Generic Hopf Bifurcation from Lines of Equilibria without Parameters

Abstract

Motivated by decoupling effects in coupled oscillators, by viscous shock profiles in systems of nonlinear hyperbolic balance laws, and by binary oscillation effects in discretizations of systems of hyperbolic balance laws, we consider vector fields with a one-dimensional line of equilibria, even in the absence of any parameters. Besides a trivial eigenvalue zero we assume that the linearization at these equilibria possesses a simple pair of nonzero eigenvalues which cross the imaginary axis transversely as we move along the equilibrium line.In normal form and under a suitable nondegeneracy condition, we distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Going beyond normal forms we present a rigorous analysis of both cases. In particular, :-and |-limit sets of nearby trajectories consist entirely of equilibria on the line.

Mathematical Analisys of a Third-order Memristor-based Chua's Oscillator

“…a kind of "bifurcation without parameter", which has yet been described in the literature and mentioned before (see e.g. [2]). This type of bifurcation is an interesting dynamical phenomenon which has also been described for memristor models in [5].…”

Bifurcations and Limit Dynamics in Adaptive Control Systems

“…2 (d). This is called the hyperbolic case in [5,6,7,8]. Similar to the saddle-type transcritical bifurcation, the hyperbolic Hopf bifurcation can only occur in the context of Rokni-Lamooki, Townley & Osinga Bifurcations and limit dynamics 25 adaptive control if another critical point exist on the positive K-axis such that the equilibria to the right of this other critical point are again stable and attract the initial conditions that escape a neighbourhood of the Hopf point.…”

“…We already determined the single-wedge, semi-centre and semi-saddle bifurcations for the closed-loop dynamics in the 1 + 1 case and the spherical-spiral and conical-spiral bifurcations in the 2 + 1 case; see also [16,20]. Secondly, we want to make connections with recent developments of so-called bifurcations without parameters, again by locally classifying the possible closed-loop dynamics; see [5,6,7,8]. Finally, we visualize the dynamics using DsTool [3] and a specially designed extension module [18].…”

“…Such a bifurcation is referred to as a bifurcation without parameters, namely the bifurcation results from the dynamics of the system rather than the system parameters. The idea of bifurcations without parameters was introduced in [7] in the context of a generic Hopf bifurcation from lines of equilibria for two-and three-dimensional centre manifolds in order to study the loss of stability for hyperbolic and elliptic cases. This type of bifurcation is also addressed in [6] where a Bogdanov-Takens bifurcation without parameters was analysed.…”

“…2 (c). This is called the elliptic case in [5,6,7,8]. Note that the cubic polynomial (ω 2 − 1 + ν 2 1 ) (ω + 1) has only one real root at ω = −1 < 0 if ν 1 > 1.…”