124:
47:
99:. Some position-velocity pairs are driven towards the center manifold, while others are flung away from it. Small perturbations that generally push them about randomly, and often push them out of the center manifold. There are, however, dramatic counterexamples to instability at the center manifold, called
1588:
All the extant theory mentioned above seeks to establish invariant manifold properties of a specific given problem. In particular, one constructs a manifold that approximates an invariant manifold of the given system. An alternative approach is to construct exact invariant manifolds for a system that
994:
In the case when the unstable manifold does not exist, center manifolds are often relevant to modelling. The center manifold emergence theorem then says that the neighborhood may be chosen so that all solutions of the system staying in the neighborhood tend exponentially quickly to some solution
373:
215:
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guarantees that these eigenvalues and eigenvectors completely characterise the system's dynamics near the equilibrium. However, if the equilibrium has eigenvalues whose real part is zero, then the corresponding (generalized) eigenvectors form the
218:
Randomly selected points of the 2D phase space converge exponentially to a 1D center manifold on which dynamics are slow (non exponential). Studying dynamics of the center manifold determines the stability of the non-hyperbolic fixed point at the
1375:
However, some applications, such as to dispersion in tubes or channels, require an infinite-dimensional center manifold. The most general and powerful theory was developed by
Aulbach and Wanner. They addressed non-autonomous dynamical systems
1446:
in infinite dimensions, with potentially infinite dimensional stable, unstable and center manifolds. Further, they usefully generalised the definition of the manifolds so that the center manifold is associated with eigenvalues such that
773:
A center manifold is of the same dimension and tangent to the center subspace. If, as is common, the eigenvalues of the center subspace are all precisely zero, rather than just real part zero, then a center manifold is often called a
65:, which act characteristically to "compress and stretch". The forces compress particle orbits into the rings, stretch particles along the rings, and ignore small shifts in ring radius. The compressing direction defines the
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As the stability of the equilibrium correlates with the "stability" of its manifolds, the existence of a center manifold brings up the question about the dynamics on the center manifold. This is analyzed by the
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on tangent bundles of
Riemann surfaces. In that case, the tangent space splits very explicitly and precisely into three parts: the unstable and stable bundles, with the neutral manifold wedged between.
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1128:. This theorem asserts that for a wide variety of initial conditions the solutions of the full system decay exponentially quickly to a solution on the relatively low dimensional center manifold.
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axis (including the origin). Moreover, all center manifolds have this potential non-uniqueness, although often the non-uniqueness only occurs in unphysical complex values of the variables.
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730:. Going beyond the linearization, when we account for perturbations by nonlinearity or forcing in the dynamical system, the center eigenspace deforms to the nearby center manifold.
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While geometrically accurate, one major difference distinguishes Saturn's rings from a physical center manifold. Like most dynamical systems, particles in the rings are governed by
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approximates the given system---called a backwards theory. The aim is to usefully apply theory to a wider range of systems, and to estimate errors and sizes of domain of validity.
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If the eigenvalues are precisely zero (as they are for the ball), rather than just real-part being zero, then the corresponding eigenspace more specifically gives rise to a
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2007:
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Aulbach, B.; Wanner, T. (1999). "Invariant foliations for
Caratheodory type differential equations in Banach spaces". In Lakshmikantham, V.; Martynyuk, A. A. (eds.).
978:
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23:
was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to
3369:
Aulbach, B.; Wanner, T. (1996). "Integral manifolds for
Caratheodory type differential equations in Banach spaces". In Aulbach, B.; Colonius, F. (eds.).
3508:
Hochs, Peter; Roberts, A.J. (2019). "Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions".
871:
times continuously differentiable), then at every equilibrium point there exists a neighborhood of some finite size in which there is at least one of
38:
because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables.
1379:
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368:{\displaystyle {\frac {d\mathbf {x} }{dt}}=\mathbf {f} (\mathbf {x} )\approx ({\mathcal {D}}\mathbf {f} )(\mathbf {x} ^{*})\mathbf {x} }
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axis, and the stable manifold is the trivial set {(0, 0)}. Any orbit not on the stable manifold satisfies an equation of the form
737:. The behavior on the center (slow) manifold is generally not determined by the linearization and thus may be difficult to construct.
254:
Mathematically, the first step when studying equilibrium points of dynamical systems is to linearize the system, and then compute its
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Potzsche and
Rasmussen established a corresponding approximation theorem for such infinite dimensional, non-autonomous systems.
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of the linearized equation may be of interest, including center-stable, center-unstable, sub-center, slow, and fast subspaces.
469:
2561:
3627:
3600:
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Aulbach, B.; Wanner, T. (2000). "The
Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces".
3221:
3173:
Aulbach, B.; Wanner, T. (2000). "The
Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces".
3487:
Roberts, A.J. (2019). "Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems".
1639:
1577:. They proved existence of these manifolds, and the emergence of a center manifold, via nonlinear coordinate transforms.
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A third theorem, the approximation theorem, asserts that if an approximate expression for such invariant manifolds, say
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270:. The (generalized) eigenvectors corresponding to eigenvalues with positive real part form the unstable eigenspace.
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1117:{\displaystyle \mathbf {x} (t)=\mathbf {y} (t)+{\mathcal {O}}(e^{-\beta t})\quad {\text{ as }}\quad t\to \infty }
389:
1958:
Fortunately, we may approximate such delays by the following trick that keeps the dimensionality finite. Define
88:. Physically speaking, the stable, unstable and neutral manifolds of Saturn's ring system do not divide up the
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Analogously, nonlinearity or forcing in the system perturbs the stable and unstable eigenspaces to a nearby
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2012:
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laws. Understanding trajectories requires modeling position and a velocity/momentum variable, to give a
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134:
3138:
Roberts, A.J. (1993). "The invariant manifold of beam deformations. Part 1: the simple circular rod".
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1961:
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3338:"The application of centre manifold theory to the evolution of systems which vary slowly in space"
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Online web services to extract center manifolds from a specified system via computer algebra:
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An invariant manifold tangent to the stable subspace and with the same dimension is the
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The unstable manifold is of the same dimension and tangent to the unstable subspace.
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capture much center-manifold geometry. Dust particles in the rings are subject to
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Potzsche, C.; Rasmussen, M. (2006). "Taylor approximation of integral manifolds".
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The center manifold existence theorem states that if the right-hand side function
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902:
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721:(that is, all eigenvalues of the linearization have nonzero real part), then the
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the unstable subspace, spanned by the generalized eigenvectors whose eigenvalues
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3091:, but the cubic nonlinearity then stabilises nearby limit cycles as in classic
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Nonlinear
Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
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3582:
3239:"The stable, center-stable, center, center-unstable and unstable manifolds"
1439:{\displaystyle {\frac {d{\textbf {x}}}{dt}}={\textbf {f}}({\textbf {x}},t)}
511:
the stable subspace, spanned by generalized eigenvectors whose eigenvalues
262:
if they occur) corresponding to eigenvalues with negative real part form a
96:
92:
for a particle's position; they analogously divide up phase space instead.
128:
111:
85:
62:
51:
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1599:
3595:. Texts in applied mathematics (2nd ed.). New York, NY: Springer.
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because interesting behavior takes place on the center manifold and in
2475:{\displaystyle {\frac {d{\textbf {u}}}{dt}}=\left{\textbf {u}}+\left.}
1529:{\displaystyle \operatorname {Re} \lambda \leq -\beta <-r\alpha }
603:{\displaystyle \operatorname {Re} \lambda \leq -\beta <-r\alpha }
3522:
3493:
95:
The center manifold typically behaves as an extended collection of
1570:{\displaystyle \operatorname {Re} \lambda \geq \beta >r\alpha }
1371:
Center manifolds of infinite-dimensional or non-autonomous systems
1178:, satisfies the differential equation for the system to residuals
699:{\displaystyle \operatorname {Re} \lambda \geq \beta >r\alpha }
3618:, Applied Mathematical Sciences, vol. 42, Berlin, New York:
755:
Corresponding to the linearized system, the nonlinear system has
1765:, we can create a center manifold by piecing together the curve
3037:{\displaystyle {\frac {ds}{dt}}=\left+{O}(\alpha ^{2}+|s|^{4})}
987:
In example applications, a nonlinear coordinate transform to a
759:, each consisting of sets of orbits of the nonlinear system.
214:
1955:. Strictly, the delay makes this DE infinite-dimensional.
1306:{\displaystyle {\textbf {x}}={\textbf {X}}({\textbf {s}})}
1171:{\displaystyle {\textbf {x}}={\textbf {X}}({\textbf {s}})}
3047:
This evolution shows the origin is linearly unstable for
1824:
Delay differential equations often have Hopf bifurcations
1482:{\displaystyle |\operatorname {Re} \lambda |\leq \alpha }
501:{\displaystyle |\operatorname {Re} \lambda |\leq \alpha }
19:
In the mathematics of evolving systems, the concept of a
2860:{\displaystyle {\textbf {u}}=\left+{O}(\alpha +|s|^{2})}
3304:
Normal forms and unfoldings for local dynamical systems
1828:
Another example shows how a center manifold models the
748:. These three types of manifolds are three cases of an
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Advances of
Stability Theory at the End of XX Century
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2015:
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Center manifold and the analysis of nonlinear systems
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Saturn's rings sit in the center manifold defined by
1696:{\displaystyle {\dot {x}}=x^{2},\quad {\dot {y}}=y.}
73:, and the neutral direction is the center manifold.
1360:{\displaystyle {\mathcal {O}}(|{\textbf {s}}|^{p})}
1225:{\displaystyle {\mathcal {O}}(|{\textbf {s}}|^{p})}
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3649:construct center manifolds for autonomous systems
3593:Ordinary differential equations with applications
3443:
1592:This approach is cognate to the well-established
1266:, then the invariant manifold is approximated by
239:of the equilibrium then consists of those nearby
3663:
658:{\displaystyle \operatorname {Re} \lambda >0}
556:{\displaystyle \operatorname {Re} \lambda <0}
3655:convert a specified ODE system to a normal form
3204:. Applied Mathematical Sciences. Vol. 35.
1948:{\displaystyle {dx}/{dt}=-ax(t-1)-2x^{2}-x^{3}}
1583:
127:Center (red) and unstable (green) manifolds of
3446:Journal of Dynamics and Differential Equations
1259:{\displaystyle {\textbf {s}}\to {\textbf {0}}}
3416:
3401:
3368:
3172:
459:{\displaystyle \operatorname {Re} \lambda =0}
3507:
3480:
2870:and the evolution on the center manifold is
2219:{\displaystyle {du_{3}}/{dt}=2(u_{2}-u_{3})}
2139:{\displaystyle {du_{2}}/{dt}=2(u_{1}-u_{2})}
991:can clearly separate these three manifolds.
817:{\displaystyle {\textbf {f}}({\textbf {x}})}
2009:and approximate the time-delayed variable,
1706:The unstable manifold at the origin is the
782:
30:Center manifolds play an important role in
3521:
3492:
3353:
3262:
1536:, and unstable manifold with eigenvalues
110:A much more sophisticated example is the
3377:. Singapore: World Scientific. pp.
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3131:
213:
122:
69:, the stretching direction defining the
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3560:
3486:
3335:
3301:
3137:
1489:, the stable manifold with eigenvalues
41:
3664:
3236:
3202:Applications of centre manifold theory
3120:Normally hyperbolic invariant manifold
3084:{\displaystyle \alpha >0\ (a>4)}
2059:{\displaystyle x(t-1)\approx u_{3}(t)}
1313:to an error of the same order, namely
710:Depending upon the application, other
3575:
3193:
1028:on the center manifold; in formulas,
3199:
1816: > 0 with the negative
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2283:
2265:is then approximated by the system
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14:
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3373:Six Lectures on Dynamical Systems
3321:Iooss, G.; Adelmeyer, M. (1992).
169:{\displaystyle {\dot {x}}=x^{2},}
3290:Guckenheimer & Holmes (1997)
3278:Guckenheimer & Holmes (1997)
1053:
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1021:{\displaystyle {\textbf {y}}(t)}
410:the center subspace, spanned by
406:defines three main subspaces:
361:
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131:equilibrium point of the system
107:of a ball is a center manifold.
3501:
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1761:. It follows that for any real
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947:and a (not necessarily unique)
3653:A more complicated service to
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2066:, by using the intermediaries
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2019:
1996:
1990:
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101:Lagrangian coherent structures
1:
3431:10.1016/S0362-546X(00)85006-3
3187:10.1016/S0362-546X(00)85006-3
3115:Lagrangian coherent structure
2548:{\displaystyle {\bar {s}}(t)}
2229:For parameter near critical,
2002:{\displaystyle u_{1}(t)=x(t)}
118:
3323:Topics in Bifurcation Theory
3264:10.1016/0022-0396(67)90016-2
1584:Alternative backwards theory
717:If the equilibrium point is
256:eigenvalues and eigenvectors
203:{\displaystyle {\dot {y}}=y}
7:
3098:
2263:delay differential equation
2254:{\displaystyle a=4+\alpha }
1860:delay differential equation
1805:{\displaystyle y=Ae^{-1/x}}
1750:{\displaystyle y=Ae^{-1/x}}
1616:
10:
3688:
3632:, corrected fifth printing
2517:and its complex conjugate
1851:{\displaystyle a\approx 4}
1832:that occurs for parameter
3540:10.1016/j.jde.2019.07.021
3510:J. Differential Equations
3466:10.1007/s10884-006-9011-8
3355:10.1017/S0334270000005968
3243:J. Differential Equations
3214:10.1007/978-1-4612-5929-9
2555:, the center manifold is
1607:center manifold reduction
3591:Chicone, Carmen (2010).
3125:
783:Center manifold theorems
626:{\displaystyle \lambda }
524:{\displaystyle \lambda }
427:{\displaystyle \lambda }
412:generalized eigenvectors
260:generalized eigenvectors
258:. The eigenvectors (and
3336:Roberts, A. J. (1988).
3302:Murdock, James (2003).
1757:for some real constant
1621:The Knowledge entry on
1596:in numerical modeling.
1594:backward error analysis
973:{\displaystyle C^{r-1}}
723:Hartman-Grobman theorem
103:. The entire unforced
3406:. Gordon & Breach.
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2003:
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36:multiscale mathematics
25:mathematical modelling
3342:J. Austral. Math. Soc
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2477:
2256:
2221:
2141:
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2004:
1950:
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1625:gives more examples.
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931:{\displaystyle C^{r}}
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895:{\displaystyle C^{r}}
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844:{\displaystyle C^{r}}
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558:
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126:
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3647:A simple service to
3610:Guckenheimer, John;
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2510:{\displaystyle s(t)}
2492:
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2070:
2013:
1962:
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1633:Consider the system
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42:Informal description
16:Mathematical concept
3532:2019JDE...267.7263H
3458:2006JDDE...18..427P
3255:1967JDE.....3..546K
3237:Kelley, A. (1967).
3200:Carr, Jack (1981).
2446:
2428:
757:invariant manifolds
712:invariant subspaces
379:, linearized about
273:Algebraically, let
105:rigid body dynamics
3419:Nonlinear Analysis
3175:Nonlinear Analysis
3152:10.1007/BF00041769
3105:Invariant manifold
3081:
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750:invariant manifold
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414:whose eigenvalues
365:
249:grow exponentially
235:of that system. A
221:
212:
200:
166:
56:
32:bifurcation theory
3672:Dynamical systems
3629:978-0-387-90819-9
3602:978-0-387-35794-2
3578:"Center manifold"
3576:Jack Carr (ed.).
3516:(12): 7263–7312.
3223:978-0-387-90577-8
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864:{\displaystyle r}
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746:unstable manifold
728:center eigenspace
665:(more generally,
563:(more generally,
466:(more generally,
381:equilibrium point
300:
233:equilibrium point
231:is based upon an
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84:structure called
71:unstable manifold
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2777:
2769:
2761:
2760:
2745:
2737:
2728:
2727:
2719:
2716:
2715:
2697:
2692:
2681:
2673:
2672:
2657:
2652:
2641:
2635:
2634:
2626:
2623:
2622:
2598:
2597:
2571:
2570:
2554:
2552:
2551:
2546:
2535:
2534:
2526:
2516:
2514:
2513:
2508:
2481:
2479:
2478:
2473:
2468:
2464:
2445:
2440:
2427:
2422:
2407:
2406:
2380:
2379:
2373:
2369:
2298:
2296:
2288:
2287:
2286:
2276:
2260:
2258:
2257:
2252:
2225:
2223:
2222:
2217:
2212:
2211:
2199:
2198:
2180:
2172:
2167:
2166:
2165:
2145:
2143:
2142:
2137:
2132:
2131:
2119:
2118:
2100:
2092:
2087:
2086:
2085:
2065:
2063:
2062:
2057:
2046:
2045:
2008:
2006:
2005:
2000:
1974:
1973:
1954:
1952:
1951:
1946:
1944:
1943:
1931:
1930:
1888:
1880:
1875:
1857:
1855:
1854:
1849:
1830:Hopf bifurcation
1811:
1809:
1808:
1803:
1801:
1800:
1796:
1756:
1754:
1753:
1748:
1746:
1745:
1741:
1702:
1700:
1699:
1694:
1683:
1682:
1674:
1667:
1666:
1654:
1653:
1645:
1629:A simple example
1576:
1574:
1573:
1568:
1535:
1533:
1532:
1527:
1488:
1486:
1485:
1480:
1472:
1458:
1445:
1443:
1442:
1437:
1426:
1425:
1416:
1415:
1406:
1404:
1396:
1395:
1394:
1384:
1366:
1364:
1363:
1358:
1353:
1352:
1347:
1341:
1340:
1334:
1326:
1325:
1312:
1310:
1309:
1304:
1299:
1298:
1289:
1288:
1279:
1278:
1265:
1263:
1262:
1257:
1255:
1254:
1245:
1244:
1231:
1229:
1228:
1223:
1218:
1217:
1212:
1206:
1205:
1199:
1191:
1190:
1177:
1175:
1174:
1169:
1164:
1163:
1154:
1153:
1144:
1143:
1127:
1123:
1121:
1120:
1115:
1103:
1100:
1094:
1093:
1075:
1074:
1056:
1039:
1027:
1025:
1024:
1019:
1008:
1007:
979:
977:
976:
971:
969:
968:
937:
935:
934:
929:
927:
926:
901:
899:
898:
893:
891:
890:
870:
868:
867:
862:
850:
848:
847:
842:
840:
839:
823:
821:
820:
815:
810:
809:
800:
799:
705:
703:
702:
697:
664:
662:
661:
656:
632:
630:
629:
624:
609:
607:
606:
601:
562:
560:
559:
554:
530:
528:
527:
522:
507:
505:
504:
499:
491:
477:
465:
463:
462:
457:
433:
431:
430:
425:
405:
396:
387:
377:dynamical system
374:
372:
371:
366:
364:
356:
355:
350:
338:
333:
332:
317:
309:
301:
299:
291:
290:
281:
229:dynamical system
209:
207:
206:
201:
193:
192:
184:
175:
173:
172:
167:
162:
161:
149:
148:
140:
90:coordinate space
82:tangent manifold
3687:
3686:
3682:
3681:
3680:
3678:
3677:
3676:
3662:
3661:
3641:
3630:
3620:Springer-Verlag
3603:
3572:
3570:Further reading
3567:
3559:
3555:
3506:
3502:
3485:
3481:
3442:
3438:
3425:(1–8): 91–104.
3415:
3411:
3400:
3396:
3389:
3367:
3363:
3334:
3330:
3319:
3315:
3308:Springer-Verlag
3300:
3296:
3292:, Theorem 3.2.1
3288:
3284:
3276:
3272:
3235:
3231:
3224:
3206:Springer-Verlag
3198:
3194:
3181:(1–8): 91–104.
3171:
3167:
3136:
3132:
3128:
3110:Stable manifold
3101:
3052:
3049:
3048:
3025:
3020:
3019:
3011:
3002:
2998:
2990:
2973:
2968:
2967:
2959:
2940:
2938:
2910:
2908:
2907:
2903:
2890:
2882:
2880:
2878:
2875:
2874:
2848:
2843:
2842:
2834:
2820:
2810:
2809:
2798:
2797:
2782:
2778:
2768:
2750:
2746:
2736:
2730:
2729:
2718:
2717:
2702:
2698:
2682:
2680:
2662:
2658:
2642:
2640:
2637:
2636:
2625:
2624:
2609:
2605:
2587:
2583:
2579:
2575:
2566:
2565:
2563:
2560:
2559:
2525:
2524:
2522:
2519:
2518:
2493:
2490:
2489:
2462:
2461:
2455:
2454:
2448:
2447:
2441:
2436:
2423:
2418:
2402:
2398:
2388:
2384:
2375:
2374:
2367:
2366:
2358:
2353:
2347:
2346:
2341:
2333:
2327:
2326:
2318:
2313:
2306:
2302:
2289:
2282:
2281:
2277:
2275:
2273:
2270:
2269:
2234:
2231:
2230:
2207:
2203:
2194:
2190:
2173:
2168:
2161:
2157:
2153:
2151:
2148:
2147:
2127:
2123:
2114:
2110:
2093:
2088:
2081:
2077:
2073:
2071:
2068:
2067:
2041:
2037:
2014:
2011:
2010:
1969:
1965:
1963:
1960:
1959:
1939:
1935:
1926:
1922:
1881:
1876:
1868:
1866:
1863:
1862:
1837:
1834:
1833:
1826:
1792:
1785:
1781:
1770:
1767:
1766:
1737:
1730:
1726:
1715:
1712:
1711:
1673:
1672:
1662:
1658:
1644:
1643:
1641:
1638:
1637:
1631:
1619:
1602:
1586:
1541:
1538:
1537:
1494:
1491:
1490:
1468:
1454:
1452:
1449:
1448:
1421:
1420:
1411:
1410:
1397:
1390:
1389:
1385:
1383:
1381:
1378:
1377:
1373:
1348:
1343:
1342:
1336:
1335:
1330:
1321:
1320:
1318:
1315:
1314:
1294:
1293:
1284:
1283:
1274:
1273:
1271:
1268:
1267:
1250:
1249:
1240:
1239:
1237:
1234:
1233:
1213:
1208:
1207:
1201:
1200:
1195:
1186:
1185:
1183:
1180:
1179:
1159:
1158:
1149:
1148:
1139:
1138:
1136:
1133:
1132:
1125:
1099:
1083:
1079:
1070:
1069:
1052:
1035:
1033:
1030:
1029:
1003:
1002:
1000:
997:
996:
958:
954:
952:
949:
948:
922:
918:
916:
913:
912:
886:
882:
880:
877:
876:
856:
853:
852:
835:
831:
829:
826:
825:
805:
804:
795:
794:
792:
789:
788:
785:
765:stable manifold
742:stable manifold
670:
667:
666:
638:
635:
634:
618:
615:
614:
568:
565:
564:
536:
533:
532:
516:
513:
512:
487:
473:
471:
468:
467:
439:
436:
435:
419:
416:
415:
394:
392:
390:Jacobian matrix
383:
360:
351:
346:
345:
334:
328:
327:
313:
305:
292:
286:
282:
280:
278:
275:
274:
266:for the stable
237:center manifold
225:center manifold
183:
182:
180:
177:
176:
157:
153:
139:
138:
136:
133:
132:
121:
67:stable manifold
44:
21:center manifold
17:
12:
11:
5:
3685:
3675:
3674:
3660:
3659:
3658:
3657:
3651:
3640:
3639:External links
3637:
3636:
3635:
3628:
3612:Holmes, Philip
3607:
3601:
3588:
3571:
3568:
3566:
3565:
3563:, p. 344.
3553:
3500:
3479:
3452:(2): 427–460.
3436:
3409:
3394:
3387:
3361:
3348:(4): 480–500.
3328:
3313:
3294:
3282:
3270:
3249:(4): 546–570.
3229:
3222:
3192:
3165:
3129:
3127:
3124:
3123:
3122:
3117:
3112:
3107:
3100:
3097:
3080:
3077:
3074:
3071:
3068:
3062:
3059:
3056:
3045:
3044:
3033:
3028:
3023:
3018:
3014:
3010:
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2976:
2971:
2966:
2962:
2956:
2952:
2949:
2946:
2943:
2937:
2934:
2931:
2926:
2922:
2919:
2916:
2913:
2906:
2902:
2896:
2893:
2888:
2885:
2868:
2867:
2856:
2851:
2846:
2841:
2837:
2833:
2830:
2827:
2823:
2819:
2815:
2805:
2802:
2794:
2791:
2788:
2785:
2781:
2775:
2772:
2767:
2764:
2759:
2756:
2753:
2749:
2743:
2740:
2735:
2732:
2731:
2725:
2722:
2714:
2711:
2708:
2705:
2701:
2695:
2691:
2688:
2685:
2679:
2676:
2671:
2668:
2665:
2661:
2655:
2651:
2648:
2645:
2639:
2638:
2632:
2629:
2621:
2618:
2615:
2612:
2608:
2604:
2601:
2596:
2593:
2590:
2586:
2582:
2581:
2578:
2574:
2544:
2541:
2538:
2532:
2529:
2506:
2503:
2500:
2497:
2485:In terms of a
2483:
2482:
2471:
2467:
2460:
2457:
2456:
2453:
2450:
2449:
2444:
2439:
2435:
2431:
2426:
2421:
2417:
2413:
2410:
2405:
2401:
2397:
2394:
2391:
2390:
2387:
2383:
2372:
2365:
2362:
2359:
2357:
2354:
2352:
2349:
2348:
2345:
2342:
2340:
2337:
2334:
2332:
2329:
2328:
2325:
2322:
2319:
2317:
2314:
2312:
2309:
2308:
2305:
2301:
2295:
2292:
2280:
2250:
2247:
2244:
2241:
2238:
2215:
2210:
2206:
2202:
2197:
2193:
2189:
2186:
2183:
2179:
2176:
2171:
2164:
2160:
2156:
2135:
2130:
2126:
2122:
2117:
2113:
2109:
2106:
2103:
2099:
2096:
2091:
2084:
2080:
2076:
2055:
2052:
2049:
2044:
2040:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1972:
1968:
1942:
1938:
1934:
1929:
1925:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1887:
1884:
1879:
1874:
1871:
1847:
1844:
1841:
1825:
1822:
1799:
1795:
1791:
1788:
1784:
1780:
1777:
1774:
1744:
1740:
1736:
1733:
1729:
1725:
1722:
1719:
1704:
1703:
1692:
1689:
1686:
1680:
1677:
1670:
1665:
1661:
1657:
1651:
1648:
1630:
1627:
1623:slow manifolds
1618:
1615:
1601:
1598:
1585:
1582:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1478:
1475:
1471:
1467:
1464:
1461:
1457:
1435:
1432:
1429:
1419:
1409:
1403:
1400:
1388:
1372:
1369:
1356:
1351:
1346:
1333:
1329:
1324:
1302:
1292:
1282:
1248:
1221:
1216:
1211:
1198:
1194:
1189:
1167:
1157:
1147:
1124:for some rate
1113:
1110:
1107:
1101: as
1097:
1092:
1089:
1086:
1082:
1078:
1073:
1068:
1065:
1062:
1059:
1055:
1051:
1048:
1045:
1042:
1038:
1017:
1014:
1011:
985:
984:
967:
964:
961:
957:
945:
925:
921:
909:
889:
885:
860:
838:
834:
813:
803:
784:
781:
780:
779:
771:
768:
708:
707:
695:
692:
689:
686:
683:
680:
677:
674:
654:
651:
648:
645:
642:
622:
611:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
552:
549:
546:
543:
540:
520:
509:
497:
494:
490:
486:
483:
480:
476:
455:
452:
449:
446:
443:
423:
363:
359:
354:
349:
344:
341:
337:
331:
326:
323:
320:
316:
312:
308:
304:
298:
295:
289:
285:
199:
196:
190:
187:
165:
160:
156:
152:
146:
143:
120:
117:
59:Saturn's rings
43:
40:
15:
9:
6:
4:
3:
2:
3684:
3673:
3670:
3669:
3667:
3656:
3652:
3650:
3646:
3645:
3643:
3642:
3631:
3625:
3621:
3617:
3613:
3608:
3604:
3598:
3594:
3589:
3585:
3584:
3579:
3574:
3573:
3562:
3557:
3549:
3545:
3541:
3537:
3533:
3529:
3524:
3519:
3515:
3511:
3504:
3495:
3490:
3483:
3475:
3471:
3467:
3463:
3459:
3455:
3451:
3447:
3440:
3432:
3428:
3424:
3420:
3413:
3405:
3398:
3390:
3388:9789810225483
3384:
3380:
3375:
3374:
3365:
3356:
3351:
3347:
3343:
3339:
3332:
3324:
3317:
3309:
3305:
3298:
3291:
3286:
3280:, Section 3.2
3279:
3274:
3265:
3260:
3256:
3252:
3248:
3244:
3240:
3233:
3225:
3219:
3215:
3211:
3207:
3203:
3196:
3188:
3184:
3180:
3176:
3169:
3161:
3157:
3153:
3149:
3145:
3141:
3134:
3130:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3102:
3096:
3094:
3075:
3072:
3069:
3060:
3057:
3054:
3026:
3016:
3008:
3003:
2999:
2991:
2987:
2983:
2979:
2974:
2964:
2954:
2950:
2947:
2944:
2941:
2935:
2932:
2929:
2924:
2920:
2917:
2914:
2911:
2904:
2900:
2894:
2891:
2886:
2883:
2873:
2872:
2871:
2849:
2839:
2831:
2828:
2821:
2817:
2813:
2800:
2792:
2789:
2786:
2783:
2779:
2773:
2770:
2765:
2762:
2757:
2754:
2751:
2747:
2741:
2738:
2733:
2720:
2712:
2709:
2706:
2703:
2699:
2693:
2689:
2686:
2683:
2677:
2674:
2669:
2666:
2663:
2659:
2653:
2649:
2646:
2643:
2627:
2619:
2616:
2613:
2610:
2606:
2602:
2599:
2594:
2591:
2588:
2584:
2576:
2572:
2558:
2557:
2556:
2539:
2527:
2501:
2495:
2488:
2469:
2465:
2458:
2451:
2442:
2437:
2433:
2429:
2424:
2419:
2415:
2411:
2408:
2403:
2399:
2395:
2392:
2385:
2381:
2370:
2363:
2360:
2355:
2350:
2343:
2338:
2335:
2330:
2323:
2320:
2315:
2310:
2303:
2299:
2293:
2290:
2278:
2268:
2267:
2266:
2264:
2248:
2245:
2242:
2239:
2236:
2227:
2208:
2204:
2200:
2195:
2191:
2184:
2181:
2177:
2174:
2169:
2162:
2158:
2154:
2128:
2124:
2120:
2115:
2111:
2104:
2101:
2097:
2094:
2089:
2082:
2078:
2074:
2050:
2042:
2038:
2034:
2028:
2025:
2022:
2016:
1993:
1987:
1984:
1978:
1970:
1966:
1956:
1940:
1936:
1932:
1927:
1923:
1919:
1916:
1910:
1907:
1904:
1898:
1895:
1892:
1889:
1885:
1882:
1877:
1872:
1869:
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776:slow manifold
772:
769:
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735:slow manifold
731:
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97:saddle points
93:
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53:
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3583:Scholarpedia
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3561:Chicone 2010
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3325:. p. 7.
3322:
3316:
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1611:bifurcations
1603:
1591:
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727:
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401:
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384:
272:
253:
236:
224:
222:
109:
94:
78:second-order
75:
63:tidal forces
57:
52:tidal forces
29:
20:
18:
989:normal form
744:and nearby
129:saddle-node
112:Anosov flow
86:phase space
3523:1906.04420
3494:1804.06998
719:hyperbolic
268:eigenspace
119:Definition
3548:184487247
3160:123743932
3055:α
3000:α
2936:−
2930:α
2829:α
2804:¯
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2734:−
2724:¯
2704:−
2647:−
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2611:−
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2430:−
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2336:−
2321:−
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2201:−
2121:−
2035:≈
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1933:−
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1843:≈
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1732:−
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1500:
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1463:
1247:→
1112:∞
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1088:β
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963:−
911:a unique
875:a unique
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353:∗
322:≈
251:quickly.
189:˙
145:˙
3666:Category
3614:(1997),
3474:59366945
3146:: 1–54.
3140:J. Elast
3099:See also
1617:Examples
942:manifold
940:unstable
906:manifold
633:satisfy
531:satisfy
434:satisfy
3528:Bibcode
3454:Bibcode
3251:Bibcode
1858:in the
219:origin.
3626:
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3385:
3381:–119.
3220:
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1126:β
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388:. The
241:orbits
3544:S2CID
3518:arXiv
3489:arXiv
3470:S2CID
3344:. B.
3156:S2CID
3126:Notes
375:be a
264:basis
245:decay
227:of a
3624:ISBN
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