Dimension Theory for Ordinary Differential Equations

In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu-Morioka system.

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“…We say that System (20) is a transformed Shimizu-Morioka system. The following assertion is valid [23][24][25][26][27].…”

Section: Lyapunov Dimension Of the Shimizu-morioka Systemmentioning

confidence: 84%

Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

Леонов

Alexeeva

Kuznetsov

2015

Entropy

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“…Consider singular values of the matrix X(t) [Hahn, 1967, Katok & Hasselblat, 1995, Boichenko et al, 2005.…”

Section: Lyapunov Exponents and Singular Valuesmentioning

confidence: 99%

Time-Varying Linearization and the Perron Effects

Леонов

Kuznetsov

2007

Int. J. Bifurcation Chaos

231

171

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“…In general convergence by the decomposition method is not guaranteed, so that the solution shall be tested by an appropriate criterion. Since standard convergence criteria do not apply in a straight forward manner for the present case, we resort to a method which is based on the reasoning of Lyapunov (Boichenko et al (2005)). While Lyapunov introduced this conception in order to test the influence of variations of the initial condition on the solution, we use a similar procedure to test the stability of convergence while starting from an approximate (initial) solution R 0 (the seed of the recursive scheme).…”

Section: Resultsmentioning

confidence: 99%