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A fundamental question is then how one can locate, out of this large family of invariant manifolds, the ones that have the highest influence on the overall system dynamics. These most influential invariant manifolds in the extended phase space of a non-autonomous dynamical systems are known as
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about an equilibrium. In dissipative systems, an invariant manifold based upon the gravest, longest lasting modes forms an effective low-dimensional, reduced, model of the dynamics.
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A. J. Roberts. The utility of an invariant manifold description of the evolution of a dynamical system. SIAM J. Math. Anal., 20:1447–1458, 1989.
798:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=ax-xy\quad {\text{and}}\quad {\frac {\mathrm {d} y}{\mathrm {d} t}}=-y+x^{2}-2y^{2}.}
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C. Chicone. Ordinary
Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34
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Hirsh M.W., Pugh C.C., Shub M., Invariant
Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977
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1557:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=f(x,t),\ x\in \mathbb {R} ^{n},\ t\in \mathbb {R} ,}
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The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.
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Typically, although by no means always, invariant manifolds are constructed as a 'perturbation' of an
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1235:{\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}\left(y-{\tfrac {x^{2}}{1+2a}}\right)}
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1903:{\displaystyle {\mathcal {M}}=\cup _{t\in \mathbb {R} }\phi _{t_{0}}^{t}(M_{0}).}
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that is invariant under the action of the dynamical system. Examples include the
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1158:. One can see this invariance by considering the time derivative
473:, defined on its maximal interval of existence, has its image in
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this parabola is the unstable manifold of the origin. For
1660:{\displaystyle x(t;t_{0},x_{0})=\phi _{t_{0}}^{t}(x_{0})}
926:{\displaystyle {\tfrac {\mathrm {d} x}{\mathrm {d} t}}=0}
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lead to solutions asymptotically approaching the origin.
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Invariant manifolds in non-autonomous dynamical systems
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governed by the pair of coupled differential equations
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1994:
Haller, G. (2015). "Lagrangian
Coherent Structures".
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http://locus.siam.org/SIMA/volume-20/art_0520094.html
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being the solution of the differential equation with
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about the origin, the stable manifold including all
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245:{\displaystyle dx/dt=f(x),\ x\in \mathbb {R} ^{n},}
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1735:{\displaystyle x(t_{0};t_{0},x_{0})=x_{0}}
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953:remains zero. This invariant manifold,
466:{\displaystyle t\mapsto \phi _{t}(x_{0})}
380:{\displaystyle S\subset \mathbb {R} ^{n}}
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106:Learn how and when to remove this message
1288:{\displaystyle y={\tfrac {x^{2}}{1+2a}}}
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1993:
1779:of such a system, any initial surface
1072:{\displaystyle x(0)=0,\ y(0)>-1/2}
303:{\displaystyle x(t)=\phi _{t}(x_{0})}
1571:, whose solutions are of the form
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2016:10.1146/annurev-fluid-010313-141322
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1996:Annual Review of Fluid Mechanics
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1569:non-autonomous dynamical system
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1742:. In the extended phase space
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1455:A differential equation
1242:and finding it is zero on
603:Simple 2D dynamical system
519:{\displaystyle x_{0}\in S}
417:{\displaystyle x_{0}\in S}
345:{\displaystyle x(0)=x_{0}}
664:{\displaystyle x(t),y(t)}
627:, consider the variables
607:For any fixed parameter
1002:{\displaystyle a\geq 0}
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1377:{\displaystyle a<0}
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1314:{\displaystyle a>0}
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132:topological manifold
55:"Invariant manifold"
40:improve this article
2008:2015AnRFM..47..137H
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1351:, more precisely a
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1340:{\displaystyle a=0}
972:{\displaystyle x=0}
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128:invariant manifold
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620:{\displaystyle a}
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559:{\displaystyle S}
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156:inertial manifold
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551:
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546:. In addition,
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144:stable manifold
140:center manifold
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2002:(1): 137–162.
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1928:Hyperbolic set
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1033:
1030:
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1021:
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965:
962:
942:
933:which ensures
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339:
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328:
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19:
9:
6:
4:
3:
2:
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2037:
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1968:
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1953:
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1703:
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1616:
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1600:
1595:
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1587:
1584:
1578:
1570:
1567:represents a
1551:
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1529:
1519:
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1510:
1504:
1501:
1498:
1492:
1489:
1483:
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1458:
1457:
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1433:
1429:
1425:
1422:
1419:
1416:
1410:
1404:
1401:
1398:
1387:
1371:
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1365:
1357:
1354:
1353:slow manifold
1350:
1334:
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1328:
1308:
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1302:
1278:
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1272:
1269:
1263:
1259:
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1187:
1179:
1145:
1122:
1119:
1116:
1113:
1106:
1100:
1096:
1092:
1089:
1082:The parabola
1081:
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27:
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38:Please help
33:verification
30:
124:mathematics
96:August 2012
1944:References
252:with flow
169:Definition
66:newspapers
1861:ϕ
1850:∈
1843:∪
1797:⊂
1762:×
1621:ϕ
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1195:−
1056:−
994:≥
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511:∈
439:ϕ
435:↦
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363:⊂
352:. A set
276:ϕ
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2030:Category
1970:Archived
1922:See also
598:Examples
592:manifold
526:lies in
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979:, is a
80:scholar
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