Stefan Liebscher

GENERIC HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS III: BINARY OSCILLATIONS

We consider discretized systems of hyperbolic balance laws. Decoupling of the flow, associated with a central difference scheme, can lead to binary oscillations — even and odd numbered grid points, separately, provide time-evolutions of two distinct, different, separate profiles. Investigating the stability of this decoupling phenomenon, we encounter Hopf-like bifurcations in the absence of parameters. With some computer-algebra assistance, we describe the qualitative behavior near these bifurcation points. In particular we observe distinct even/odd profiles which oscillate periodically in time and, for arbitrarily fine discretization, exhibit preferred, nonzero phase relationships between adjacent discretization points.

Generic hopf bifurcation from lines of equilibria without parameters: II. systems of viscous hyperbolic balance laws

We investigate viscous shock profiles of the Riemann problem for systems of hyper- bolic balance laws. Even strictly hyperbolic flux terms together with a nonoscillating kinetic part can lead to oscillating viscous shock profiles. They appear near a Hopf-like bifurcation point of the traveling wave equation.

Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

We consider cosmological models of Bianchi type. In particular, we are interested in the α-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity.We prove the existence of a codimension-one family of solutions that limit, for t → −∞, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.

Travelling Waves in Systems of Hyperbolic Balance Laws

We discuss several results on the existence of continuous travelling wave solutions in systems of conservation laws with nonlinear source terms.

Plane Kolmogorov flows and Takens–Bogdanov bifurcation without parameters: The singly reversible case

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we use the Kirchgassner reduction to obtain a spatial dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. In particular we describe the complete set B of all small bounded solutions. In the classical Kolmogorov case, B consists of periodic profiles, homoclinic pulses and a heteroclinic front-back pair. This is a consequence of the symmetry of the external force.

Poincaré-Andronov-Hopf Bifurcation

Without parameters, no periodic orbits bifurcate. Depending on the drift condition, two cases appear. Both are discussed in this chapter.

Double-Hopf Bifurcation

The final bifurcation of codimension 2 is characterized by the intersection of 2 curves of Poincare-Andronov-Hopf points on a two-dimensional surface of equilibria. As we shall see, the drift direction at the Hopf lines play an important role. In the case of a parameter-dependent fixed line of equilibria, drifts at both Hopf-lines can be opposite and spiraling orbits appear, see Sect. 12.1. In the generic case with a plane of equilibria without parameters, both drifts are transverse and generate a Lyapunov function. Only heteroclinic orbits arise. See Sect. 12.2.

Application: Decoupling in Networks

Networks are an important structure in many applications ranging from chemistry and biology to engineering. Pattern formation in networks has caught an ever growing interest in recent years [57]. The main focus is usually the synchronization of the cells of the network. Here, we study the converse phenomenon: under suitable symmetry assumptions, networks can decouple and continua of states emerge where all couplings cancel out each other and several pairs of cells can have arbitrary phase differences.

Degenerate Poincaré-Andronov-Hopf Bifurcation

In this chapter we study the Poincare-Andronov-Hopf bifurcation without parameters, see Chap. 5, with an additional degeneracy of the drift or transversality due to an additional parameter or an additional dimension of the primary manifold of equilibria.

Zero-Hopf Bifurcation

In this chapter we study a bifurcation characterized by a zero eigenvalue and a pair of nonzero purely imaginary eigenvalues of the linearization transverse to a plane of equilibria. It turns out that instead we can study a one-parameter family of lines in a system depending on one parameter. Indeed, the rescaled normal form (11.6) is the same in both cases.