Anomalous scaling behavior in Takens-Bogdanov bifurcations

Physica D: Nonlinear Phenomena, 2000

This paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurcation with a homoclinic orbit of the Hopf equilibrium. Our work is motivated by our previous analysis of a Hodgkin-Huxley neuron model which possesses a subcritical Hopf bifurcation . In this model, the Hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium, that is, the Hopf bifurcation appears to be close to a homoclinic bifurcation as well. This model of the lateral pyloric (LP) cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike-frequency adaptation , in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent. The presence of a subcritical Hopf bifurcation in this model was one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non-oscillatory solution. The analysis presented here explains the apparent proximity of homoclinic and Hopf bifurcations. We also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical Hopf bifurcation that may be close to a codimension two subcritical Hopf-homclinic bifurcation.

arXiv (Cornell University), 2024

We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time dependent parameter p(t). The combined dynamics can be considered as a dynamical systems where p is a slowly evolving parameter. Here, we investigate settings where the parameter features an overshoot. It crosses the bifurcation threshold for some finite duration t e and up to an amplitude R, before it returns to its initial value. We denote the overshoot as safe when the dynamical system returns to its initial state. Otherwise, one encounters runaway trajectories (tipping), and the overshoot is unsafe. For shallow overshoots (small R) safe and unsafe overshoots are discriminated by an inverse square-root border, t e ∝ R -1/2 , as reported in earlier literature. However, for larger overshoots we here establish a crossover to another power law with an exponent that depends on the asymptotics of p(t). For overshoots with a finite support we find that t e ∝ R -1 , and we provide examples for overshoots with exponents in the range [-1, -1/2]. All results are substantiated by numerical simulations, and it is discussed how the analytic and numeric results pave the way towards improved risks assessments separating safe from unsafe overshoots in climate, ecology and nonlinear dynamics.

recercat.net

We consider a Hamiltonian of three degrees of freedom and a family of periodic orbits with a transition from stability to complex instability, such that there is an irrational collision of the Floquet eigenvalues of opposite sign. We analyze the local dynamics and the bifurcation phenomena linked to this transition. We study the resulting Hamiltonian Hopf-like bifurcation from an analytical point of view by means of normal forms. The existence of a bifurcating family of 2D tori is derived and both cases (direct and inverse bifurcation) are described.

Scholarpedia, 2006

Preface These lecture notes are, like many websites, " under construction ". Some parts are based on material obtained from Dr. (Utrecht University). I am grateful to these colleagues for providing me their material.

Computers & Structures, 2004

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order-m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation (m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.

New Advances in Celestial Mechanics and Hamiltonian Systems: HAMSYS 2001, 2004

We consider a Hamiltonian of three degrees of freedom and a family of periodic orbits with a transition from stability to complex instability, such that there is an irrational collision of the Floquet mulipliers of opposite sign. We analyze the local dynamics and the bifurcation phenomena linked to this transition. We study the resulting Hamiltonian Hopf-like bifurcation from an analytical point of view by means of normal forms. The existence of a bifurcating family of 2D tori is derived and both cases (direct and inverse bifurcation) are described.

International Journal of Bifurcation and Chaos

A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despite the eigenvalues of the fixed points before and after the bifurcation being within the unit circle. In this paper, we show that similar bifurcation phenomena also occur when the stable orbit in question is of a higher periodicity or is chaotic. Accordingly, we propose a generalized definition of dangerous bifurcation suitable for any kind of attracting sets. We report two types of dangerous border collision bifurcations and show that, in addition to the originally reported mechanism typically involving singleton saddle cycles, there exists one more situation where the basin boundary is formed by a repelling closed invariant curve.

International Journal of Bifurcation and Chaos, 2008

A generic stationary instability that arises in quasi-reversible systems is studied. It is characterized by the confluence of three eigenvalues at the origin of complex plane with only one eigenfunction. We characterize the dynamics through the normal form that exhibits in particular, Shilnikov chaos, for which we give an analytical prediction. We construct a simple mechanical system, Shilnikov particle, which exhibits this quasi-reversal instability and displays its chaotic behavior.

Physical Review Letters

We study a two-dimensional low-dissipation dynamical system with a control parameter that is swept linearly in time across a transcritical bifurcation. We investigate the relaxation time of a perturbation applied to a variable of the system and we show that critical slowing down may occur at a parameter value well above the bifurcation point. We test experimentally the occurrence of critical slowing down by applying a perturbation to the accessible control parameter and we find that this perturbation leaves the system behavior unaltered, thus providing no useful information on the occurrence of critical slowing down. The theoretical analysis reveals the reasons why these tests fail in predicting an incoming bifurcation.