[Paper Review] Mathematical remarks on transcritical bifurcation in Hamiltonian systems

Klaus Jaenich|ArXiv.org|Oct 18, 2007

Quantum chaos and dynamical systems4 references3 citations

TL;DR

This paper provides a mathematical framework for analyzing transcritical bifurcations in Hamiltonian systems by focusing on rank-1 bifurcations in symplectic maps. It introduces adapted coordinates and defines regular, definite, and transcritical cross-bifurcations using the Hessian of the Poincaré map's $P$-component, proving that such bifurcations are destroyed under generic perturbations via a non-degeneracy condition on the Poincaré map's Jacobian and perturbation terms.

ABSTRACT

This article is meant as a mathematical appendix or comment on [BT]. We first consider the notion of transcritical bifurcations of fixed points of general area-preserving maps, and then adress some questions related to [BT] on bifurcation in Poincaré maps of 2-dimensional Hamiltonian systems. [BT] M. Brack and K. Tanaka, arXiv:0705.0753

Motivation & Objective

• To formalize the concept of transcritical bifurcations in symplectic maps of 2D Hamiltonian systems, particularly in the context of Poincaré return maps.
• To define and classify rank-1 bifurcations in symplectic families, distinguishing between regular, definite, and transcritical types using Hessian analysis.
• To establish a criterion for the destruction of transcritical bifurcations under perturbations, ensuring structural instability under generic conditions.
• To provide a computable condition based on the Poincaré map's linearized dynamics and perturbation terms to determine whether a bifurcation persists or is destroyed.

Proposed method

• Introduces adapted coordinates in which the Jacobian of the symplectic map at the bifurcation point takes a canonical form with $Q_q - 1 = 0$, $Q_p eq 0$, and $Q_ u = 0$.
• Defines a regular rank-1 bifurcation by requiring the Hessian of $P$ with respect to $(q, u)$ to be non-degenerate at the origin.
• Introduces the concept of a cross-bifurcation as an indefinite regular rank-1 bifurcation, and a transcritical bifurcation as one where $P_{qq} eq 0$.
• Analyzes the fixed point set $F$ as the zero set of $P - p$ restricted to the surface $X = \{Q = q\}$, using the Hessian of this restriction to study local structure.
• Applies Lagrange multipliers to compute the Hessian of the $ u$-coordinate function on the fixed point set, proving that the singularity at the bifurcation point is isolated.
• Derives a criterion for bifurcation destruction using the Poincaré map's linearization and perturbation terms, reducing the problem to a non-vanishing condition on a vector derived from integrals of the perturbation along the unperturbed orbit.

Experimental results

Research questions

• RQ1 Under what conditions does a transcritical bifurcation in a Hamiltonian system persist under small perturbations?
• RQ2 How can the local structure of the fixed point set be characterized near a rank-1 bifurcation point in symplectic maps?
• RQ3 What role does the Hessian of the $P$-component of the Poincaré map play in classifying the type of bifurcation (definite, indefinite, transcritical)?
• RQ4 Can a generic perturbation destroy a transcritical bifurcation, and if so, what is the precise mathematical condition for this destruction?
• RQ5 How can the Poincaré map's Jacobian and perturbation terms be used to algorithmically test for bifurcation destruction in numerical applications?

Key findings

• A transcritical bifurcation in a symplectic family is structurally unstable under generic perturbations, meaning it typically does not persist when the system is slightly altered.
• The fixed point set $F$ near a transcritical bifurcation point is locally a smooth 1-dimensional submanifold, but the $ u$-coordinate function on $F$ has an isolated singularity at the bifurcation point.
• The Hessian of the $ u$-coordinate function restricted to the fixed point set is non-degenerate, implying the singularity is isolated and thus the bifurcation is not part of a continuous family of bifurcations.
• A necessary and sufficient condition for the destruction of a transcritical bifurcation is that the vector $(c_1(T), c_2(T))$ defined by integrals along the unperturbed orbit does not lie in the kernel of the matrix $\left(\begin{array}{cc} \varphi(T)-1 & \psi(T) \\ \dot\varphi(T) & \dot\psi(T)-1 \end{array}\right)$, where $\varphi, \psi$ are solutions of the linearized variational equation.
• For the specific perturbation $F(x,y,p_x,p_y) = x p_y - y p_x$, the destruction criterion reduces to a non-vanishing condition on a vector formed from integrals of $y(t) \dot\psi(t)$ and $y(t) \dot\varphi(t)$ over the period $T$.
• The Poincaré map's Jacobian at the bifurcation point is given by the fundamental solution matrix of the linearized system, enabling explicit computation of the destruction condition via standard ODE theory.