[Paper Review] Lyapunov coefficients for Hopf bifurcations in systems with piecewise smooth nonlinearity

Miriam Steinherr Zazo, Jens D. M. Rademacher|arXiv (Cornell University)|Jun 12, 2020

Advanced Differential Equations and Dynamical Systems17 references6 citations

TL;DR

This paper derives explicit formulas for the first Lyapunov coefficient in nonsmooth dynamical systems with piecewise smooth nonlinearities, particularly of the form $ u_i|u_j| $, generalizing classical Hopf bifurcation theory. It establishes a normal form for planar systems using averaging theory and proves the existence of a unique periodic orbit branch at bifurcation, with the coefficient $ \sigma\#$ determining super- or subcriticality despite non-smoothness, applicable to models like ship maneuvering and shimmying wheels.

ABSTRACT

Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothing of the vector field. We focus on nonsmooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part. In addition, we discuss some codimension-one degeneracies and apply the results to a model of a shimmying wheel.

Motivation & Objective

• To generalize the classical first Lyapunov coefficient to systems with piecewise smooth nonlinearities, particularly of the form $ u_i|u_j| $, which arise in mechanical and control systems.
• To determine the direction of Hopf bifurcation (supercritical vs. subcritical) in the absence of smoothness, where standard formulas fail.
• To develop a normal form theory for planar and higher-dimensional systems with Lipschitz-continuous, piecewise smooth nonlinearities.
• To analyze codimension-one degeneracies, including the case $ \sigma\# = 0 $, and extend results to systems with additional nonhyperbolic modes.
• To apply the theory to a concrete model of a shimmying wheel and validate results via numerical continuation.

Proposed method

• Formulates a general $ n $-dimensional ODE system with linear part $ A(\mu)u $ and piecewise smooth nonlinear term $ G(u) $, where $ G $ is Lipschitz and $ C^2 $ away from switching surfaces.
• Applies Lyapunov-Schmidt reduction to the periodic boundary value problem to derive a reduced bifurcation equation, referred to as the 'direct method'.
• Uses averaging theory to derive a compact normal form for planar systems with $ p_+ = -p_- = 1 $, yielding explicit expressions for $ \sigma\# $ and the second Lyapunov coefficient $ \sigma_2 $.
• Derives integral formulas for the first Lyapunov coefficient $ \sigma\# $ in terms of system parameters and integrals over $ [0, 2\pi] $, with explicit evaluation via trigonometric symmetry and Taylor expansion.
• Considers the case of three imaginary eigenvalues (one zero, one complex pair) and proves that either no periodic orbits or two curves bifurcate, depending on a coefficient combination.
• Performs detailed symbolic computation of coefficients $ \gamma_{ij}, \delta_{ij} $ via ODEs derived from Taylor expansion of the vector field, with asymptotic expansion in $ \mu $ near bifurcation.

Experimental results

Research questions

• RQ1 How can the first Lyapunov coefficient be generalized to systems with piecewise smooth nonlinearities of the form $ u_i|u_j| $, where the nonlinearity is not twice differentiable?
• RQ2 What determines the direction of the Hopf bifurcation (supercritical vs. subcritical) in such nonsmooth systems, given that standard smooth bifurcation theory does not apply?
• RQ3 How do smooth quadratic and cubic terms interact with nonsmooth terms in determining the criticality of the bifurcation when $ \sigma\# = 0 $?
• RQ4 What happens to the bifurcation structure when the linearization has a triple imaginary eigenvalue structure (one zero, one complex pair), and how does this affect the number of bifurcating periodic orbits?
• RQ5 Can a normal form be derived for planar systems with piecewise smooth nonlinearities that captures the dynamics near the Hopf bifurcation with explicit, computable coefficients?

Key findings

• The first Lyapunov coefficient $ \sigma\# $ is derived as $ \sigma\# = \frac{1}{2\pi} \int_0^{2\pi} \chi_2(\phi) \, d\phi $, with explicit evaluation yielding $ \int_0^{2\pi} \chi_2(\phi) \, d\phi = \frac{4}{3} \sigma\# $, where $ \chi_2 $ combines smooth and nonsmooth terms.
• For the planar system with $ p_+ = -p_- = 1 $, the first Lyapunov coefficient is given by $ \sigma\# = \frac{1}{3} (a_{11} + b_{22}) + \frac{1}{6} (a_{21} + b_{12}) $, with $ a_{ij}, b_{ij} $ from the nonlinear terms $ f, g $, and the sign of $ \sigma\# $ determines bifurcation direction.
• When $ \sigma\# = 0 $, the second Lyapunov coefficient $ \sigma_2 $ becomes critical and is given by $ \sigma_2 = \frac{1}{4} \int_0^{2\pi} \chi_3(\phi) \, d\phi - \frac{1}{4} \int_0^{2\pi} \chi_2(\phi) \Omega_1(\phi) \, d\phi $, with $ \Omega_1 $ arising from the nonsmooth part.
• In the case of three imaginary eigenvalues (one zero, one complex pair), the system bifurcates into two periodic orbit curves if a certain coefficient combination is nonzero, otherwise no periodic orbits emerge.
• The normal form for the planar case is derived using averaging theory, yielding a compact expression for the radial component: $ \dot{u} = \mu u + \sigma\# u|u| $, with $ \sigma\# $ explicitly computable from system parameters.
• Numerical continuation of the model (1.5)–(1.6) confirms the theoretical predictions: $ \sigma\# > 0 $ yields subcritical bifurcation (unstable periodic orbits), $ \sigma\# < 0 $ yields supercritical (stable periodic orbits), matching the theoretical bifurcation diagrams in Figure 2.