Lyapunov dimension

Consider a dynamical system

$\backslash big(\backslash \{\backslash varphi^t\backslash \}\_\{t\backslash geq0\},\; (U\backslash subseteq\; \backslash mathbb\{R\}^n,\; \backslash |\backslash cdot\backslash |)\backslash big)$, where $\backslash varphi^t$ is the shift operator along the solutions:

$\backslash varphi^t(u\_0)\; =\; u(t,u\_0)$,

of ODE $\backslash dot\{u\}\; =\; f(\{u\})$, $t\; \backslash leq\; 0$,

or difference equation $\{u\}(t+1)\; =\; f(\{u\}(t))$, $t=0,1,...$,

with continuously differentiable vector-function $f$.

Then $D\backslash varphi^t(u)$ is the fundamental matrix of solutions of linearized system

and denote by $\backslash sigma\_i(t,u)\; =\; \backslash sigma\_i(D\backslash varphi^t(u)),\; \backslash \; i\; =\; 1...n$,

singular values with respect to their algebraic multiplicity,

ordered by decreasing for any $u$ and $t$.

The concept of finite-time Lyapunov dimension and related definition of the Lyapunov dimension, developed in the works by N. Kuznetsov,{{Cite journal |first=N.V. |last=Kuznetsov |title=The Lyapunov dimension and its estimation via the Leonov method |journal=Physics Letters A |volume=380 |year=2016 |issue=25–26 |pages=2142–2149 |doi= 10.1016/j.physleta.2016.04.036|bibcode =2016PhLA..380.2142K |arxiv=1602.05410|s2cid=118467839 }}{{Cite journal

|first1=N.V. |last1=Kuznetsov

|first2=G.A. |last2=Leonov

|first3=T.N. |last3=Mokaev

|first4=A. |last4=Prasad

|first5=M.D. |last5=Shrimali

|title=Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

|journal=Nonlinear Dynamics

|volume=92 | issue=2 |year=2018 |pages=267–285 |doi=10.1007/s11071-018-4054-z|arxiv=1504.04723|s2cid=254888463

}} is convenient for the numerical experiments where only finite time can be observed.

Consider an analog of the Kaplan–Yorke formula for the finite-time Lyapunov exponents:

:$$

d_{\rm KY}(\{ {\rm LE}_i(t,u)\}_{i=1}^n)=j(t,u) +

\frac{ {\rm LE}_1(t,u) + \cdots + {\rm LE}_{j(t,u)}(t,u)}

{\rm LE}_{j(t,u)+1}(t,u)

,

:$$

j(t,u) = \max\{m: \sum_{i=1}^m {\rm LE}_i(t,u) \geq 0\},

with respect to the ordered set of finite-time Lyapunov exponents

$\backslash \{\{\backslash rm\; LE\}\_i(t,u)\backslash \}\_\{i=1\}^n\; =\; \backslash \{\backslash frac\{1\}\{t\}\backslash ln\backslash sigma\_i(t,u)\backslash \}\_\{i=1\}^n$ at the point $u$.

The finite-time Lyapunov dimension of dynamical system with respect

to invariant set $K$

is defined as follows

:$$

\dim_{\rm L}(t, K) = \sup\limits_{u \in K}

d_{\rm KY}(\{{\rm LE}_i(t,u)\}_{i=1}^n).

In this approach the use of the analog of Kaplan–Yorke formula

is rigorously justified by the Douady–Oesterlè theorem,{{Cite journal

|first1=A. |last1=Douady

|first2=J. |last2=Oesterle

|title=Dimension de Hausdorff des attracteurs

|journal=Comptes Rendus de l'Académie des Sciences, Série A

|volume=290 | issue=24 |year=1980 |pages=1135–1138}} which proves that for any fixed $t\; >\; 0$

the finite-time Lyapunov dimension for a closed bounded invariant set $K$

is an upper estimate of the Hausdorff dimension:

:$$

\dim_{\rm H} K \leq \dim_{\rm L}(t, K).

Looking for best such estimation

$$

\inf_{t>0} \dim_{\rm L} (t, K)

= \liminf_{t \to +\infty}\sup\limits_{u \in K} \dim_{\rm L}(t,u)

, the Lyapunov dimension is defined as follows:

:$$

\dim_{\rm L} K = \liminf_{t \to +\infty}\sup\limits_{u \in K} \dim_{\rm L}(t,u).

The possibilities of changing the order of the time limit and the supremum over set is discussed, e.g., in.{{Cite journal

|first1=P. |last1=Constantin

|first2=C. |last2=Foias

|first3=R. | last3=Temam

|title=Attractors representing turbulent flows

|journal=Memoirs of the American Mathematical Society

|volume=53 | issue=314 |year=1985 |pages=1–67|doi=10.1090/memo/0314

}}{{Cite journal

|first1=A. |last1=Eden

|first2=C. |last2=Foias

|first3=R. | last3=Temam

|title=Local and global Lyapunov exponents

|journal=Journal of Dynamics and Differential Equations

|volume=3 | issue=1 |year=1991 |pages=133–177 |doi=10.1007/bf01049491|bibcode=1991JDDE....3..133E

|s2cid=119490212

}}

Note that the above defined Lyapunov dimension is invariant under Lipschitz diffeomorphisms.{{Cite journal

|first1=N. |last1=Kuznetsov

|first2=T. |last2=Alexeeva

|first3=G. |last3=Leonov

|title=Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

|journal=Nonlinear Dynamics

|volume=85 | issue=1 |year=2016 |pages=195–201 |doi=10.1007/s11071-016-2678-4|arxiv=1410.2016 |s2cid=254894000

}}

Let the Jacobian matrix $Df(u\_\backslash text\{eq\})$ at one of the equilibria have simple real eigenvalues:

$\backslash \{\backslash lambda\_i(u\_\backslash text\{eq\})\backslash \}\_\{i=1\}^n,\; \backslash lambda\_\{i\}(u\_\backslash text\{eq\})\; \backslash geq\; \backslash lambda\_\{i+1\}(u\_\backslash text\{eq\})$,

then

:$$

\dim_{\rm L}u_\text{eq} = d_{\rm KY}(\{\lambda_i(u_\text{eq})\}_{i=1}^n).

If the supremum of local Lyapunov dimensions on the global attractor, which involves all equilibria, is achieved at an equilibrium point, then this allows one to get analytical formula of the exact Lyapunov dimension of the global attractor (see corresponding Eden’s conjecture).

Following the statistical physics approach and assuming the ergodicity

the Lyapunov dimension of attractor is estimated by

limit value of the local Lyapunov dimension $\backslash lim\_\{t\backslash to+\backslash infty\}\backslash dim\_\{\backslash rm\; L\}\; (t,\; u\_0)$

of a typical trajectory, which belongs to the attractor.

In this case $\backslash \{\backslash lim\backslash limits\_\{t\backslash to+\backslash infty\}\{\backslash rm\; LE\}\_i(t,u\_0)\backslash \}\_\{i\}^n\; =\; \backslash \{\; \{\backslash rm\; LE\}\_i(u\_0)\backslash \}\_1^n$

and $\backslash dim\_\{\backslash rm\; L\}u\_0=\; d\_\{\backslash rm\; KY\}(\backslash \{\; \{\backslash rm\; LE\}\_i(u\_0)\backslash \}\_\{i=1\}^n)=j(u\_0)\; +\; \backslash frac\{\; \{\backslash rm\; LE\}\_1(u\_0)\; +\; \backslash cdots\; +\; \{\backslash rm\; LE\}\_\{j(u\_0)\}(u\_0)\}$

{\rm LE}_{j(u_0)+1}(u_0)

.

From a practical point of view, the rigorous use of ergodic Oseledec theorem,

verification that the considered trajectory $u(t,u\_0)$ is a typical trajectory,

and the use of corresponding Kaplan–Yorke formula is a challenging task

(see, e.g. discussions in{{cite book

|author1= P. Cvitanovic |author2=R. Artuso |author3=R. Mainieri

|author4= G. Tanner |author5= G. Vattay

|name-list-style=amp | year = 2017

| title = Chaos: Classical and Quantum

| publisher = Niels Bohr Institute

| url = http://chaosbook.org/version15/chapters/Lyapunov.pdf#page=6

}}).

The exact limit values of finite-time Lyapunov exponents,

if they exist and are the same for all $u\_0\; \backslash in\; U$,

are called the absolute ones $\backslash \{\backslash lim\backslash limits\_\{t\backslash to+\backslash infty\}\{\backslash rm\; LE\}\_i(t,u\_0)\backslash \}\_\{i\}^n\; =\; \backslash \{\; \{\backslash rm\; LE\}\_i(u\_0)\backslash \}\_1^n\; \backslash equiv\; \backslash \{\; \{\backslash rm\; LE\}\_i\; \backslash \}\_1^n$ and used in the Kaplan–Yorke formula.

Examples of the rigorous use of the ergodic theory for the computation of the Lyapunov exponents and dimension can be found in.{{cite journal

| last1=Ledrappier | first1=F.

| title=Some relations between dimension and Lyapounov exponents

| journal=Communications in Mathematical Physics

| volume=81 | issue=2 | pages=229–238

| year=1981

| doi=10.1007/bf01208896

| bibcode=1981CMaPh..81..229L

| s2cid=122105442

| url=http://projecteuclid.org/euclid.cmp/1103920241

}}{{cite journal

| last1=Benedicks | first1=M.

| last2=Young | first2=L.-S.

| title=Sinai–Bowen–Ruelle measures for certain Henon maps

| journal=Inventiones Mathematicae

| volume=112 | issue=1 | pages=541–576

| year=1993 | doi=10.1007/bf01232446

| bibcode=1993InMat.112..541B

}}{{cite book | first1= Nikolay | last1=Kuznetsov |

first2=Volker | last2=Reitmann | year = 2021| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|

publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}

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Category:Dynamical systems