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I have basic questions to understand the invariant sets of dynamical systems. Let me define a dynamical system $\left\{ {T, X, \phi^{t}}\right\}$. Here an orbit with a starting value $x_{0}$ is defined by $or(x_{0})=\left\{ {x\in X: x=\phi^{t}x_{0}, t\in T }\right\}$.

An invariant set $S\subset X$ of this dynamical system consist of $x_{0}\in S$ which implies $\phi^{t}x_{0} \in S$ for all $t\in T$. Because of these definitions any individual orbit is an invariant set.

Could you please give me other simple examples for invariant sets and non-invariant sets? Another silly question of mine is: orbits are ordered subsets of state spaces.Why do we not define them $S\subset X \rightarrow X$? Is it not possible to start in $S$ and leave it through the evolution operator?

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  • $\begingroup$ Is $T$ a group or monoid? Is $S$ an invariant subset of $X$ if $\phi^t(S)=S$ for all $t\in T$, or is it when $\phi^t(S)\subset S$ for all $t\in T$? $\endgroup$ – Dan Rust Aug 14 '13 at 13:35
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I'll assume $T$ is a group. An invariant set can be partitioned in to its orbits (as any set with a group action on it can be) and also any union of orbits is an invariant set. In particular, there is a bijective correspondence between the set of all sets of orbits of the system and the set of invariant subsets of the system.

Some simple examples would be:

  • the empty set as a subset of $X$.
  • any fixed point of the group action.
  • the entire space X
  • the eigenspaces (and unions thereof) of any invertible $n\times n$ matrix acting on $\mathbb{R}^n$ by linear transformation.
  • any circles centered at the origin of the plane for the system given by rotation about the origin.
  • any stright line through the origin of the place for the system given by expansion by a linear factor.
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You are right in saying that every orbit is an invariant set. However, in general (but not always) when we talk about invariant sets, we talk about sets of nonzero measure. Examples are easy to find. If you take a 2D nonlinear ODE with a sink type fixed point, it will have a basin of attraction that will form an invariant set.

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Fixed and periodic points come in flavors: elliptic, parabolic, and hyperbolic. For the latter, there are two very important invariant sets called the stable and unstable manifolds. If $x$ is the fixed (periodic) point, the set $\{y: \lim_{t \rightarrow \infty} \phi^{t}y = x\}$ is called the stable manifold. The unstable manifold is $\{y: \lim_{t \rightarrow \infty} \phi^{-t}y = x\}$.

Part of the reason these are important comes from points that are either homoclinic, they are elements of both sets for a particular fixed or periodic orbit, or heteroclinic, they are elements of the stable manifold of one orbit and the unstable manifold of the other. A lot of chaotic behavior is either caused by transverse homoclinic points (homoclinic points for which the manifolds cross instead of smoothly meeting), or is very close to a system created that way.

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