[Paper Review] Finite-time and exact Lyapunov dimension of the Henon map

Н. В. Кузнецов, Г. А. Леонов+1 more|arXiv (Cornell University)|Dec 3, 2017

Nonlinear Dynamics and Pattern Formation58 references3 citations

TL;DR

This paper presents an adaptive algorithm for computing finite-time Lyapunov dimension in the Hénon map, extending analytical estimates and proving the exact Lyapunov dimension formula for self-excited attractors to cases with negative shrinking parameter $ b \in (-1, 0) $. It confirms that the Lyapunov dimension of self-excited attractors does not exceed that of the unstable equilibrium $ O_- $, with exact values derived via symmetrized Jacobian and localization techniques, yielding $ \dim_L \approx 1.495 $ for $ a=1.4, b=0.3 $.

ABSTRACT

This work is devoted to further consideration of the Henon map with negative values of the shrinking parameter and the study of transient oscillations, multistability, and possible existence of hidden attractors. The computation of the finite-time Lyapunov exponents by different algorithms is discussed. A new adaptive algorithm for the finite-time Lyapunov dimension computation in studying the dynamics of dimension is used. Analytical estimates of the Lyapunov dimension using the localization of attractors are given. A proof of the conjecture on the Lyapunov dimension of self-excited attractors and derivation of the exact Lyapunov dimension formula are revisited.

Motivation & Objective

• To extend the analytical estimation of Lyapunov dimension to the Hénon map with negative shrinking parameter $ b \in (-1, 0) $, where transient dynamics and multistability are prominent.
• To develop and apply a new adaptive algorithm for finite-time Lyapunov dimension computation to study the dynamics of dimension in transient and chaotic regimes.
• To prove the conjecture that the Lyapunov dimension of a self-excited attractor does not exceed the Lyapunov dimension of the unstable equilibrium whose unstable manifold intersects the attractor's basin.
• To derive an exact formula for the Lyapunov dimension of self-excited attractors using symmetrized Jacobian matrices and localization of attractors in phase space.
• To investigate the coexistence of multiple attractors (multistability) and the existence of hidden attractors in the Hénon map under varying parameters.

Proposed method

• An adaptive algorithm is proposed for finite-time Lyapunov dimension computation, dynamically adjusting to the system's transient behavior to improve accuracy and convergence.
• The Lyapunov dimension is estimated using the symmetrized Jacobian matrix $ SJ S^{-1} $, where $ S = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{|b|} \end{pmatrix} $, enabling the computation of singular values $ \sigma_1, \sigma_2 $ that bound the dimension.
• The exact Lyapunov dimension is derived via the Kaplan-Yorke formula: $ \dim_L = 1 + \frac{1}{1 - \frac{\ln |b|}{\ln \sigma_1((x_-,x_-), S)}} $, evaluated at the unstable equilibrium $ O_- $.
• Attractor localization is achieved using Feit's analytical bounds: $ |x| \leq \max\left\{ -r, m, \sqrt{b(m+a)+a-r} \right\} $, where $ r $ and $ m $ are bounds on the attractor's projection.
• The invariance of Lyapunov dimension under linear transformations is leveraged to equate the dimension at the equilibrium to the dimension of the attractor if the maximum local dimension is achieved there.
• The method is validated numerically and analytically for the canonical Hénon map parameters $ a=1.4, b=0.3 $, with convergence to $ \dim_L \approx 1.495 $.

Experimental results

Research questions

• RQ1 Does the conjecture that the Lyapunov dimension of a self-excited attractor is bounded by the dimension at its associated unstable equilibrium hold for negative values of the shrinking parameter $ b $?
• RQ2 Can an adaptive algorithm reliably compute the finite-time Lyapunov dimension in transient and chaotic regimes of the Hénon map with $ b < 0 $?
• RQ3 What analytical bounds can be derived for the Lyapunov dimension using attractor localization and symmetrized Jacobian singular values?
• RQ4 How does the Lyapunov dimension of the Hénon attractor change when $ b $ is negative, and does it remain consistent with the Kaplan-Yorke formula?
• RQ5 What is the exact value of the Lyapunov dimension for the self-excited chaotic attractor in the Hénon map with $ a=1.4, b=0.3 $?

Key findings

• The exact Lyapunov dimension of the self-excited chaotic attractor in the Hénon map with $ a=1.4, b=0.3 $ is $ \dim_L = 1 + \frac{1}{1 - \frac{\ln 0.3}{\ln \sigma_1((x_-,x_-), S)}} \approx 1.495 $, derived from the equilibrium $ O_- $.
• The conjecture that the Lyapunov dimension of a self-excited attractor does not exceed the dimension at the associated unstable equilibrium is proven for $ b < 0 $, extending prior results.
• The adaptive finite-time Lyapunov dimension algorithm enables accurate tracking of dimension dynamics during transient oscillations, improving reliability in chaotic regimes.
• For $ a=1.4, b=0.3 $, the analytical upper bound on the Hausdorff dimension using Feit's localization is $ \dim_H K \leq 1.5319 $, slightly higher than the exact value.
• The Lyapunov dimension remains invariant under linear transformations, allowing the use of symmetrized Jacobians to derive exact formulas without requiring full attractor reconstruction.
• Multistability and the coexistence of self-excited and hidden attractors are confirmed for $ a=1.49, b=-0.138 $, demonstrating the relevance of the method in complex dynamics.