies it converges to a limit. But it does not say whether or not
3116:
7502:
Lyapunov Stability and Feedback Control of Two-Stream Plasma Systems
4648:
is the only equilibrium point. Let us choose as a Lyapunov function
1703:{\displaystyle \|x(t)-x_{e}\|\leq \alpha \|x(0)-x_{e}\|e^{-\beta t}}
7400:
6685:
are bounded. But it does not say anything about the convergence of
4203:
where the (generally time-dependent) input u(t) may be viewed as a
2355:{\displaystyle {\dot {y}}=f(t,y+\phi (t))-{\dot {\phi }}(t)=g(t,y)}
7189:"Lyapunov functions for the problem of Lur'e in automatic control"
3646:
are negative. This condition is equivalent to the following one:
137:
7617:
This article incorporates material from asymptotically stable on
6217:{\displaystyle \textstyle \int _{0}^{t}f(\tau )\mathrm {d} \tau }
6150:
be a uniformly continuous function with values in a Banach space
1740:
Conceptually, the meanings of the above terms are the following:
4720:{\displaystyle V={\frac {1}{2}}\left(x_{1}^{2}+x_{2}^{2}\right)}
4069:
is asymptotically stable (in fact, exponentially stable) if the
948:{\displaystyle x(t)\in {\mathcal {D}}\subseteq \mathbb {R} ^{n}}
7632:
7387:
Goh, B. S. (1977). "Global stability in many-species systems".
2968:
and the system is stable in the sense of Lyapunov. (Note that
36:
7593:
Introduction to Applied Nonlinear Dynamical Systems and Chaos
7321:
Braun, Philipp; Grune, Lars; Kellett, Christopher M. (2021).
1485:{\displaystyle \lim _{t\rightarrow \infty }\|x(t)-x_{e}\|=0}
1037:
1021:{\displaystyle f:{\mathcal {D}}\rightarrow \mathbb {R} ^{n}}
998:
968:
925:
5128:
2558:{\displaystyle V:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
2110:
of the dynamical system at an equilibrium happens to be a
1764:
from it) remain "close enough" forever (within a distance
650:
is Lyapunov stable and all solutions that start out near
7159:
Stability by Lyapunov's Second Method with Applications
5536:{\displaystyle f(t)=\sin \left(t^{2}\right)/t,\;t>0}
3618:
is a finite matrix, is asymptotically stable (in fact,
1548:{\displaystyle \alpha >0,~\beta >0,~\delta >0}
755:(ISS) applies Lyapunov notions to systems with inputs.
60:
7513:(Third ed.). Berlin: Springer. pp. 407–428.
6177:
5129:
Barbalat's lemma and stability of time-varying systems
3842:{\displaystyle {\textbf {x}}_{t+1}=A{\textbf {x}}_{t}}
7564:
Ordinary Differential Equations and Dynamical Systems
7530:
IMA Journal of Mathematical Control & Information
6985:
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2671:
2636:
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2170:
2128:
2057:
2028:
1999:
1943:
1891:
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1287:
1261:
1207:
1181:
1155:
1105:
1078:
1058:
1034:
989:
965:
907:
890:{\displaystyle {\dot {x}}=f(x(t)),\;\;\;\;x(0)=x_{0}}
818:
710:
683:
656:
629:
598:
571:
544:
8059:
8028:
7938:
7871:
7840:
7833:
7795:
7734:
7673:
7666:
3587:{\displaystyle {\dot {\textbf {x}}}=A{\textbf {x}}}
780:The interest in it suddenly skyrocketed during the
7076:, Academic Press, New-York & London, 1966 (2)
6997:
6971:
6945:
6910:
6890:
6870:
6846:
6775:
6746:
6726:
6697:
6677:
6657:
6637:
6591:
6537:
6488:
6459:
6436:
6379:
6303:
6277:
6242:
6216:
6162:
6142:
6106:
6077:
6042:
5985:
5937:
5904:
5858:
5832:
5788:
5759:
5730:
5704:
5663:
5637:
5602:
5567:
5535:
5458:
5414:
5388:
5356:
5284:
5258:
5229:
5174:
5145:
5117:
5090:
5063:
5034:
4996:
4973:
4719:
4640:
4582:
4419:
4352:
4192:
4117:
4058:
3923:
3864:
3841:
3774:
3724:
3682:
3638:
3610:
3586:
3527:
3360:
3104:
3060:
2995:
2948:
2922:
2878:
2852:
2683:
2657:
2621:
2595:
2557:
2514:
2488:
2444:Lyapunov, in his original 1892 work, proposed two
2428:
2384:
2354:
2241:
2200:
2156:
2072:
2043:
2011:
1985:
1921:
1872:
1799:
1776:
1756:
1728:
1702:
1601:
1547:
1495:The equilibrium of the above system is said to be
1484:
1414:
1360:
1338:The equilibrium of the above system is said to be
1327:
1273:
1247:
1193:
1167:
1133:
1091:
1064:
1044:
1020:
975:
947:
889:
767:, a Russian mathematician who defended the thesis
723:
696:
669:
642:
611:
584:
557:
8046:List of nonlinear ordinary differential equations
1873:{\displaystyle \alpha \|x(0)-x_{e}\|e^{-\beta t}}
8051:List of nonlinear partial differential equations
7623:Creative Commons Attribution/Share-Alike License
4133:A system with inputs (or controls) has the form
3442:
1430:
7481:Bhatia, Nam Parshad; Szegő, Giorgio P. (2002).
3131:can be found to satisfy the above constraints.
1499:if it is asymptotically stable and there exist
8041:List of linear ordinary differential equations
7078:The General Problem of the Stability of Motion
7070:The General Problem of the Stability of Motion
6467:is a function of time. Assume that the input
6043:{\displaystyle {\dot {f}}\in L^{q}(0,\infty )}
1986:{\displaystyle \|x(t)-\phi (t)\|\rightarrow 0}
7644:
7549:Slotine, Jean-Jacques E.; Weiping Li (1991).
7435:Slotine, Jean-Jacques E.; Weiping Li (1991).
2084:if this property holds for all trajectories.
499:
8:
6847:{\displaystyle {\ddot {V}}=-4e(-e+g\cdot w)}
4112:
4080:
4053:
4021:
3918:
3886:
1974:
1944:
1851:
1823:
1681:
1653:
1644:
1616:
1590:
1562:
1473:
1445:
1403:
1375:
1316:
1288:
1236:
1208:
7102:Устойчивость нелинейных регулируемых систем
2439:
1328:{\displaystyle \|x(t)-x_{e}\|<\epsilon }
7837:
7670:
7651:
7637:
7629:
7430:
7428:
7327:. Springer. pp. 19–20, Example 2.18.
7292:. Springer. pp. 191–194, Section 40.
7062:
7060:
5523:
5357:{\displaystyle f(t)=\sin(\ln(t)),\;t>0}
5344:
3528:{\displaystyle \exists \delta >0\left.}
1784:from it). Note that this must be true for
1342:if it is Lyapunov stable and there exists
861:
860:
859:
858:
769:The General Problem of Stability of Motion
506:
492:
94:
7224:
7214:
7108:] (in Russian). Moscow: Gostekhizdat.
6984:
6958:
6926:
6925:
6923:
6903:
6883:
6863:
6797:
6796:
6794:
6783:=0) and the dynamics are non-autonomous.
6762:
6761:
6759:
6739:
6713:
6712:
6710:
6690:
6670:
6650:
6606:
6592:{\displaystyle {\dot {V}}=-2e^{2}\leq 0.}
6577:
6553:
6552:
6550:
6529:
6516:
6504:
6472:
6452:
6447:This is non-autonomous because the input
6396:
6395:
6393:
6380:{\displaystyle {\dot {e}}=-e+g\cdot w(t)}
6336:
6335:
6333:
6290:
6255:
6229:
6205:
6187:
6182:
6175:
6155:
6126:
6090:
6055:
6019:
6001:
6000:
5998:
5962:
5950:
5912:
5879:
5845:
5804:
5803:
5801:
5775:
5774:
5772:
5746:
5745:
5743:
5717:
5688:
5650:
5618:
5617:
5615:
5583:
5582:
5580:
5551:
5512:
5502:
5471:
5430:
5429:
5427:
5401:
5372:
5297:
5271:
5242:
5201:
5200:
5198:
5161:
5160:
5158:
5138:
5109:
5103:
5082:
5076:
5050:
5049:
5047:
5020:
5015:
5009:
5004:is positive, stability is asymptotic for
4989:
4961:
4956:
4951:
4934:
4929:
4923:
4910:
4905:
4900:
4883:
4878:
4872:
4860:
4850:
4837:
4827:
4814:
4803:
4802:
4795:
4782:
4771:
4770:
4763:
4745:
4744:
4742:
4706:
4701:
4688:
4683:
4664:
4656:
4626:
4604:
4598:
4561:
4556:
4542:
4537:
4531:
4514:
4498:
4487:
4486:
4471:
4458:
4447:
4446:
4440:
4438:
4406:
4405:
4396:
4377:
4371:
4328:
4327:
4313:
4302:
4301:
4298:
4270:
4269:
4267:
4259:equation with the friction term changed:
4181:
4180:
4171:
4170:
4161:
4160:
4146:
4144:
4143:
4141:
4106:
4087:
4078:
4047:
4028:
4010:
4005:
3991:
3985:
3984:
3975:
3970:
3950:
3944:
3943:
3941:
3939:
3912:
3893:
3884:
3857:
3833:
3827:
3826:
3807:
3801:
3800:
3797:
3760:
3759:
3758:
3737:
3716:
3715:
3714:
3702:
3662:
3661:
3660:
3654:
3631:
3603:
3578:
3577:
3560:
3558:
3557:
3555:
3491:
3469:
3445:
3395:
3324:
3302:
3282:
3194:
3076:
3075:
3073:
3050:
3042:
3028:
3008:
2973:
2935:
2894:
2893:
2891:
2865:
2805:
2792:
2774:
2768:
2757:
2723:
2700:
2699:
2697:
2670:
2635:
2608:
2573:
2551:
2550:
2541:
2537:
2536:
2527:
2501:
2460:
2459:
2457:
2397:
2371:
2311:
2310:
2260:
2259:
2257:
2213:
2169:
2148:
2127:
2056:
2027:
1998:
1942:
1890:
1858:
1845:
1818:
1792:
1769:
1749:
1715:
1688:
1675:
1638:
1614:
1602:{\displaystyle \|x(0)-x_{e}\|<\delta }
1584:
1560:
1504:
1467:
1433:
1427:
1415:{\displaystyle \|x(0)-x_{e}\|<\delta }
1397:
1373:
1347:
1310:
1286:
1260:
1248:{\displaystyle \|x(0)-x_{e}\|<\delta }
1230:
1206:
1180:
1154:
1116:
1104:
1083:
1077:
1057:
1036:
1035:
1033:
1012:
1008:
1007:
997:
996:
988:
967:
966:
964:
939:
935:
934:
924:
923:
906:
881:
820:
819:
817:
715:
709:
688:
682:
661:
655:
634:
628:
603:
597:
576:
570:
549:
543:
83:Learn how and when to remove this message
7005:. This proves that the error converges.
6437:{\displaystyle {\dot {g}}=-e\cdot w(t).}
4420:{\displaystyle x_{1}=y,x_{2}={\dot {y}}}
3785:Correspondingly, a time-discrete linear
7056:
3539:Stability for linear state space models
983:an open set containing the origin, and
397:
312:
116:
102:
7106:Stability of Nonlinear Control Systems
5870:An alternative version is as follows:
4118:{\displaystyle \{A_{1},\dots ,A_{m}\}}
3924:{\displaystyle \{A_{1},\dots ,A_{m}\}}
3775:{\displaystyle V(x)=x^{\textsf {T}}Mx}
2440:Lyapunov's second method for stability
801:Definition for continuous-time systems
522:may be discussed for the solutions of
7483:Stability theory of dynamical systems
6734:is only negative semi-definite (note
5986:{\displaystyle f\in L^{p}(0,\infty )}
3732:. (The relevant Lyapunov function is
3379:if it belongs to the interior of its
2249:, obeying the differential equation:
7:
8036:List of named differential equations
4430:so that the corresponding system is
3135:Definition for discrete-time systems
2012:{\displaystyle t\rightarrow \infty }
7961:Method of undetermined coefficients
7742:Dependent and independent variables
4182:
4172:
4162:
4147:
3986:
3945:
3828:
3802:
3683:{\displaystyle A^{\textsf {T}}M+MA}
3579:
3561:
3003:is required; otherwise for example
2446:methods for demonstrating stability
6966:
6645:by first two conditions and hence
6298:
6237:
6206:
6098:
6034:
5977:
5929:
5896:
5853:
5833:{\displaystyle {\dot {f}}(t)\to 0}
5725:
5658:
5459:{\displaystyle {\dot {f}}(t)\to 0}
5409:
5279:
5230:{\displaystyle {\dot {f}}(t)\to 0}
5098:, and will be 0 everywhere on the
3452:
3397:
3273:
3226:
3211:
3196:
2923:{\displaystyle {\dot {V}}(x)<0}
2886:. Note: for asymptotic stability,
2823:
2785:
2777:
2006:
1440:
763:Lyapunov stability is named after
25:
4641:{\displaystyle x_{1}=0,\ x_{2}=0}
4129:Stability for systems with inputs
3725:{\displaystyle M=M^{\textsf {T}}}
2496:having a point of equilibrium at
278:Kepler's laws of planetary motion
7858:Carathéodory's existence theorem
5938:{\displaystyle q\in (1,\infty ]}
5905:{\displaystyle p\in [1,\infty )}
5603:{\displaystyle {\dot {f}}\leq 0}
3283:
1028:is a continuous vector field on
108:
41:
6946:{\displaystyle {\dot {V}}\to 0}
5638:{\displaystyle {\dot {f}}\to 0}
5035:{\displaystyle x_{2}^{2}<3.}
4984:It seems that if the parameter
4000:
3105:{\displaystyle {\dot {x}}(t)=x}
2489:{\displaystyle {\dot {x}}=f(x)}
2091:belongs to the interior of its
1145:This equilibrium is said to be
765:Aleksandr Mikhailovich Lyapunov
7621:, which is licensed under the
7020:LaSalle's invariance principle
6989:
6963:
6937:
6841:
6820:
6632:
6626:
6617:
6611:
6483:
6477:
6428:
6422:
6374:
6368:
6295:
6269:
6266:
6260:
6234:
6202:
6196:
6137:
6131:
6095:
6069:
6066:
6060:
6037:
6025:
5980:
5968:
5932:
5920:
5899:
5887:
5850:
5824:
5821:
5815:
5722:
5699:
5693:
5655:
5629:
5575:lower bounded and decreasing (
5562:
5556:
5482:
5476:
5450:
5447:
5441:
5406:
5383:
5377:
5338:
5335:
5329:
5320:
5308:
5302:
5276:
5253:
5247:
5221:
5218:
5212:
4187:
4167:
3748:
3742:
3693:is negative definite for some
3503:
3497:
3481:
3475:
3449:
3438:
3429:
3417:
3336:
3330:
3314:
3308:
3270:
3261:
3249:
3093:
3087:
3061:{\displaystyle V(x)=1/(1+|x|)}
3055:
3051:
3043:
3033:
3019:
3013:
2984:
2978:
2911:
2905:
2841:
2835:
2817:
2811:
2747:
2741:
2717:
2711:
2646:
2640:
2584:
2578:
2547:
2483:
2477:
2423:
2417:
2408:
2402:
2349:
2337:
2328:
2322:
2304:
2301:
2295:
2277:
2236:
2230:
2195:
2189:
2180:
2174:
2138:
2132:
2067:
2061:
2038:
2032:
2003:
1977:
1971:
1965:
1956:
1950:
1916:
1910:
1901:
1895:
1835:
1829:
1665:
1659:
1628:
1622:
1574:
1568:
1457:
1451:
1437:
1387:
1381:
1300:
1294:
1220:
1214:
1168:{\displaystyle \epsilon >0}
1122:
1109:
1045:{\displaystyle {\mathcal {D}}}
1003:
976:{\displaystyle {\mathcal {D}}}
917:
911:
871:
865:
852:
849:
843:
837:
1:
7573:American Mathematical Society
7250:Annals of Operations Research
6638:{\displaystyle V(t)\leq V(0)}
6538:{\displaystyle V=e^{2}+g^{2}}
6107:{\displaystyle t\to \infty .}
2429:{\displaystyle x(t)=\phi (t)}
2201:{\displaystyle x(t)=\phi (t)}
1922:{\displaystyle x(t)=\phi (t)}
7686:Notation for differentiation
7509:Gandolfo, Giancarlo (1996).
7125:Journal of Basic Engineering
6972:{\displaystyle t\to \infty }
6304:{\displaystyle t\to \infty }
6243:{\displaystyle t\to \infty }
5859:{\displaystyle t\to \infty }
5731:{\displaystyle t\to \infty }
5664:{\displaystyle t\to \infty }
5415:{\displaystyle t\to \infty }
5285:{\displaystyle t\to \infty }
4997:{\displaystyle \varepsilon }
2242:{\displaystyle y=x-\phi (t)}
1807:that one may want to choose.
1361:{\displaystyle \delta >0}
1194:{\displaystyle \delta >0}
7782:Exact differential equation
7504:(PhD). Columbia University.
7161:. New York: Academic Press.
5789:{\displaystyle {\ddot {f}}}
3622:) if all real parts of the
2051:that start close enough to
809:nonlinear dynamical system
432:Tsiolkovsky rocket equation
8189:
7596:(2nd ed.). New York:
7110:English tr. Princeton 1961
6918:are bounded. This implies
6776:{\displaystyle {\dot {V}}}
6727:{\displaystyle {\dot {V}}}
5760:{\displaystyle {\dot {f}}}
5175:{\displaystyle {\dot {V}}}
5064:{\displaystyle {\dot {V}}}
2157:{\displaystyle x(t)=x_{e}}
1134:{\displaystyle f(x_{e})=0}
774:Nikolay Gur'yevich Chetaev
401:Engineering and efficiency
220:Bi-elliptic transfer orbit
29:
8092:Józef Maria Hoene-Wroński
8072:Gottfried Wilhelm Leibniz
7863:Cauchy–Kowalevski theorem
7551:Applied Nonlinear Control
7437:Applied Nonlinear Control
7333:10.1007/978-3-030-76317-6
7298:10.1007/978-3-642-50085-5
7176:. I Nov 1962 II Dec 1962.
6317:Applied Nonlinear Control
6278:{\displaystyle f(t)\to 0}
6078:{\displaystyle f(t)\to 0}
5184:the invariant set theorem
5042:But this is wrong, since
2658:{\displaystyle V(x)>0}
1800:{\displaystyle \epsilon }
1777:{\displaystyle \epsilon }
7986:Finite difference method
7500:Chervin, Robert (1971).
7364:Vinograd, R. E. (1957).
7030:Markus–Yamabe conjecture
6858:This is bounded because
6786:Using Barbalat's lemma:
4241:input-to-state stability
2073:{\displaystyle \phi (t)}
753:Input-to-state stability
427:Propellant mass fraction
326:Gravitational influences
55:may need to be rewritten
7966:Variation of parameters
7956:Separation of variables
7853:Peano existence theorem
7848:Picard–Lindelöf theorem
7735:Attributes of variables
7389:The American Naturalist
7040:Hartman–Grobman theorem
7025:Lyapunov–Malkin theorem
5396:approaching a limit as
2949:{\displaystyle x\neq 0}
2879:{\displaystyle x\neq 0}
2684:{\displaystyle x\neq 0}
1757:{\displaystyle \delta }
1729:{\displaystyle t\geq 0}
1274:{\displaystyle t\geq 0}
298:Specific orbital energy
8127:Carl David Tolmé Runge
7701:Differential-algebraic
7660:Differential equations
7542:10.1093/imamci/9.4.275
7194:Proc Natl Acad Sci USA
7187:Kalman, R. E. (1963).
6999:
6998:{\displaystyle e\to 0}
6973:
6947:
6912:
6892:
6872:
6848:
6777:
6748:
6728:
6699:
6679:
6659:
6639:
6593:
6539:
6490:
6461:
6438:
6381:
6305:
6279:
6244:
6224:has a finite limit as
6218:
6164:
6144:
6108:
6079:
6044:
5987:
5939:
5906:
5860:
5834:
5790:
5761:
5732:
5712:has a finite limit as
5706:
5665:
5639:
5604:
5569:
5537:
5460:
5416:
5390:
5358:
5286:
5260:
5231:
5176:
5147:
5119:
5092:
5065:
5036:
4998:
4975:
4721:
4642:
4584:
4421:
4354:
4257:Van der Pol oscillator
4194:
4119:
4060:
3925:
3866:
3843:
3776:
3726:
3684:
3640:
3612:
3588:
3529:
3362:
3106:
3062:
2997:
2996:{\displaystyle V(0)=0}
2950:
2924:
2880:
2854:
2773:
2685:
2659:
2623:
2597:
2596:{\displaystyle V(x)=0}
2559:
2522:. Consider a function
2516:
2490:
2450:Lyapunov function V(x)
2430:
2386:
2356:
2243:
2202:
2158:
2101:homoclinic connections
2074:
2045:
2013:
1987:
1923:
1874:
1801:
1778:
1758:
1730:
1704:
1603:
1549:
1486:
1416:
1362:
1329:
1275:
1249:
1195:
1169:
1135:
1093:
1072:has an equilibrium at
1066:
1046:
1022:
977:
949:
891:
725:
698:
671:
644:
613:
586:
559:
524:differential equations
215:Hohmann transfer orbit
8112:Augustin-Louis Cauchy
8107:Joseph-Louis Lagrange
8001:Finite element method
7991:Crank–Nicolson method
7925:Numerical integration
7904:Exponential stability
7796:Relation to processes
7681:Differential operator
7370:Doklady Akademii Nauk
7216:10.1073/pnas.49.2.201
7100:Letov, A. M. (1955).
7035:Libration point orbit
7000:
6974:
6948:
6913:
6893:
6873:
6849:
6778:
6754:can be non-zero when
6749:
6729:
6700:
6680:
6660:
6640:
6594:
6540:
6491:
6462:
6439:
6382:
6324:non-autonomous system
6306:
6280:
6245:
6219:
6165:
6145:
6109:
6080:
6045:
5988:
5940:
5907:
5861:
5835:
5791:
5762:
5733:
5707:
5666:
5640:
5605:
5570:
5538:
5461:
5417:
5391:
5359:
5287:
5261:
5232:
5177:
5148:
5120:
5118:{\displaystyle x_{1}}
5093:
5091:{\displaystyle x_{1}}
5066:
5037:
4999:
4976:
4722:
4643:
4585:
4422:
4355:
4195:
4125:is smaller than one.
4120:
4071:joint spectral radius
4061:
3926:
3867:
3844:
3777:
3727:
3685:
3641:
3613:
3589:
3530:
3377:asymptotically stable
3363:
3107:
3063:
2998:
2951:
2925:
2881:
2855:
2753:
2686:
2660:
2624:
2598:
2560:
2517:
2491:
2431:
2387:
2357:
2244:
2203:
2159:
2097:asymptotically stable
2075:
2046:
2022:for all trajectories
2014:
1988:
1924:
1875:
1802:
1779:
1759:
1731:
1705:
1604:
1550:
1487:
1417:
1363:
1340:asymptotically stable
1330:
1276:
1250:
1196:
1170:
1136:
1094:
1092:{\displaystyle x_{e}}
1067:
1047:
1023:
978:
950:
892:
744:exponential stability
734:asymptotically stable
726:
724:{\displaystyle x_{e}}
699:
697:{\displaystyle x_{e}}
672:
670:{\displaystyle x_{e}}
645:
643:{\displaystyle x_{e}}
623:. More strongly, if
614:
612:{\displaystyle x_{e}}
587:
585:{\displaystyle x_{e}}
560:
558:{\displaystyle x_{e}}
411:Preflight engineering
143:Argument of periapsis
32:exponential stability
18:Asymptotically stable
8168:Lagrangian mechanics
8006:Finite volume method
7930:Dirac delta function
7899:Asymptotic stability
7841:Existence/uniqueness
7706:Integro-differential
7590:Wiggins, S. (2003).
7553:. NJ: Prentice Hall.
7439:. NJ: Prentice Hall.
6983:
6957:
6922:
6902:
6882:
6862:
6793:
6758:
6738:
6709:
6689:
6669:
6649:
6605:
6549:
6503:
6489:{\displaystyle w(t)}
6471:
6451:
6392:
6332:
6289:
6254:
6228:
6174:
6154:
6143:{\displaystyle f(t)}
6125:
6089:
6054:
5997:
5949:
5911:
5878:
5844:
5800:
5771:
5742:
5716:
5705:{\displaystyle f(t)}
5687:
5649:
5614:
5579:
5568:{\displaystyle f(t)}
5550:
5470:
5426:
5422:does not imply that
5400:
5389:{\displaystyle f(t)}
5371:
5296:
5270:
5259:{\displaystyle f(t)}
5241:
5237:does not imply that
5197:
5157:
5137:
5102:
5075:
5046:
5008:
4988:
4741:
4734:. Its derivative is
4655:
4597:
4437:
4370:
4266:
4140:
4077:
3938:
3883:
3856:
3796:
3736:
3701:
3653:
3630:
3620:exponentially stable
3602:
3554:
3394:
3193:
3072:
3007:
2972:
2934:
2890:
2864:
2696:
2669:
2634:
2607:
2572:
2526:
2500:
2456:
2396:
2370:
2256:
2212:
2168:
2126:
2118:System of deviations
2055:
2044:{\displaystyle x(t)}
2026:
1997:
1941:
1889:
1817:
1791:
1768:
1748:
1714:
1613:
1559:
1503:
1497:exponentially stable
1426:
1372:
1346:
1285:
1259:
1205:
1179:
1153:
1103:
1076:
1056:
1032:
987:
963:
905:
816:
749:structural stability
708:
681:
654:
627:
596:
569:
542:
528:difference equations
467:Propulsive maneuvers
8016:Perturbation theory
7996:Runge–Kutta methods
7976:Integral transforms
7909:Rate of convergence
7805:(discrete analogue)
7289:Stability of Motion
7207:1963PNAS...49..201K
7074:Stability of Motion
7045:Perturbation theory
6192:
5071:does not depend on
5025:
4966:
4939:
4915:
4888:
4711:
4693:
4547:
4229:control engineering
3172:continuous function
3139:The definition for
3068:would "prove" that
2622:{\displaystyle x=0}
2515:{\displaystyle x=0}
2385:{\displaystyle y=0}
2082:globally attractive
957:system state vector
739:asymptotic analysis
444:Efficiency measures
347:Sphere of influence
316:Celestial mechanics
98:Part of a series on
8137:Sofya Kovalevskaya
7971:Integrating factor
7894:Lyapunov stability
7814:Stochastic partial
7262:10.1007/BF02031940
6995:
6969:
6943:
6908:
6888:
6868:
6844:
6773:
6744:
6724:
6695:
6675:
6655:
6635:
6589:
6535:
6486:
6457:
6434:
6377:
6301:
6275:
6240:
6214:
6213:
6178:
6160:
6140:
6104:
6075:
6040:
5983:
5935:
5902:
5856:
5830:
5796:is bounded), then
5786:
5757:
5728:
5702:
5661:
5635:
5600:
5565:
5533:
5456:
5412:
5386:
5354:
5282:
5256:
5227:
5172:
5143:
5115:
5088:
5061:
5032:
5011:
4994:
4971:
4952:
4925:
4901:
4874:
4717:
4697:
4679:
4638:
4580:
4578:
4533:
4417:
4350:
4190:
4115:
4056:
3921:
3876:smaller than one.
3862:
3839:
3772:
3722:
3680:
3636:
3608:
3584:
3525:
3456:
3358:
3102:
3058:
2993:
2946:
2920:
2876:
2860:for all values of
2850:
2681:
2655:
2619:
2593:
2555:
2512:
2486:
2426:
2382:
2352:
2239:
2198:
2154:
2070:
2041:
2009:
1983:
1919:
1870:
1797:
1774:
1754:
1726:
1700:
1599:
1545:
1482:
1444:
1412:
1358:
1325:
1271:
1245:
1191:
1165:
1131:
1089:
1062:
1042:
1018:
973:
945:
887:
721:
694:
667:
640:
609:
582:
555:
536:Aleksandr Lyapunov
263:Dynamical friction
8173:Three-body orbits
8163:Dynamical systems
8145:
8144:
8024:
8023:
7829:
7828:
7607:978-0-387-00177-7
7582:978-0-8218-8328-0
7520:978-3-540-60988-9
7511:Economic Dynamics
7492:978-3-540-42748-3
7342:978-3-030-76316-9
7307:978-3-642-50087-9
7137:10.1115/1.3662604
7015:Lyapunov function
6934:
6911:{\displaystyle w}
6891:{\displaystyle g}
6871:{\displaystyle e}
6805:
6770:
6747:{\displaystyle g}
6721:
6698:{\displaystyle e}
6678:{\displaystyle g}
6658:{\displaystyle e}
6561:
6460:{\displaystyle w}
6404:
6344:
6163:{\displaystyle E}
6009:
5812:
5783:
5754:
5626:
5591:
5438:
5209:
5169:
5146:{\displaystyle V}
5058:
4943:
4892:
4811:
4779:
4753:
4732:positive definite
4730:which is clearly
4672:
4621:
4551:
4495:
4455:
4414:
4336:
4322:
4310:
4278:
4245:nonlinear systems
4184:
4174:
4164:
4154:
4149:
3988:
3947:
3865:{\displaystyle A}
3830:
3804:
3762:
3718:
3695:positive definite
3664:
3639:{\displaystyle A}
3611:{\displaystyle A}
3581:
3568:
3563:
3441:
3289:
3240:
3225:
3210:
3129:Lyapunov function
3084:
2966:Lyapunov function
2902:
2799:
2736:
2708:
2468:
2319:
2268:
1535:
1520:
1429:
1065:{\displaystyle f}
828:
791:Lyapunov exponent
741:). The notion of
532:dynamical systems
518:Various types of
516:
515:
366:Lagrangian points
303:Vis-viva equation
273:Kepler's equation
120:Orbital mechanics
93:
92:
85:
65:lead layout guide
16:(Redirected from
8180:
8158:Stability theory
8122:Phyllis Nicolson
8102:Rudolf Lipschitz
7939:Solution methods
7914:Series solutions
7838:
7671:
7653:
7646:
7639:
7630:
7611:
7586:
7554:
7545:
7524:
7505:
7496:
7468:
7465:
7459:
7456:
7450:
7447:
7441:
7440:
7432:
7423:
7419:
7413:
7412:
7395:(977): 135–143.
7384:
7378:
7377:
7361:
7355:
7354:
7318:
7312:
7311:
7280:
7274:
7273:
7245:
7239:
7238:
7228:
7218:
7184:
7178:
7177:
7169:
7163:
7162:
7147:
7141:
7140:
7117:
7111:
7109:
7097:
7091:
7087:
7081:
7064:
7004:
7002:
7001:
6996:
6978:
6976:
6975:
6970:
6952:
6950:
6949:
6944:
6936:
6935:
6927:
6917:
6915:
6914:
6909:
6897:
6895:
6894:
6889:
6877:
6875:
6874:
6869:
6853:
6851:
6850:
6845:
6807:
6806:
6798:
6782:
6780:
6779:
6774:
6772:
6771:
6763:
6753:
6751:
6750:
6745:
6733:
6731:
6730:
6725:
6723:
6722:
6714:
6704:
6702:
6701:
6696:
6684:
6682:
6681:
6676:
6664:
6662:
6661:
6656:
6644:
6642:
6641:
6636:
6598:
6596:
6595:
6590:
6582:
6581:
6563:
6562:
6554:
6544:
6542:
6541:
6536:
6534:
6533:
6521:
6520:
6495:
6493:
6492:
6487:
6466:
6464:
6463:
6458:
6443:
6441:
6440:
6435:
6406:
6405:
6397:
6386:
6384:
6383:
6378:
6346:
6345:
6337:
6310:
6308:
6307:
6302:
6284:
6282:
6281:
6276:
6249:
6247:
6246:
6241:
6223:
6221:
6220:
6215:
6209:
6191:
6186:
6170:and assume that
6169:
6167:
6166:
6161:
6149:
6147:
6146:
6141:
6113:
6111:
6110:
6105:
6084:
6082:
6081:
6076:
6049:
6047:
6046:
6041:
6024:
6023:
6011:
6010:
6002:
5992:
5990:
5989:
5984:
5967:
5966:
5944:
5942:
5941:
5936:
5909:
5908:
5903:
5865:
5863:
5862:
5857:
5839:
5837:
5836:
5831:
5814:
5813:
5805:
5795:
5793:
5792:
5787:
5785:
5784:
5776:
5766:
5764:
5763:
5758:
5756:
5755:
5747:
5737:
5735:
5734:
5729:
5711:
5709:
5708:
5703:
5670:
5668:
5667:
5662:
5644:
5642:
5641:
5636:
5628:
5627:
5619:
5609:
5607:
5606:
5601:
5593:
5592:
5584:
5574:
5572:
5571:
5566:
5542:
5540:
5539:
5534:
5516:
5511:
5507:
5506:
5465:
5463:
5462:
5457:
5440:
5439:
5431:
5421:
5419:
5418:
5413:
5395:
5393:
5392:
5387:
5363:
5361:
5360:
5355:
5291:
5289:
5288:
5283:
5265:
5263:
5262:
5257:
5236:
5234:
5233:
5228:
5211:
5210:
5202:
5181:
5179:
5178:
5173:
5171:
5170:
5162:
5152:
5150:
5149:
5144:
5124:
5122:
5121:
5116:
5114:
5113:
5097:
5095:
5094:
5089:
5087:
5086:
5070:
5068:
5067:
5062:
5060:
5059:
5051:
5041:
5039:
5038:
5033:
5024:
5019:
5003:
5001:
5000:
4995:
4980:
4978:
4977:
4972:
4967:
4965:
4960:
4944:
4938:
4933:
4924:
4916:
4914:
4909:
4893:
4887:
4882:
4873:
4865:
4864:
4855:
4854:
4842:
4841:
4832:
4831:
4819:
4818:
4813:
4812:
4804:
4800:
4799:
4787:
4786:
4781:
4780:
4772:
4768:
4767:
4755:
4754:
4746:
4726:
4724:
4723:
4718:
4716:
4712:
4710:
4705:
4692:
4687:
4673:
4665:
4647:
4645:
4644:
4639:
4631:
4630:
4619:
4609:
4608:
4589:
4587:
4586:
4581:
4579:
4572:
4568:
4567:
4566:
4565:
4552:
4546:
4541:
4532:
4519:
4518:
4503:
4502:
4497:
4496:
4488:
4483:
4476:
4475:
4463:
4462:
4457:
4456:
4448:
4443:
4426:
4424:
4423:
4418:
4416:
4415:
4407:
4401:
4400:
4382:
4381:
4359:
4357:
4356:
4351:
4343:
4339:
4338:
4337:
4329:
4323:
4318:
4317:
4312:
4311:
4303:
4299:
4280:
4279:
4271:
4221:forcing function
4199:
4197:
4196:
4191:
4186:
4185:
4176:
4175:
4166:
4165:
4156:
4155:
4150:
4145:
4124:
4122:
4121:
4116:
4111:
4110:
4092:
4091:
4065:
4063:
4062:
4057:
4052:
4051:
4033:
4032:
4017:
4016:
4015:
4014:
3996:
3995:
3990:
3989:
3982:
3981:
3980:
3979:
3962:
3961:
3960:
3949:
3948:
3930:
3928:
3927:
3922:
3917:
3916:
3898:
3897:
3871:
3869:
3868:
3863:
3848:
3846:
3845:
3840:
3838:
3837:
3832:
3831:
3818:
3817:
3806:
3805:
3781:
3779:
3778:
3773:
3765:
3764:
3763:
3731:
3729:
3728:
3723:
3721:
3720:
3719:
3689:
3687:
3686:
3681:
3667:
3666:
3665:
3645:
3643:
3642:
3637:
3617:
3615:
3614:
3609:
3593:
3591:
3590:
3585:
3583:
3582:
3570:
3569:
3564:
3559:
3534:
3532:
3531:
3526:
3521:
3517:
3510:
3506:
3496:
3495:
3474:
3473:
3455:
3367:
3365:
3364:
3359:
3354:
3350:
3343:
3339:
3329:
3328:
3307:
3306:
3287:
3286:
3238:
3223:
3208:
3111:
3109:
3108:
3103:
3086:
3085:
3077:
3067:
3065:
3064:
3059:
3054:
3046:
3032:
3002:
3000:
2999:
2994:
2955:
2953:
2952:
2947:
2929:
2927:
2926:
2921:
2904:
2903:
2895:
2885:
2883:
2882:
2877:
2859:
2857:
2856:
2851:
2810:
2809:
2800:
2798:
2797:
2796:
2783:
2775:
2772:
2767:
2737:
2735:
2724:
2710:
2709:
2701:
2690:
2688:
2687:
2682:
2664:
2662:
2661:
2656:
2628:
2626:
2625:
2620:
2602:
2600:
2599:
2594:
2564:
2562:
2561:
2556:
2554:
2546:
2545:
2540:
2521:
2519:
2518:
2513:
2495:
2493:
2492:
2487:
2470:
2469:
2461:
2435:
2433:
2432:
2427:
2391:
2389:
2388:
2383:
2361:
2359:
2358:
2353:
2321:
2320:
2312:
2270:
2269:
2261:
2248:
2246:
2245:
2240:
2207:
2205:
2204:
2199:
2163:
2161:
2160:
2155:
2153:
2152:
2112:stability matrix
2079:
2077:
2076:
2071:
2050:
2048:
2047:
2042:
2018:
2016:
2015:
2010:
1992:
1990:
1989:
1984:
1928:
1926:
1925:
1920:
1879:
1877:
1876:
1871:
1869:
1868:
1850:
1849:
1806:
1804:
1803:
1798:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1735:
1733:
1732:
1727:
1709:
1707:
1706:
1701:
1699:
1698:
1680:
1679:
1643:
1642:
1608:
1606:
1605:
1600:
1589:
1588:
1554:
1552:
1551:
1546:
1533:
1518:
1491:
1489:
1488:
1483:
1472:
1471:
1443:
1421:
1419:
1418:
1413:
1402:
1401:
1367:
1365:
1364:
1359:
1334:
1332:
1331:
1326:
1315:
1314:
1280:
1278:
1277:
1272:
1254:
1252:
1251:
1246:
1235:
1234:
1200:
1198:
1197:
1192:
1174:
1172:
1171:
1166:
1140:
1138:
1137:
1132:
1121:
1120:
1098:
1096:
1095:
1090:
1088:
1087:
1071:
1069:
1068:
1063:
1051:
1049:
1048:
1043:
1041:
1040:
1027:
1025:
1024:
1019:
1017:
1016:
1011:
1002:
1001:
982:
980:
979:
974:
972:
971:
954:
952:
951:
946:
944:
943:
938:
929:
928:
896:
894:
893:
888:
886:
885:
830:
829:
821:
786:guidance systems
730:
728:
727:
722:
720:
719:
703:
701:
700:
695:
693:
692:
676:
674:
673:
668:
666:
665:
649:
647:
646:
641:
639:
638:
618:
616:
615:
610:
608:
607:
591:
589:
588:
583:
581:
580:
564:
562:
561:
556:
554:
553:
508:
501:
494:
473:Orbital maneuver
422:Payload fraction
402:
383:Lissajous orbits
317:
288:Orbital velocity
235:Hyperbolic orbit
131:Orbital elements
121:
112:
95:
88:
81:
77:
74:
68:
61:improve the lead
45:
44:
37:
21:
8188:
8187:
8183:
8182:
8181:
8179:
8178:
8177:
8148:
8147:
8146:
8141:
8082:Jacob Bernoulli
8055:
8020:
8011:Galerkin method
7934:
7872:Solution topics
7867:
7825:
7791:
7730:
7662:
7657:
7615:
7608:
7598:Springer Verlag
7589:
7583:
7557:
7548:
7527:
7521:
7508:
7499:
7493:
7480:
7477:
7475:Further reading
7472:
7471:
7466:
7462:
7457:
7453:
7448:
7444:
7434:
7433:
7426:
7420:
7416:
7386:
7385:
7381:
7363:
7362:
7358:
7343:
7320:
7319:
7315:
7308:
7282:
7281:
7277:
7247:
7246:
7242:
7186:
7185:
7181:
7171:
7170:
7166:
7149:
7148:
7144:
7119:
7118:
7114:
7099:
7098:
7094:
7088:
7084:
7067:Lyapunov, A. M.
7065:
7058:
7053:
7011:
6981:
6980:
6955:
6954:
6920:
6919:
6900:
6899:
6880:
6879:
6860:
6859:
6791:
6790:
6756:
6755:
6736:
6735:
6707:
6706:
6687:
6686:
6667:
6666:
6647:
6646:
6603:
6602:
6601:This says that
6573:
6547:
6546:
6525:
6512:
6501:
6500:
6469:
6468:
6449:
6448:
6390:
6389:
6330:
6329:
6287:
6286:
6252:
6251:
6226:
6225:
6172:
6171:
6152:
6151:
6123:
6122:
6087:
6086:
6052:
6051:
6015:
5995:
5994:
5958:
5947:
5946:
5876:
5875:
5842:
5841:
5798:
5797:
5769:
5768:
5740:
5739:
5714:
5713:
5685:
5684:
5647:
5646:
5612:
5611:
5577:
5576:
5548:
5547:
5498:
5494:
5468:
5467:
5466:. For example,
5424:
5423:
5398:
5397:
5369:
5368:
5294:
5293:
5292:. For example,
5268:
5267:
5266:has a limit at
5239:
5238:
5195:
5194:
5155:
5154:
5135:
5134:
5131:
5105:
5100:
5099:
5078:
5073:
5072:
5044:
5043:
5006:
5005:
4986:
4985:
4856:
4846:
4833:
4823:
4801:
4791:
4769:
4759:
4739:
4738:
4678:
4674:
4653:
4652:
4622:
4600:
4595:
4594:
4577:
4576:
4557:
4530:
4526:
4510:
4485:
4481:
4480:
4467:
4445:
4435:
4434:
4392:
4373:
4368:
4367:
4300:
4297:
4293:
4264:
4263:
4253:
4227:and applied in
4138:
4137:
4131:
4102:
4083:
4075:
4074:
4043:
4024:
4006:
4001:
3983:
3971:
3966:
3942:
3936:
3935:
3908:
3889:
3881:
3880:
3854:
3853:
3825:
3799:
3794:
3793:
3754:
3734:
3733:
3710:
3699:
3698:
3656:
3651:
3650:
3628:
3627:
3600:
3599:
3552:
3551:
3541:
3487:
3465:
3464:
3460:
3413:
3409:
3392:
3391:
3320:
3298:
3297:
3293:
3245:
3241:
3191:
3190:
3184:Lyapunov stable
3137:
3070:
3069:
3005:
3004:
2970:
2969:
2932:
2931:
2888:
2887:
2862:
2861:
2801:
2788:
2784:
2776:
2728:
2694:
2693:
2667:
2666:
2665:if and only if
2632:
2631:
2605:
2604:
2603:if and only if
2570:
2569:
2535:
2524:
2523:
2498:
2497:
2454:
2453:
2442:
2394:
2393:
2368:
2367:
2254:
2253:
2210:
2209:
2166:
2165:
2144:
2124:
2123:
2120:
2093:stable manifold
2053:
2052:
2024:
2023:
1995:
1994:
1939:
1938:
1887:
1886:
1884:The trajectory
1854:
1841:
1815:
1814:
1789:
1788:
1766:
1765:
1746:
1745:
1712:
1711:
1684:
1671:
1634:
1611:
1610:
1580:
1557:
1556:
1501:
1500:
1463:
1424:
1423:
1393:
1370:
1369:
1344:
1343:
1306:
1283:
1282:
1257:
1256:
1255:then for every
1226:
1203:
1202:
1177:
1176:
1175:there exists a
1151:
1150:
1147:Lyapunov stable
1112:
1101:
1100:
1079:
1074:
1073:
1054:
1053:
1030:
1029:
1006:
985:
984:
961:
960:
933:
903:
902:
877:
814:
813:
803:
761:
711:
706:
705:
684:
679:
678:
657:
652:
651:
630:
625:
624:
621:Lyapunov stable
599:
594:
593:
572:
567:
566:
545:
540:
539:
512:
483:
482:
478:Orbit insertion
468:
460:
459:
445:
437:
436:
412:
404:
400:
393:
392:
388:Lyapunov orbits
379:
378:
362:
352:
351:
327:
319:
315:
308:
307:
293:Surface gravity
268:Escape velocity
258:
250:
249:
230:Parabolic orbit
226:
225:
192:
190:
187:two-body orbits
178:
177:
168:Semi-major axis
133:
123:
119:
89:
78:
72:
69:
58:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8186:
8184:
8176:
8175:
8170:
8165:
8160:
8150:
8149:
8143:
8142:
8140:
8139:
8134:
8129:
8124:
8119:
8114:
8109:
8104:
8099:
8097:Ernst Lindelöf
8094:
8089:
8084:
8079:
8077:Leonhard Euler
8074:
8069:
8063:
8061:
8060:Mathematicians
8057:
8056:
8054:
8053:
8048:
8043:
8038:
8032:
8030:
8026:
8025:
8022:
8021:
8019:
8018:
8013:
8008:
8003:
7998:
7993:
7988:
7983:
7978:
7973:
7968:
7963:
7958:
7953:
7948:
7942:
7940:
7936:
7935:
7933:
7932:
7927:
7922:
7916:
7911:
7906:
7901:
7896:
7891:
7886:
7884:Phase portrait
7881:
7875:
7873:
7869:
7868:
7866:
7865:
7860:
7855:
7850:
7844:
7842:
7835:
7831:
7830:
7827:
7826:
7824:
7823:
7818:
7817:
7816:
7806:
7799:
7797:
7793:
7792:
7790:
7789:
7787:On jet bundles
7784:
7779:
7774:
7769:
7764:
7759:
7754:
7752:Nonhomogeneous
7749:
7744:
7738:
7736:
7732:
7731:
7729:
7728:
7723:
7718:
7713:
7708:
7703:
7698:
7693:
7688:
7683:
7677:
7675:
7668:
7667:Classification
7664:
7663:
7658:
7656:
7655:
7648:
7641:
7633:
7613:
7612:
7606:
7587:
7581:
7555:
7546:
7536:(4): 275–303.
7525:
7519:
7506:
7497:
7491:
7476:
7473:
7470:
7469:
7460:
7451:
7442:
7424:
7414:
7401:10.1086/283144
7379:
7372:(in Russian).
7356:
7341:
7313:
7306:
7284:Hahn, Wolfgang
7275:
7240:
7201:(2): 201–205.
7179:
7164:
7151:LaSalle, J. P.
7142:
7131:(2): 371–393.
7112:
7092:
7082:
7055:
7054:
7052:
7049:
7048:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7010:
7007:
6994:
6991:
6988:
6968:
6965:
6962:
6942:
6939:
6933:
6930:
6907:
6887:
6867:
6856:
6855:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6804:
6801:
6769:
6766:
6743:
6720:
6717:
6694:
6674:
6654:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6588:
6585:
6580:
6576:
6572:
6569:
6566:
6560:
6557:
6532:
6528:
6524:
6519:
6515:
6511:
6508:
6485:
6482:
6479:
6476:
6456:
6445:
6444:
6433:
6430:
6427:
6424:
6421:
6418:
6415:
6412:
6409:
6403:
6400:
6387:
6376:
6373:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6343:
6340:
6313:
6312:
6300:
6297:
6294:
6274:
6271:
6268:
6265:
6262:
6259:
6239:
6236:
6233:
6212:
6208:
6204:
6201:
6198:
6195:
6190:
6185:
6181:
6159:
6139:
6136:
6133:
6130:
6115:
6114:
6103:
6100:
6097:
6094:
6074:
6071:
6068:
6065:
6062:
6059:
6039:
6036:
6033:
6030:
6027:
6022:
6018:
6014:
6008:
6005:
5982:
5979:
5976:
5973:
5970:
5965:
5961:
5957:
5954:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5868:
5867:
5855:
5852:
5849:
5829:
5826:
5823:
5820:
5817:
5811:
5808:
5782:
5779:
5753:
5750:
5727:
5724:
5721:
5701:
5698:
5695:
5692:
5673:
5672:
5660:
5657:
5654:
5634:
5631:
5625:
5622:
5599:
5596:
5590:
5587:
5564:
5561:
5558:
5555:
5544:
5532:
5529:
5526:
5522:
5519:
5515:
5510:
5505:
5501:
5497:
5493:
5490:
5487:
5484:
5481:
5478:
5475:
5455:
5452:
5449:
5446:
5443:
5437:
5434:
5411:
5408:
5405:
5385:
5382:
5379:
5376:
5365:
5353:
5350:
5347:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5281:
5278:
5275:
5255:
5252:
5249:
5246:
5226:
5223:
5220:
5217:
5214:
5208:
5205:
5168:
5165:
5142:
5130:
5127:
5112:
5108:
5085:
5081:
5057:
5054:
5031:
5028:
5023:
5018:
5014:
4993:
4982:
4981:
4970:
4964:
4959:
4955:
4950:
4947:
4942:
4937:
4932:
4928:
4922:
4919:
4913:
4908:
4904:
4899:
4896:
4891:
4886:
4881:
4877:
4871:
4868:
4863:
4859:
4853:
4849:
4845:
4840:
4836:
4830:
4826:
4822:
4817:
4810:
4807:
4798:
4794:
4790:
4785:
4778:
4775:
4766:
4762:
4758:
4752:
4749:
4728:
4727:
4715:
4709:
4704:
4700:
4696:
4691:
4686:
4682:
4677:
4671:
4668:
4663:
4660:
4637:
4634:
4629:
4625:
4618:
4615:
4612:
4607:
4603:
4591:
4590:
4575:
4571:
4564:
4560:
4555:
4550:
4545:
4540:
4536:
4529:
4525:
4522:
4517:
4513:
4509:
4506:
4501:
4494:
4491:
4484:
4482:
4479:
4474:
4470:
4466:
4461:
4454:
4451:
4444:
4442:
4428:
4427:
4413:
4410:
4404:
4399:
4395:
4391:
4388:
4385:
4380:
4376:
4361:
4360:
4349:
4346:
4342:
4335:
4332:
4326:
4321:
4316:
4309:
4306:
4296:
4292:
4289:
4286:
4283:
4277:
4274:
4252:
4249:
4237:linear systems
4233:BIBO stability
4225:control theory
4209:external input
4201:
4200:
4189:
4179:
4169:
4159:
4153:
4130:
4127:
4114:
4109:
4105:
4101:
4098:
4095:
4090:
4086:
4082:
4067:
4066:
4055:
4050:
4046:
4042:
4039:
4036:
4031:
4027:
4023:
4020:
4013:
4009:
4004:
3999:
3994:
3978:
3974:
3969:
3965:
3959:
3956:
3953:
3920:
3915:
3911:
3907:
3904:
3901:
3896:
3892:
3888:
3861:
3850:
3849:
3836:
3824:
3821:
3816:
3813:
3810:
3771:
3768:
3757:
3753:
3750:
3747:
3744:
3741:
3713:
3709:
3706:
3691:
3690:
3679:
3676:
3673:
3670:
3659:
3635:
3607:
3596:
3595:
3576:
3573:
3567:
3540:
3537:
3536:
3535:
3524:
3520:
3516:
3513:
3509:
3505:
3502:
3499:
3494:
3490:
3486:
3483:
3480:
3477:
3472:
3468:
3463:
3459:
3454:
3451:
3448:
3444:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3412:
3408:
3405:
3402:
3399:
3369:
3368:
3357:
3353:
3349:
3346:
3342:
3338:
3335:
3332:
3327:
3323:
3319:
3316:
3313:
3310:
3305:
3301:
3296:
3292:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3244:
3237:
3234:
3231:
3228:
3222:
3219:
3216:
3213:
3207:
3204:
3201:
3198:
3182:is said to be
3136:
3133:
3101:
3098:
3095:
3092:
3089:
3083:
3080:
3057:
3053:
3049:
3045:
3041:
3038:
3035:
3031:
3027:
3024:
3021:
3018:
3015:
3012:
2992:
2989:
2986:
2983:
2980:
2977:
2958:
2957:
2945:
2942:
2939:
2919:
2916:
2913:
2910:
2907:
2901:
2898:
2875:
2872:
2869:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2808:
2804:
2795:
2791:
2787:
2782:
2779:
2771:
2766:
2763:
2760:
2756:
2752:
2749:
2746:
2743:
2740:
2734:
2731:
2727:
2722:
2719:
2716:
2713:
2707:
2704:
2691:
2680:
2677:
2674:
2654:
2651:
2648:
2645:
2642:
2639:
2629:
2618:
2615:
2612:
2592:
2589:
2586:
2583:
2580:
2577:
2553:
2549:
2544:
2539:
2534:
2531:
2511:
2508:
2505:
2485:
2482:
2479:
2476:
2473:
2467:
2464:
2441:
2438:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2381:
2378:
2375:
2364:
2363:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2318:
2315:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2267:
2264:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2119:
2116:
2069:
2066:
2063:
2060:
2040:
2037:
2034:
2031:
2020:
2019:
2008:
2005:
2002:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1882:
1881:
1867:
1864:
1861:
1857:
1853:
1848:
1844:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1811:
1808:
1796:
1773:
1753:
1738:
1737:
1725:
1722:
1719:
1697:
1694:
1691:
1687:
1683:
1678:
1674:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1641:
1637:
1633:
1630:
1627:
1624:
1621:
1618:
1598:
1595:
1592:
1587:
1583:
1579:
1576:
1573:
1570:
1567:
1564:
1544:
1541:
1538:
1532:
1529:
1526:
1523:
1517:
1514:
1511:
1508:
1493:
1481:
1478:
1475:
1470:
1466:
1462:
1459:
1456:
1453:
1450:
1447:
1442:
1439:
1436:
1432:
1411:
1408:
1405:
1400:
1396:
1392:
1389:
1386:
1383:
1380:
1377:
1357:
1354:
1351:
1336:
1324:
1321:
1318:
1313:
1309:
1305:
1302:
1299:
1296:
1293:
1290:
1270:
1267:
1264:
1244:
1241:
1238:
1233:
1229:
1225:
1222:
1219:
1216:
1213:
1210:
1190:
1187:
1184:
1164:
1161:
1158:
1130:
1127:
1124:
1119:
1115:
1111:
1108:
1086:
1082:
1061:
1039:
1015:
1010:
1005:
1000:
995:
992:
970:
942:
937:
932:
927:
922:
919:
916:
913:
910:
899:
898:
884:
880:
876:
873:
870:
867:
864:
857:
854:
851:
848:
845:
842:
839:
836:
833:
827:
824:
802:
799:
760:
757:
731:is said to be
718:
714:
691:
687:
664:
660:
637:
633:
606:
602:
592:forever, then
579:
575:
552:
548:
514:
513:
511:
510:
503:
496:
488:
485:
484:
481:
480:
475:
469:
466:
465:
462:
461:
458:
457:
452:
450:Gravity assist
446:
443:
442:
439:
438:
435:
434:
429:
424:
419:
413:
410:
409:
406:
405:
398:
395:
394:
391:
390:
385:
377:
376:
368:
364:
363:
358:
357:
354:
353:
350:
349:
344:
339:
334:
328:
325:
324:
321:
320:
313:
310:
309:
306:
305:
300:
295:
290:
285:
283:Orbital period
280:
275:
270:
265:
259:
256:
255:
252:
251:
248:
247:
245:Decaying orbit
242:
237:
232:
224:
223:
217:
210:
208:Transfer orbit
206:
205:
204:
202:Elliptic orbit
199:
197:Circular orbit
193:
184:
183:
180:
179:
176:
175:
170:
165:
160:
155:
150:
145:
140:
134:
129:
128:
125:
124:
117:
114:
113:
105:
104:
100:
99:
91:
90:
50:The article's
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8185:
8174:
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8155:
8153:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8064:
8062:
8058:
8052:
8049:
8047:
8044:
8042:
8039:
8037:
8034:
8033:
8031:
8027:
8017:
8014:
8012:
8009:
8007:
8004:
8002:
7999:
7997:
7994:
7992:
7989:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7957:
7954:
7952:
7949:
7947:
7944:
7943:
7941:
7937:
7931:
7928:
7926:
7923:
7920:
7917:
7915:
7912:
7910:
7907:
7905:
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7882:
7880:
7877:
7876:
7874:
7870:
7864:
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7845:
7843:
7839:
7836:
7832:
7822:
7819:
7815:
7812:
7811:
7810:
7807:
7804:
7801:
7800:
7798:
7794:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7768:
7765:
7763:
7760:
7758:
7755:
7753:
7750:
7748:
7745:
7743:
7740:
7739:
7737:
7733:
7727:
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7694:
7692:
7689:
7687:
7684:
7682:
7679:
7678:
7676:
7672:
7669:
7665:
7661:
7654:
7649:
7647:
7642:
7640:
7635:
7634:
7631:
7627:
7626:
7624:
7620:
7609:
7603:
7599:
7595:
7594:
7588:
7584:
7578:
7574:
7570:
7566:
7565:
7560:
7556:
7552:
7547:
7543:
7539:
7535:
7531:
7526:
7522:
7516:
7512:
7507:
7503:
7498:
7494:
7488:
7484:
7479:
7478:
7474:
7464:
7461:
7455:
7452:
7446:
7443:
7438:
7431:
7429:
7425:
7418:
7415:
7410:
7406:
7402:
7398:
7394:
7390:
7383:
7380:
7376:(2): 239–240.
7375:
7371:
7367:
7360:
7357:
7352:
7348:
7344:
7338:
7334:
7330:
7326:
7325:
7317:
7314:
7309:
7303:
7299:
7295:
7291:
7290:
7285:
7279:
7276:
7271:
7267:
7263:
7259:
7255:
7251:
7244:
7241:
7236:
7232:
7227:
7222:
7217:
7212:
7208:
7204:
7200:
7196:
7195:
7190:
7183:
7180:
7175:
7168:
7165:
7160:
7156:
7155:Lefschetz, S.
7152:
7146:
7143:
7138:
7134:
7130:
7126:
7122:
7121:Kalman, R. E.
7116:
7113:
7107:
7103:
7096:
7093:
7086:
7083:
7079:
7075:
7071:
7068:
7063:
7061:
7057:
7050:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7012:
7008:
7006:
6992:
6986:
6960:
6940:
6931:
6928:
6905:
6885:
6865:
6838:
6835:
6832:
6829:
6826:
6823:
6817:
6814:
6811:
6808:
6802:
6799:
6789:
6788:
6787:
6784:
6767:
6764:
6741:
6718:
6715:
6692:
6672:
6652:
6629:
6623:
6620:
6614:
6608:
6599:
6586:
6583:
6578:
6574:
6570:
6567:
6564:
6558:
6555:
6530:
6526:
6522:
6517:
6513:
6509:
6506:
6497:
6480:
6474:
6454:
6431:
6425:
6419:
6416:
6413:
6410:
6407:
6401:
6398:
6388:
6371:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6341:
6338:
6328:
6327:
6326:
6325:
6320:
6318:
6292:
6272:
6263:
6257:
6231:
6210:
6199:
6193:
6188:
6183:
6179:
6157:
6134:
6128:
6120:
6119:
6118:
6101:
6092:
6072:
6063:
6057:
6031:
6028:
6020:
6016:
6012:
6006:
6003:
5974:
5971:
5963:
5959:
5955:
5952:
5926:
5923:
5917:
5914:
5893:
5890:
5884:
5881:
5873:
5872:
5871:
5847:
5827:
5818:
5809:
5806:
5780:
5777:
5751:
5748:
5719:
5696:
5690:
5682:
5681:
5680:
5678:
5652:
5632:
5623:
5620:
5597:
5594:
5588:
5585:
5559:
5553:
5545:
5530:
5527:
5524:
5520:
5517:
5513:
5508:
5503:
5499:
5495:
5491:
5488:
5485:
5479:
5473:
5453:
5444:
5435:
5432:
5403:
5380:
5374:
5366:
5351:
5348:
5345:
5341:
5332:
5326:
5323:
5317:
5314:
5311:
5305:
5299:
5273:
5250:
5244:
5224:
5215:
5206:
5203:
5192:
5191:
5190:
5187:
5185:
5166:
5163:
5140:
5126:
5110:
5106:
5083:
5079:
5055:
5052:
5029:
5026:
5021:
5016:
5012:
4991:
4968:
4962:
4957:
4953:
4948:
4945:
4940:
4935:
4930:
4926:
4920:
4917:
4911:
4906:
4902:
4897:
4894:
4889:
4884:
4879:
4875:
4869:
4866:
4861:
4857:
4851:
4847:
4843:
4838:
4834:
4828:
4824:
4820:
4815:
4808:
4805:
4796:
4792:
4788:
4783:
4776:
4773:
4764:
4760:
4756:
4750:
4747:
4737:
4736:
4735:
4733:
4713:
4707:
4702:
4698:
4694:
4689:
4684:
4680:
4675:
4669:
4666:
4661:
4658:
4651:
4650:
4649:
4635:
4632:
4627:
4623:
4616:
4613:
4610:
4605:
4601:
4573:
4569:
4562:
4558:
4553:
4548:
4543:
4538:
4534:
4527:
4523:
4520:
4515:
4511:
4507:
4504:
4499:
4492:
4489:
4477:
4472:
4468:
4464:
4459:
4452:
4449:
4433:
4432:
4431:
4411:
4408:
4402:
4397:
4393:
4389:
4386:
4383:
4378:
4374:
4366:
4365:
4364:
4347:
4344:
4340:
4333:
4330:
4324:
4319:
4314:
4307:
4304:
4294:
4290:
4287:
4284:
4281:
4275:
4272:
4262:
4261:
4260:
4258:
4250:
4248:
4246:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4214:
4210:
4206:
4177:
4157:
4151:
4136:
4135:
4134:
4128:
4126:
4107:
4103:
4099:
4096:
4093:
4088:
4084:
4072:
4048:
4044:
4040:
4037:
4034:
4029:
4025:
4018:
4011:
4007:
4002:
3997:
3992:
3976:
3972:
3967:
3963:
3957:
3954:
3951:
3934:
3933:
3932:
3913:
3909:
3905:
3902:
3899:
3894:
3890:
3877:
3875:
3859:
3834:
3822:
3819:
3814:
3811:
3808:
3792:
3791:
3790:
3788:
3783:
3769:
3766:
3755:
3751:
3745:
3739:
3711:
3707:
3704:
3696:
3677:
3674:
3671:
3668:
3657:
3649:
3648:
3647:
3633:
3625:
3621:
3605:
3574:
3571:
3565:
3550:
3549:
3548:
3546:
3538:
3522:
3518:
3514:
3511:
3507:
3500:
3492:
3488:
3484:
3478:
3470:
3466:
3461:
3457:
3446:
3435:
3432:
3426:
3423:
3420:
3414:
3410:
3406:
3403:
3400:
3390:
3389:
3388:
3386:
3382:
3378:
3374:
3355:
3351:
3347:
3344:
3340:
3333:
3325:
3321:
3317:
3311:
3303:
3299:
3294:
3290:
3279:
3276:
3267:
3264:
3258:
3255:
3252:
3246:
3242:
3235:
3232:
3229:
3220:
3217:
3214:
3205:
3202:
3199:
3189:
3188:
3187:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3144:
3142:
3141:discrete-time
3134:
3132:
3130:
3125:
3123:
3118:
3113:
3099:
3096:
3090:
3081:
3078:
3047:
3039:
3036:
3029:
3025:
3022:
3016:
3010:
2990:
2987:
2981:
2975:
2967:
2963:
2943:
2940:
2937:
2917:
2914:
2908:
2899:
2896:
2873:
2870:
2867:
2847:
2844:
2838:
2832:
2829:
2826:
2820:
2814:
2806:
2802:
2793:
2789:
2780:
2769:
2764:
2761:
2758:
2754:
2750:
2744:
2738:
2732:
2729:
2725:
2720:
2714:
2705:
2702:
2692:
2678:
2675:
2672:
2652:
2649:
2643:
2637:
2630:
2616:
2613:
2610:
2590:
2587:
2581:
2575:
2568:
2567:
2566:
2542:
2532:
2529:
2509:
2506:
2503:
2480:
2474:
2471:
2465:
2462:
2451:
2447:
2437:
2420:
2414:
2411:
2405:
2399:
2379:
2376:
2373:
2346:
2343:
2340:
2334:
2331:
2325:
2316:
2313:
2307:
2298:
2292:
2289:
2286:
2283:
2280:
2274:
2271:
2265:
2262:
2252:
2251:
2250:
2233:
2227:
2224:
2221:
2218:
2215:
2192:
2186:
2183:
2177:
2171:
2149:
2145:
2141:
2135:
2129:
2117:
2115:
2113:
2109:
2104:
2102:
2098:
2094:
2090:
2085:
2083:
2064:
2058:
2035:
2029:
2000:
1980:
1968:
1962:
1959:
1953:
1947:
1937:
1936:
1935:
1933:
1930:is (locally)
1929:
1913:
1907:
1904:
1898:
1892:
1865:
1862:
1859:
1855:
1846:
1842:
1838:
1832:
1826:
1820:
1812:
1809:
1794:
1787:
1771:
1751:
1743:
1742:
1741:
1723:
1720:
1717:
1695:
1692:
1689:
1685:
1676:
1672:
1668:
1662:
1656:
1650:
1647:
1639:
1635:
1631:
1625:
1619:
1596:
1593:
1585:
1581:
1577:
1571:
1565:
1555:such that if
1542:
1539:
1536:
1530:
1527:
1524:
1521:
1515:
1512:
1509:
1506:
1498:
1494:
1479:
1476:
1468:
1464:
1460:
1454:
1448:
1434:
1409:
1406:
1398:
1394:
1390:
1384:
1378:
1368:such that if
1355:
1352:
1349:
1341:
1337:
1322:
1319:
1311:
1307:
1303:
1297:
1291:
1268:
1265:
1262:
1242:
1239:
1231:
1227:
1223:
1217:
1211:
1201:such that if
1188:
1185:
1182:
1162:
1159:
1156:
1149:if for every
1148:
1144:
1143:
1142:
1128:
1125:
1117:
1113:
1106:
1084:
1080:
1059:
1013:
993:
990:
958:
940:
930:
920:
914:
908:
882:
878:
874:
868:
862:
855:
846:
840:
834:
831:
825:
822:
812:
811:
810:
808:
800:
798:
796:
792:
787:
783:
778:
775:
770:
766:
758:
756:
754:
750:
746:
745:
740:
736:
735:
716:
712:
689:
685:
662:
658:
635:
631:
622:
604:
600:
577:
573:
550:
546:
537:
533:
529:
525:
521:
509:
504:
502:
497:
495:
490:
489:
487:
486:
479:
476:
474:
471:
470:
464:
463:
456:
455:Oberth effect
453:
451:
448:
447:
441:
440:
433:
430:
428:
425:
423:
420:
418:
415:
414:
408:
407:
403:
396:
389:
386:
384:
381:
380:
374:
370:
369:
367:
361:
360:N-body orbits
356:
355:
348:
345:
343:
342:Perturbations
340:
338:
335:
333:
330:
329:
323:
322:
318:
311:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
264:
261:
260:
254:
253:
246:
243:
241:
238:
236:
233:
231:
228:
227:
221:
218:
216:
212:
211:
209:
203:
200:
198:
195:
194:
188:
182:
181:
174:
171:
169:
166:
164:
163:Orbital nodes
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
135:
132:
127:
126:
122:
115:
111:
107:
106:
103:Astrodynamics
101:
97:
96:
87:
84:
76:
73:December 2021
66:
63:and read the
62:
56:
53:
48:
39:
38:
33:
19:
8132:Martin Kutta
8087:Émile Picard
8067:Isaac Newton
7981:Euler method
7951:Substitution
7893:
7616:
7614:
7592:
7563:
7550:
7533:
7529:
7510:
7501:
7485:. Springer.
7482:
7463:
7454:
7445:
7436:
7417:
7392:
7388:
7382:
7373:
7369:
7359:
7323:
7316:
7288:
7278:
7256:(1): 59–79.
7253:
7249:
7243:
7198:
7192:
7182:
7173:
7167:
7158:
7145:
7128:
7124:
7115:
7105:
7101:
7095:
7085:
7077:
7073:
7069:
6857:
6785:
6705:to zero, as
6600:
6498:
6496:is bounded.
6446:
6321:
6316:
6314:
6116:
5869:
5674:
5188:
5132:
4983:
4729:
4592:
4429:
4362:
4254:
4220:
4216:
4212:
4208:
4204:
4202:
4132:
4068:
3878:
3851:
3784:
3692:
3597:
3542:
3384:
3376:
3372:
3371:We say that
3370:
3183:
3179:
3175:
3167:
3163:
3159:
3156:metric space
3151:
3147:
3145:
3138:
3126:
3114:
2964:is called a
2961:
2959:
2956:is required.
2449:
2443:
2365:
2121:
2105:
2096:
2088:
2087:That is, if
2086:
2081:
2021:
1931:
1885:
1883:
1785:
1739:
1496:
1339:
1146:
955:denotes the
900:
805:Consider an
804:
795:chaos theory
779:
768:
762:
742:
733:
732:
677:converge to
620:
517:
387:
240:Radial orbit
191:eccentricity
173:True anomaly
158:Mean anomaly
148:Eccentricity
79:
70:
59:Please help
54:
52:lead section
7889:Phase space
7747:Homogeneous
6322:Consider a
5675:Barbalat's
4593:The origin
4243:(ISS) (for
4217:disturbance
4073:of the set
3787:state space
3624:eigenvalues
3545:state space
1052:. Suppose
530:describing
373:Halo orbits
337:Hill sphere
153:Inclination
8152:Categories
8117:John Crank
7946:Inspection
7809:Stochastic
7803:Difference
7777:Autonomous
7721:Non-linear
7711:Fractional
7674:Operations
7619:PlanetMath
7569:Providence
7559:Teschl, G.
7051:References
6979:and hence
3381:stable set
3174:. A point
2565:such that
1932:attractive
807:autonomous
565:stay near
417:Mass ratio
332:Barycenter
7921:solutions
7879:Wronskian
7834:Solutions
7762:Decoupled
7726:Holonomic
7351:237964551
6990:→
6967:∞
6964:→
6938:→
6932:˙
6836:⋅
6824:−
6812:−
6803:¨
6768:˙
6719:˙
6621:≤
6584:≤
6568:−
6559:˙
6417:⋅
6411:−
6402:˙
6363:⋅
6351:−
6342:˙
6299:∞
6296:→
6270:→
6238:∞
6235:→
6211:τ
6200:τ
6180:∫
6099:∞
6096:→
6070:→
6035:∞
6013:∈
6007:˙
5978:∞
5956:∈
5930:∞
5918:∈
5897:∞
5885:∈
5854:∞
5851:→
5825:→
5810:˙
5781:¨
5752:˙
5726:∞
5723:→
5659:∞
5656:→
5630:→
5624:˙
5595:≤
5589:˙
5492:
5451:→
5436:˙
5410:∞
5407:→
5327:
5318:
5280:∞
5277:→
5222:→
5207:˙
5167:˙
5056:˙
4992:ε
4949:ε
4946:−
4921:ε
4898:ε
4895:−
4870:ε
4844:−
4809:˙
4777:˙
4751:˙
4554:−
4524:ε
4508:−
4493:˙
4453:˙
4412:˙
4334:˙
4325:−
4308:˙
4291:ε
4288:−
4276:¨
4152:˙
4097:…
4038:…
4019:∈
3903:…
3566:˙
3543:A linear
3453:∞
3450:→
3439:⇒
3436:δ
3401:δ
3398:∃
3348:ϵ
3280:∈
3274:∀
3271:⇒
3268:δ
3233:∈
3227:∀
3215:δ
3212:∃
3200:ϵ
3197:∀
3122:attractor
3082:˙
2941:≠
2900:˙
2871:≠
2845:≤
2830:⋅
2824:∇
2786:∂
2778:∂
2755:∑
2706:˙
2676:≠
2548:→
2466:˙
2415:ϕ
2317:˙
2314:ϕ
2308:−
2293:ϕ
2266:˙
2228:ϕ
2225:−
2187:ϕ
2059:ϕ
2007:∞
2004:→
1978:→
1975:‖
1963:ϕ
1960:−
1945:‖
1908:ϕ
1863:β
1860:−
1852:‖
1839:−
1824:‖
1821:α
1795:ϵ
1772:ϵ
1752:δ
1721:≥
1693:β
1690:−
1682:‖
1669:−
1654:‖
1651:α
1648:≤
1645:‖
1632:−
1617:‖
1597:δ
1591:‖
1578:−
1563:‖
1537:δ
1522:β
1507:α
1474:‖
1461:−
1446:‖
1441:∞
1438:→
1410:δ
1404:‖
1391:−
1376:‖
1350:δ
1323:ϵ
1317:‖
1304:−
1289:‖
1266:≥
1243:δ
1237:‖
1224:−
1209:‖
1183:δ
1157:ϵ
1004:→
931:⊆
921:∈
826:˙
520:stability
257:Equations
185:Types of
8029:Examples
7919:Integral
7691:Ordinary
7561:(2012).
7409:84826590
7286:(1967).
7270:14034490
7235:16591048
7157:(1961).
7009:See also
4213:stimulus
3162: :
2108:Jacobian
2095:, it is
1710:for all
1281:we have
1099:so that
782:Cold War
7757:Coupled
7696:Partial
7203:Bibcode
7174:Control
6499:Taking
6250:. Then
6050:, then
5738:and if
5546:Having
5367:Having
5193:Having
4251:Example
4205:control
3874:modulus
3872:have a
3697:matrix
3154:) be a
2106:If the
759:History
704:, then
7772:Degree
7716:Linear
7604:
7579:
7517:
7489:
7407:
7349:
7339:
7304:
7268:
7233:
7226:299777
7223:
7090:pages.
6545:gives
5679:says:
4620:
4239:) and
3789:model
3598:where
3547:model
3288:
3239:
3224:
3209:
3186:, if,
3117:energy
2080:, and
1534:
1519:
901:where
7821:Delay
7767:Order
7405:S2CID
7347:S2CID
7266:S2CID
7104:[
5945:. If
5677:Lemma
5153:with
4235:(for
4219:, or
3146:Let (
2960:Then
1609:then
1422:then
1141:then
737:(see
138:Apsis
7602:ISBN
7577:ISBN
7515:ISBN
7487:ISBN
7337:ISBN
7302:ISBN
7231:PMID
6898:and
6665:and
6121:Let
5993:and
5874:Let
5528:>
5349:>
5027:<
4363:Let
3433:<
3404:>
3387:if,
3385:i.e.
3345:<
3265:<
3218:>
3203:>
3158:and
2962:V(x)
2930:for
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2650:>
1993:as
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1525:>
1510:>
1407:<
1353:>
1320:<
1240:<
1186:>
1160:>
7538:doi
7397:doi
7393:111
7374:114
7329:doi
7294:doi
7258:doi
7221:PMC
7211:doi
7133:doi
6953:as
6285:as
6085:as
5840:as
5683:If
5645:as
5489:sin
5315:sin
3782:.)
3626:of
3443:lim
3375:is
3178:in
2436:.
2103:.)
1934:if
1786:any
1431:lim
619:is
526:or
189:by
8154::
7600:.
7575:.
7571::
7567:.
7532:.
7427:^
7403:.
7391:.
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7345:.
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7300:.
7264:.
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7252:.
7229:.
7219:.
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7197:.
7191:.
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7127:.
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6319:.
5324:ln
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4247:)
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4211:,
4207:,
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3170:a
3166:→
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959:,
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7638:v
7625:.
7610:.
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6800:V
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6630:0
6627:(
6624:V
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5480:t
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5312:=
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5216:t
5213:(
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5022:2
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4969:.
4963:2
4958:2
4954:x
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4918:=
4912:2
4907:2
4903:x
4890:3
4885:4
4880:2
4876:x
4867:+
4862:2
4858:x
4852:1
4848:x
4839:2
4835:x
4829:1
4825:x
4821:=
4816:2
4806:x
4797:2
4793:x
4789:+
4784:1
4774:x
4765:1
4761:x
4757:=
4748:V
4714:)
4708:2
4703:2
4699:x
4695:+
4690:2
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4681:x
4676:(
4670:2
4667:1
4662:=
4659:V
4636:0
4633:=
4628:2
4624:x
4617:,
4614:0
4611:=
4606:1
4602:x
4574:.
4570:)
4563:2
4559:x
4549:3
4544:3
4539:2
4535:x
4528:(
4521:+
4516:1
4512:x
4505:=
4500:2
4490:x
4478:,
4473:2
4469:x
4465:=
4460:1
4450:x
4409:y
4403:=
4398:2
4394:x
4390:,
4387:y
4384:=
4379:1
4375:x
4345:=
4341:)
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4320:3
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4295:(
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4273:y
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4163:f
4158:=
4148:x
4113:}
4108:m
4104:A
4100:,
4094:,
4089:1
4085:A
4081:{
4054:}
4049:m
4045:A
4041:,
4035:,
4030:1
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4022:{
4012:t
4008:i
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3993:t
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3964:=
3958:1
3955:+
3952:t
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3919:}
3914:m
3910:A
3906:,
3900:,
3895:1
3891:A
3887:{
3860:A
3835:t
3829:x
3823:A
3820:=
3815:1
3812:+
3809:t
3803:x
3770:x
3767:M
3761:T
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3752:=
3749:)
3746:x
3743:(
3740:V
3717:T
3712:M
3708:=
3705:M
3678:A
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3672:+
3669:M
3663:T
3658:A
3634:A
3606:A
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3580:x
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3572:=
3562:x
3523:.
3519:]
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3512:=
3508:)
3504:)
3501:y
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3415:d
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3407:0
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3352:]
3341:)
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3331:(
3326:n
3322:f
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3315:)
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3304:n
3300:f
3295:(
3291:d
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3277:n
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3253:x
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3247:d
3243:[
3236:X
3230:y
3221:0
3206:0
3180:X
3176:x
3168:X
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3152:d
3148:X
3100:x
3097:=
3094:)
3091:t
3088:(
3079:x
3056:)
3052:|
3048:x
3044:|
3040:+
3037:1
3034:(
3030:/
3026:1
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3020:)
3017:x
3014:(
3011:V
2991:0
2988:=
2985:)
2982:0
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2976:V
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2897:V
2874:0
2868:x
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2842:)
2839:x
2836:(
2833:f
2827:V
2821:=
2818:)
2815:x
2812:(
2807:i
2803:f
2794:i
2790:x
2781:V
2770:n
2765:1
2762:=
2759:i
2751:=
2748:)
2745:x
2742:(
2739:V
2733:t
2730:d
2726:d
2721:=
2718:)
2715:x
2712:(
2703:V
2679:0
2673:x
2653:0
2647:)
2644:x
2641:(
2638:V
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2611:x
2591:0
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2585:)
2582:x
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2576:V
2552:R
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2538:R
2533::
2530:V
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2504:x
2484:)
2481:x
2478:(
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2472:=
2463:x
2424:)
2421:t
2418:(
2412:=
2409:)
2406:t
2403:(
2400:x
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2377:=
2374:y
2362:.
2350:)
2347:y
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2341:t
2338:(
2335:g
2332:=
2329:)
2326:t
2323:(
2305:)
2302:)
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2296:(
2290:+
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2278:(
2275:f
2272:=
2263:y
2237:)
2234:t
2231:(
2222:x
2219:=
2216:y
2196:)
2193:t
2190:(
2184:=
2181:)
2178:t
2175:(
2172:x
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2146:x
2142:=
2139:)
2136:t
2133:(
2130:x
2089:x
2068:)
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2062:(
2039:)
2036:t
2033:(
2030:x
2001:t
1981:0
1972:)
1969:t
1966:(
1957:)
1954:t
1951:(
1948:x
1917:)
1914:t
1911:(
1905:=
1902:)
1899:t
1896:(
1893:x
1880:.
1866:t
1856:e
1847:e
1843:x
1836:)
1833:0
1830:(
1827:x
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1724:0
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1673:x
1666:)
1663:0
1660:(
1657:x
1640:e
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909:x
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