[Paper Review] Lyapunov Coefficients for Degenerate Hopf Bifurcations

Jorge Sotomayor, Luis Fernando Mello+1 more|ArXiv.org|Sep 25, 2007

Advanced Differential Equations and Dynamical SystemsMathematics9 references19 citations

TL;DR

This paper derives algebraic expressions for the first through fourth Lyapunov coefficients in degenerate Hopf bifurcations using computer-assisted symbolic computation. It establishes orbital normal forms for codimensions 1 to 4, showing that when the first four Lyapunov coefficients vanish up to $ l_4 \neq 0 $, the system exhibits a codimension-4 Hopf bifurcation with an $ |w|^8 $-nonlinear term, enabling classification of complex bifurcation scenarios in dynamical systems.

ABSTRACT

In this paper are studied the codimensions one, two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients. Algebraic expressions obtained with computer assisted calculations are displayed.

Motivation & Objective

• To systematically compute the first through fourth Lyapunov coefficients for degenerate Hopf bifurcations, where standard methods fail due to higher-order degeneracies.
• To provide explicit algebraic expressions for these coefficients using computer-assisted symbolic calculations, particularly via Mathematica 5.
• To characterize the dynamic behavior near codimension-1 to -4 Hopf bifurcations through orbital normal forms involving higher-order nonlinearities.
• To establish a framework for analyzing complex bifurcation structures in systems with multiple parameters, such as the Watt governor with a spring.
• To extend existing theory by deriving the fourth Lyapunov coefficient, which is absent in the current literature.

Proposed method

• The method employs Taylor expansions of the vector field up to ninth order, using the Jacobian and multilinear forms of higher-order derivatives of $ F({\bf x}) $.
• Center manifold reduction is applied to reduce the system to a two-dimensional complex normal form in terms of $ w \in \mathbb{C} $, with $ w $ representing the complex amplitude of the critical mode.
• The coefficients $ h_{jk} $ of the center manifold immersion $ H(w, \bar{w}) $ are computed recursively by solving linear systems derived from the normal form equation $ H_w w' + H_{\bar{w}} \bar{w}' = F(H(w, \bar{w})) $.
• The Lyapunov coefficients $ l_1, l_2, l_3, l_4 $ are extracted from the normal form $ w' = (\eta + i\omega_0)w + \tau w|w|^2 + \nu w|w|^4 + \sigma w|w|^6 + l_4 w|w|^8 $, with $ l_k $ computed from the multilinear terms of $ F $.
• Symbolic computation in Mathematica 5 is used to derive and verify the long algebraic expressions for $ l_1 $ to $ l_4 $, with intermediate steps published online.
• Theoretical results are validated through the construction of orbital normal forms for codimension-1 to -4 Hopf bifurcations, with $ l_k \neq 0 $ at the $ k $-th degeneracy level.

Experimental results

Research questions

• RQ1 What are the explicit algebraic expressions for the first through fourth Lyapunov coefficients in degenerate Hopf bifurcations?
• RQ2 How can the dynamic behavior of a system be classified near a codimension-4 Hopf bifurcation point?
• RQ3 What is the structure of the orbital normal form when the first four Lyapunov coefficients vanish but the fifth is non-zero?
• RQ4 How can symbolic computation be used to derive high-order Lyapunov coefficients that are not available in the existing literature?
• RQ5 What are the necessary and sufficient conditions for a Hopf bifurcation to be of codimension 4?

Key findings

• The first through fourth Lyapunov coefficients are derived algebraically using symbolic computation, with expressions provided for systems with up to ninth-order Taylor terms.
• The fourth Lyapunov coefficient $ l_4 $ is explicitly computed and shown to be non-zero at a codimension-4 Hopf bifurcation point, enabling classification of the bifurcation structure.
• For a codimension-4 Hopf bifurcation, the orbital normal form is $ w' = (\eta + i\omega_0)w + \tau w|w|^2 + \nu w|w|^4 + \sigma w|w|^6 + l_4 w|w|^8 $, with $ l_4 \neq 0 $, confirming the presence of an $ |w|^8 $-nonlinearity.
• The genericity condition that the map $ \mu \mapsto (\eta, l_1, l_2, l_3) $ is regular at $ \mu = 0 $ ensures that the unfolding of the bifurcation is transversal and structurally stable.
• The results are applicable to the Watt governor system with a spring, where higher-order degeneracies can now be analyzed using the derived coefficients.
• The paper provides the first complete derivation of the fourth Lyapunov coefficient in the literature, filling a critical gap in the theory of degenerate Hopf bifurcations.