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Lyapunov Dimension for Dynamical Systems in Euclidean Spaces

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Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 38))

Abstract

Nowadays there is a number of surveys and theoretical works devoted to Lyapunov exponents and Lyapunov dimension, however most of them are devoted to infinite dimensional systems or rely on special ergodic properties of a system. At the same time the provided illustrative examples are often finite dimensional systems and the rigorous proof of their ergodic properties can be a difficult task. Also the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. This chapter is devoted to the finite dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985–90), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method, 1991), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension. In this chapter a concise overview of the classical results is presented, various definitions of Lyapunov exponents and Lyapunov dimension are discussed. An effective analytical method for the estimation of the Lyapunov dimension is presented, its application to self-excited and hidden attractors of well-known dynamical systems is demonstrated, and analytical formulas of exact Lyapunov dimension are obtained.

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Notes

  1. 1.

    Considering additional properties of the dynamical system and the singular value function, one could get \(\lim _{t \rightarrow +\infty }\) instead of \(\liminf _{t \rightarrow +\infty }\), but we do not need it for our further consideration.

  2. 2.

    If \(f : \mathbb {R}^n \rightarrow \mathbb {R}\) is a measurable subadditive function, then for every \(u \in \mathbb {R}^n\) there exists the limit \(\lim \limits _{t\rightarrow \infty }{f(tu) \over t}\).

  3. 3.

    This is not a dimension in a rigorous sense (see, e.g. [35, 36, 43]).

  4. 4.

    While inf and sup give the same values for \(\omega _{d}(D\varphi ^t(u))\) in (6.28) and (6.29), for \(\inf _{t > 0}\max \limits _{u \in \mathcal{K}}\omega _{d}(D\varphi ^t(u))\) we need consider

    \(\sup \{d \in [0,n]: \forall \widetilde{d} \in [0,d] \ \inf _{t> 0}\max \limits _{u \in \mathcal{K}}\omega _{\widetilde{d}}(D\varphi ^t(u)) \ge 1 \} = \inf \{d \in [0,n]: \inf _{t > 0}\max \limits _{u \in \mathcal{K}}\omega _{d}(D\varphi ^t(u)) < 1 \}. \).

  5. 5.

    If there exists \(\lim _{t \rightarrow +\infty }\max \limits _{u \in \mathcal{K}}\omega _{d}(D\varphi ^t(u))=0\) then it is interesting to study the existence of a critical point \(u_0 \in \mathcal{K}\) such that \( \lim _{t \rightarrow +\infty }\max \limits _{u \in \mathcal{K}}\omega _{d}(D\varphi ^t(u)) = \limsup _{t \rightarrow +\infty }\omega _{d}(D\varphi ^t(u_0)), \) and to compare \( \inf \{d \in [0,n]: \sup \limits _{u \in \mathcal{K}}\limsup _{t \rightarrow +\infty }\omega _{d}(D\varphi ^t(u)) < 1\}\) or \( \sup \limits _{u \in \mathcal{K}}\limsup _{t \rightarrow +\infty }\sup \{d \in [0,n]: \omega _{d}(D\varphi ^t(u)) \ge 1\}\) with \(\dim _{H}\mathcal{K}\). Remark, it is clear that \( \lim _{t \rightarrow +\infty }\max \limits _{u \in \mathcal{K}}\omega _{d}(D\varphi ^t(u)) \ge \sup \limits _{u \in \mathcal{K}}\limsup _{t \rightarrow +\infty }\omega _{d}(D\varphi ^t(u)) \). From (6.30) it follows the existence of a critical point \(u_{L}(t)\) such that \( \dim _{{}_L}(\varphi ^t,u_{L}(t)) = \max _{u \in \mathcal{K}}\dim _{{}_L}(\varphi ^t,u). \) Taking into account (6.32) we can consider a sequence \(t_k \rightarrow +\infty \) such that \(\dim _{{}_L}(\varphi ^{t_k},u_{L}(t_k))\) is monotonically converging to \(\inf _{t\ge 0}\max _{u \in \mathcal{K}}\dim _{{}_L}(\varphi ^t,u)\). Since \(\mathcal{K}\) is a compact set, we can consider a subsequence \(t_m = t_{k_m} \rightarrow +\infty \) such that there exists a limit critical point \(u^{cr}_{L}\): \(u_{L}(t_m) \rightarrow u^{cr}_{L} \in \mathcal{K}\) as \(t_m \rightarrow +\infty \). Thus we have \(\dim _{{}_L}(\varphi ^{t_m},u_{L}(t_m)) \searrow \dim _{{}_L}(\{\varphi ^t\}_{t\ge 0},\mathcal{K})\) and \(u_{L}(t_m) \rightarrow u^{cr}_{L} \in \mathcal{K}\) as \(m \rightarrow +\infty \).

  6. 6.

    In [15] Constantin, Foias, Temam stated that if \( \sup _{u \in \mathcal{K}}\limsup _{t\rightarrow +\infty } \big (\omega _{d}(D\varphi ^t(u))\big )^{1/t} < 1 \) or \( \limsup _{t\rightarrow +\infty } \sup _{u \in \mathcal{K}}\big (\omega _{d}(D\varphi ^t(u))\big )^{1/t}<1, \) then \(\dim _{{}_H}\mathcal{K}\le d\). In [25] Eden considered the value \(\dim _{{}_L}^\mathrm{DO}(\mathcal{K}) = \inf \{ d>0:\, \sup _{u \in \mathcal{K}} \omega _{d}(D\varphi ^t(u)) \text{ converges } \text{ to } \text{ zero } \text{ exponentially } \text{ as } t \rightarrow \infty \}\) and called it the Douady-Oesterlé dimension of \(\mathcal{K}\).

  7. 7.

    Comparing the expressions in the definitions (6.33) and (6.40), remark that we can change \({1 \over t}\) in (6.40) to another scalar positive monotonically decreasing function q(t) such that \(\inf _{t>0}q(t)\max _{u \in \mathcal{K}}\omega _d(D\varphi ^{t}(u))= \lim _{t\rightarrow +\infty }q(t)\max _{u \in \mathcal{K}}\omega _d(D\varphi ^{t}(u))\). The last relation is important from a computational point of view.

  8. 8.

    We add “of singular value” to distinguish this definition from other definitions of Lyapunov exponents; if the differences in the definitions are not significant for the presentation, we use the term “Lyapunov exponents” or “LEs”.

  9. 9.

    Often they are called Lyapunov characteristic exponents (LCE) [60]. In [75] these values are defined with the opposite sign and called characteristic exponents at the point u.

  10. 10.

    For example [46], for the matrix \( u(t)=\left( \begin{array}{cc} 1 &{} g(t)-g^{-1}(t) \\ 0 &{} 1 \\ \end{array} \right) \) we have the following ordered values: \( {{\,\mathrm{\nu ^{L}}\,}}_1 = \mathrm{max}\big (\limsup \limits _{t \rightarrow +\infty }{1 \over t}\log |g(t)|, \limsup \limits _{t \rightarrow +\infty }{1 \over t}\log |g^{-1}(t)|\big ), {{\,\mathrm{\nu ^{L}}\,}}_2 = 0\); \( {{\,\mathrm{\nu }\,}}_{1,2} = \mathrm{max, min} \big ( \limsup \limits _{t \rightarrow +\infty }{1 \over t}\log |g(t)|, \limsup \limits _{t \rightarrow +\infty }{1 \over t}\log |g^{-1}(t)| \big ). \).

  11. 11.

    Let \({{\,\mathrm{\nu }\,}}_1(t,u) = (e^u)^t\), \({{\,\mathrm{\nu }\,}}_2(t,u) = ({1 \over 2}(1-u))^t\) for all \(u \in \mathcal{K}=[0,1]\). Thus \({{\,\mathrm{\nu }\,}}_1(u)={{\,\mathrm{\widetilde{\nu }}\,}}_1(u) = u\), \({{\,\mathrm{\nu }\,}}(u)={{\,\mathrm{\widetilde{\nu }}\,}}_2 = \log (1-u)-\log 2\); \({{\,\mathrm{\widetilde{\nu }}\,}}_1(\mathcal{K}) = 1\), \({{\,\mathrm{\widetilde{\nu }}\,}}_2 =-1-\log 2\). Here \(u^{cr}(1)=1\): \({{\,\mathrm{\widetilde{\nu }}\,}}_1(1)=={{\,\mathrm{\widetilde{\nu }}\,}}_1(\mathcal{K})=1\); \(u^{cr}(2)=0\): \({{\,\mathrm{\widetilde{\nu }}\,}}_1(0)+{{\,\mathrm{\widetilde{\nu }}\,}}_2(0)={{\,\mathrm{\widetilde{\nu }}\,}}_1(\mathcal{K})+{{\,\mathrm{\widetilde{\nu }}\,}}_2(\mathcal{K})=-\log 2\). Then \(\sup _{u \in [0,1]}\dim _{{}_L}^\mathrm{KY}(\{{{\,\mathrm{\widetilde{\nu }}\,}}_i(u)\}_1^2)={u \over \log 2-\log (1-u)} < 1+{1 \over 1+\log 2}= \dim _{{}_L}^\mathrm{KY}(\{{{\,\mathrm{\widetilde{\nu }}\,}}_i(\mathcal{K})\}_1^2)\).

  12. 12.

    The expression in (6.64) corresponds to the expressions considered in [54] for \(p(u)=Dh(u)\), [55] and [56] for \(Q(u)=Dh(u)\).

  13. 13.

    In [78] and in Chap. 7 it is interpreted as changes of Riemannian metrics.

  14. 14.

    The numerical search of hidden attractors can be complicated by the small size of the basin of attraction with respect to the considered set of parameters \(p \in P\) and subset of the phase space \(\mathcal{U}_0\subset \mathcal{U}\): following [9, 103], the attractor may be called a rare attractor if the measure \(\mu \) of the basin of attractors \(\beta (\mathcal{K}_p)\) for the considered set of parameters \(p \in P\) is small with respect to the considered part of the phase space \(\mathcal{U}_0 \subset \mathcal{U}\), i.e. \({\int _{p \in P} \mu (\beta (\mathcal{K}_p)\cap \mathcal{U}_0) \over \mu (\mathcal{U}_0)}<< 1\). Also computational difficulties may be caused by the shape of basin of attraction, e.g. by Wada and riddled basins.

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    Nikolay Kuznetsov & Volker Reitmann

  2. Faculty of Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Nikolay Kuznetsov

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Kuznetsov, N., Reitmann, V. (2021). Lyapunov Dimension for Dynamical Systems in Euclidean Spaces. In: Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Emergence, Complexity and Computation, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-50987-3_6

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