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Definition

Consider the differential equation d x / d t = f ( x ) ,   x ∈ R n , {\displaystyle dx/dt=f(x),\ x\in \mathbb {R} ^{n},} with flow x ( t ) = ϕ t ( x 0 ) {\displaystyle x(t)=\phi _{t}(x_{0})} being the solution of the differential equation with x ( 0 ) = x 0 {\displaystyle x(0)=x_{0}} . A set S ⊂ R n {\displaystyle S\subset \mathbb {R} ^{n}} is called an invariant set for the differential equation if, for each x 0 ∈ S {\displaystyle x_{0}\in S} , the solution t ↦ ϕ t ( x 0 ) {\displaystyle t\mapsto \phi _{t}(x_{0})} , defined on its maximal interval of existence, has its image in S {\displaystyle S} . Alternatively, the orbit passing through each x 0 ∈ S {\displaystyle x_{0}\in S} lies in S {\displaystyle S} . In addition, S {\displaystyle S} is called an invariant manifold if S {\displaystyle S} is a manifold.³

Examples

Simple 2D dynamical system

For any fixed parameter a {\displaystyle a} , consider the variables x ( t ) , y ( t ) {\displaystyle x(t),y(t)} governed by the pair of coupled differential equations

d x d t = a x − x y and d y d t = − y + x 2 − 2 y 2 . {\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}.}

The origin is an equilibrium. This system has two invariant manifolds of interest through the origin.

• The vertical line x = 0 {\displaystyle x=0} is invariant as when x = 0 {\displaystyle x=0} the x {\displaystyle x} -equation becomes d x d t = 0 {\displaystyle {\tfrac {\mathrm {d} x}{\mathrm {d} t}}=0} which ensures x {\displaystyle x} remains zero. This invariant manifold, x = 0 {\displaystyle x=0} , is a stable manifold of the origin (when a ≥ 0 {\displaystyle a\geq 0} ) as all initial conditions x ( 0 ) = 0 ,   y ( 0 ) > − 1 / 2 {\displaystyle x(0)=0,\ y(0)>-1/2} lead to solutions asymptotically approaching the origin.
• The parabola y = x 2 / ( 1 + 2 a ) {\displaystyle y=x^{2}/(1+2a)} is invariant for all parameter a {\displaystyle a} . One can see this invariance by considering the time derivative d d t ( y − x 2 1 + 2 a ) {\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}\left(y-{\tfrac {x^{2}}{1+2a}}\right)} and finding it is zero on y = x 2 1 + 2 a {\displaystyle y={\tfrac {x^{2}}{1+2a}}} as required for an invariant manifold. For a > 0 {\displaystyle a>0} this parabola is the unstable manifold of the origin. For a = 0 {\displaystyle a=0} this parabola is a center manifold, more precisely a slow manifold, of the origin.
• For a < 0 {\displaystyle a<0} there is only an invariant stable manifold about the origin, the stable manifold including all ( x , y ) ,   y > − 1 / 2 {\displaystyle (x,y),\ y>-1/2} .

Invariant manifolds in non-autonomous dynamical systems

A differential equation

d x d t = f ( x , t ) ,   x ∈ R n ,   t ∈ R , {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=f(x,t),\ x\in \mathbb {R} ^{n},\ t\in \mathbb {R} ,}

represents a non-autonomous dynamical system, whose solutions are of the form x ( t ; t 0 , x 0 ) = ϕ t 0 t ( x 0 ) {\displaystyle x(t;t_{0},x_{0})=\phi _{t_{0}}^{t}(x_{0})} with x ( t 0 ; t 0 , x 0 ) = x 0 {\displaystyle x(t_{0};t_{0},x_{0})=x_{0}} . In the extended phase space R n × R {\displaystyle \mathbb {R} ^{n}\times \mathbb {R} } of such a system, any initial surface M 0 ⊂ R n {\displaystyle M_{0}\subset \mathbb {R} ^{n}} generates an invariant manifold

M = ∪ t ∈ R ϕ t 0 t ( M 0 ) . {\displaystyle {\mathcal {M}}=\cup _{t\in \mathbb {R} }\phi _{t_{0}}^{t}(M_{0}).}

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 Lagrangian Coherent Structures.⁴

See also

• Hyperbolic set
• Lagrangian coherent structure
• Spectral submanifold

References

1. Hirsh M.W., Pugh C.C., Shub M., Invariant Manifolds, Lect. Notes. Math., 583, Springer, Berlin — Heidelberg, 1977 ↩

2. 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. http://locus.siam.org/SIMA/volume-20/art_0520094.html Archived 2008-08-20 at the Wayback Machine http://locus.siam.org/SIMA/volume-20/art_0520094.html ↩

3. C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006, p.34 ↩

4. Haller, G. (2015). "Lagrangian Coherent Structures". Annual Review of Fluid Mechanics. 47 (1): 137–162. Bibcode:2015AnRFM..47..137H. doi:10.1146/annurev-fluid-010313-141322. /wiki/Bibcode_(identifier) ↩