Shadowing lemma for flows

 Kenneth James Palmer (2009), Scholarpedia, 4(4):7918. doi:10.4249/scholarpedia.7918 revision #91761 [link to/cite this article]
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Curator: Kenneth James Palmer

Shadowing describes the situation where a true orbit of a dynamical system such as a differential equation or a map lies uniformly near (that is, shadows) a pseudo (or approximate) orbit. The shadowing lemma for discrete dynamical systems has been described in Barreto (2008). In this entry the analogous results for continuous dynamical systems, that is, flows, are described.

Two shadowing lemmas for flows

The shadowing lemma for flows is rather more complicated than that for diffeomorphisms. For flows it is not even clear what the definition of a pseudo orbit should be. Should it be a sequence or a function of time? Shadowing lemmas have been proved for both possibilities.

As with diffeomorphisms, shadowing is a property of hyperbolic sets, which are defined as follows (conf. Hasselblatt and Pesin (2008)).

Definition of hyperbolic set. A compact set $$S\subset R^n$$ is said to be hyperbolic for the autonomous differential equation $\tag{1} \dot x=F(x),\quad x \in I\!\! R^n$

with associated flow $$\phi^t$$ if

(i) $$F(x)\neq 0$$ for all $$x$$ in $$S\ ;$$

(ii) $$S$$ is invariant, that is, $$\phi ^{t} (S)=S$$ for all $$t\ ;$$

(iii) there is a splitting

$I\!\! R^{n} =E^{0}(x)\oplus E^{s} (x)\oplus E^{u} (x) \;\;\; {\rm for} \; x\in S$ with $$E^{0}(x)={\rm span}\{ F(x)\} \; {\rm and} \; {\rm dim}\, E^{s}(x), \; {\rm dim} \, E^{u}(x)$$ constant, such that for all $$t$$ and $$x$$ in $$S$$ $D\phi ^{t}(x)(E^{s}(x))=E^{s}(\phi^{t}(x)), \;\; D\phi^{t}(x)(E^{u}(x))=E^{u}(\phi^{t}(x)),$ and such that there are positive constants $$K_{1}\ ,$$ $$K_{2}\ ,$$ $$\alpha_{1}\ ,$$ $$\alpha _{2}$$ with the property that for all $$t\geq 0$$ and $$x$$ in $$S$$ $|D\phi ^{t}(x)\xi |\leq K_{1} e^{-\alpha_{1}t} |\xi| \;\;\; {\rm for} \; \xi \in E^{s}(x),$ $|D\phi ^{-t}(x)\xi|\leq K_{2} e^{-\alpha_{2} t}|\xi| \;\;\; {\rm for} \; \xi \in E^{u} (x).$

First we state a "discrete" version of the shadowing lemma (see also the entry shadowing). To this end we first define a discrete pseudo orbit and the appropriate notion of shadowing.

Definition of discrete pseudo orbit. If $$\delta$$ is a positive number, a sequence of points $$\{y_k\}^{\infty}_{k=-\infty}$$ is said to be a discrete $$\delta$$ pseudo orbit for Eq. (1) if there is a sequence $$\{h_k\}^{\infty}_{k=-\infty}$$ of positive times with $$\sup h_k<\infty\ ,$$ $$\inf h_k>0$$ such that $|y_{k+1}-\phi^{h_k}(y_k)|\leq\delta \quad{\rm for}\quad k\in Z\!\!\!\!Z.$

Definition of shadowing. A discrete $$\delta$$ pseudo orbit $$\{y_{k}\}_{k=-\infty}^{\infty}$$ of Eq. (1) with associated times $$\{h_k\}_{k=-\infty}^{\infty}$$ is said to be $$\varepsilon$$-shadowed by a true orbit of Eq. (1) if there are sequences $$\{x_k\}_{k=-\infty}^{\infty}$$ and $$\{t_k\}_{k=-\infty}^{\infty}$$ such that $$x_{k+1}=\phi ^{t_k}(x_{k})$$ and $|x_k-y_k| \leq \varepsilon \;\;{\rm and}\;\; | t_k-h_k|\leq \varepsilon$ for all $$k\ .$$

Now we state a "discrete" shadowing lemma for flows (Palmer (2000)). Note that to ensure uniqueness of the shadowing orbit, an "anchor" condition is imposed.

Discrete Shadowing Lemma. Let $$F:I\!\! R^n\mapsto I\!\! R^n$$ be a $$C^1$$ vector field and suppose $$S$$ is a compact hyperbolic set for Eq. (1). Let $$\{y_k\}_{k=-\infty}^{\infty}$$ be a discrete $$\delta$$ pseudo orbit of Eq. (1) in $$S$$ with associated times $$\{h_k\}_{k=-\infty}^{\infty}$$ satisfying $0 < h_{min} \leq h_k \leq h_{max}.$ Then there exist positive constants $$\delta_0$$ and $$M$$ depending only on $$F\ ,$$ $$S\ ,$$ $$h_{min}$$ and $$h_{max}$$ such that if $$\delta\leq\delta_0\ ,$$ the discrete $$\delta$$ pseudo orbit $$\{y_k\}^{\infty}_{k=-\infty}$$ is $$\varepsilon$$-shadowed by a unique true orbit $$\{x_k\}_{k=-\infty}^{\infty}$$ of Eq. (1) with $$\varepsilon =M\delta$$ and such that $\langle\, x_k-y_k, F(y_k)\,\rangle\,=0\;\;{\rm for\; all}\;\;k.$

Next we state a "continuous" shadowing lemma (Pilyugin (1999)). In this theorem (this is just one possible definition of a "continuous" pseudo orbit) a $$\delta$$ pseudo orbit is a real vector-valued function $$y(t)$$ such that for any real $$\tau$$ $|y(t+\tau)-\phi^t(y(\tau))| \le\delta,\quad |t|\le 1.$

Continuous Shadowing Lemma. Let $$F:I\!\! R^n\mapsto I\!\! R^n$$ be a continuously differentiable vector field and suppose $$S$$ is a compact hyperbolic set for Eq. (1). Then there exists a neighbourhood $$W$$ of $$S$$ and numbers $$\delta_0\ ,$$ $$L>0$$ such that for any $$\delta$$ pseudo orbit $$y(t)$$ in $$W$$ with $$\delta\le\delta_0$$ there is a point $$x$$ and a homeomorphism $$\alpha:R\to R^n$$ such that $$\alpha(t)-t$$ has Lipschitz constant $$L\delta$$ and for all $$t$$ $|y(t)-\phi^{\alpha(t)}(x)| \le L\delta.$

Chaos

Shadowing has many applications in the theory of dynamical systems especially to the existence of topological conjugacies (Kuznetsov (2007)). An application to chaos is described here. In the flow context, the analogous theorem to Smale's (Smale and Shub (2007)) is that of Sil'nikov, which describes the chaotic dynamics in the neighbourhood of a transversal homoclinic orbit associated with a hyperbolic periodic orbit of a flow. One can apply Smale's theorem to an appropriately defined return map to verify chaotic behaviour in the neighbourhood of such a transversal periodic-to-periodic homoclinic orbit. However we can also use the discrete shadowing theorem above to verify the chaotic dynamics. Corresponding to a sequence of symbols, a pseudo orbit is constructed and then the shadowing lemma is used to prove the existence of a nearby true orbit. In this fashion we can describe all the orbits which stay in a neighbourhood of the homoclinic orbit (Palmer (2000)).

Since a computer generated orbit of Eq. (1) is a (finite) discrete $$\delta$$ pseudo orbit, it is natural to use shadowing techniques to try to prove the existence of a nearby true orbit. However, in general, such orbits may not be hyperbolic. So the theorems stated above are not directly applicable. Nonetheless, even though these orbits are not hyperbolic, they are often close to being so and even though the theory is not directly applicable it does give us a guide in our study of the systems. For instance, in these systems infinite pseudo orbits may not be shadowed by true orbits. However, they may still be shadowed for long times by true orbits. The key idea is the construction of a right inverse of small norm for a linear operator similar to the one used for infinite time shadowing. The choice of this right inverse is guided by the infinite time case: one takes the formula for the inverse in the infinite time case and truncates it appropriately. Similar techniques can be used to give computer-assisted proofs of the existence of true periodic (resp. homoclinic) orbits near computed approximated periodic (resp. homoclinic) orbits (conf. Palmer (2000) and Pilyugin (1999)).

Generalizations

Shadowing lemmas can also be proved for ordinary differential equations on manifolds (Katok and Hasselblatt (1995)). Shadowing lemmas have been developed for nonautonomous systems where the time dependence is not periodic, skew product flows, differential inclusions and ordinary differential equations in Banach space. Pilyugin (1999) has proved a shadowing lemma for structurally stable flows with equilibria. Other types of shadowing have been studied such as limit shadowing. Shadowing has also been studied in the context of metric spaces. A flow on a compact metric space is said to have the pseudo orbit tracing property if the shadowing lemma holds for it. The consequences of this assumption have received much study.