Claude Baesens

Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Abstract Many systems exhibit behaviour which can be described by three coupled oscillators. Provided the coupling is not too strong, such systems can be modelled by maps of the two-torus to itself. In this paper we describe and explain those aspects of the bifurcation diagram for two-parameter families of torus maps ƒ that involve change of mode-locking type. We introduce the concepts of partial and full mode-locking . For the coupled oscillators, these notions correspond to the presence of one or two rational relations between the frequencies, respectively. Numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations . In order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time-1 map of a flow on the torus. Then we analyse those condimension-1 and -2 bifurcations for flows on the torus which change the set of frequency ratios. We find a large variety of bifurcation diagrams, in particular many involving homotopically non-trivial saddle connections. Next we ask how the picture changes when the time-1 maps of a family of flows are perturbed to a general family of diffeomorphisms. We find toroidal chaos in the neighbourhood of all these bifurcations, meaning that there exist orbits which perform a pseudo-random sequence of rotations in different directions around the torus. The corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk. Finally, we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps, which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation.

Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation

Abstract We investigate analytically the effect on a period-doubling cascade of slowly sweeping the bifurcation parameter, by means of asymptotic calculations. First we analyse the behaviour of the orbits when sweeping through one period-doubling bifurcation. There is still an effective bifurcation called a dynamic bifurcation , but which is delayed. We compute the delay. Sweeping in forward and backward directions yields qualitatively different results. The presence of noise in the system, even of small intensity, turns out to play a key role. We analyse the various behaviours as a function of the relative sizes of the sweep rate and the noise intensity. In order to tackle the sweep through a whole cascade we extend the renormalisation theory for period doubling of one-dimensional maps to non-autonomous maps. The particular we show that a slow linear of the parameter results in a new unstable eigenvalue which provides a scaling law for the sweep rate. Taking also into account the effect of noise, we then derive an asymptotic scaling for the successive period-doubling bifurcations. As a result of the sweep (and the noise), only a finite number of doublings can be observed: we estimate this maximum number. We also show how to estimate the location of the real bifurcation of the real points from results of experiments with non-zero sweep velocity. When the functions involved to be analytic we show that the delay of a period-doubling bifurcation due to sweep depends crucially on their degree of smoothness. In both the analytic and non-analytic cases, we give insight into the geometry of some invariant curves reminiscent of an accordion. Our analysis explains numerical and experimental results on the problem.

Gradient dynamics of tilted Frenkel-Kontorova models

Complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics. AMS classification number: 58F22

A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel–Kontorova chains

We introduce a novel partial order on the state space for cooperative networks of units with overdamped second order dynamics, which is preserved under the forward dynamics. Among other things, we use it to prove that for chains of particles in a tilted periodic potential with overdamped inertial dynamics, for each class of spatially periodic states if there are no rotationally ordered equilibria then there is a globally attracting periodically sliding solution.

Cantori for multiharmonic maps

Abstract We compute all the cantori and their gap and turnstile structures, for area-preserving twist maps near non-degenerate anti-integrable limits with arbitrarily many wells per period. The results imply a rich bifurcation diagram for cantori of families of maps containing degenerate anti-integrable limits. We conjecture the structure of this bifurcation diagram in the case of the two-harmonic family.

Improved proof of existence of chaotic polaronic and bipolaronic states for the adiabatic Holstein model and generalizations

We simplify the proofs and extend the scope of the main theorems of Aubry, Abramovici and Raimbault on existence of polaronic and bipolaronic states near the anti-integrable limit for the adiabatic Holstein model of electron-phonon systems and generalizations. In particular, we obtain sharper estimates and allow exponential lattices, arbitrary square-integrable electron hopping, and some amounts of inhomogeneity and dispersion.

Gevrey series and dynamic bifurcations for analytic slow-fast mappings

The surprising effects of slow sweep of a parameter in a family of dynamical systems exhibiting bifurcations are known as dynamic bifurcations. In this paper we study dynamic bifurcations in the context of analytic slow-fast mappings of the form Fv(x, lambda )=(f(x, lambda ), lambda +v) for small v in the neighbourhood of an analytic curve x=y( lambda ) of fixed points of f(., lambda ). Firstly, we construct a formal power series expansion U in powers of v for invariant curves lambda to U( lambda ,v) close to the curve of fixed points, and prove it is Gevrey-1. Secondly we show how the behaviour of the orbits, and in particular the existence of a delay for the dynamic bifurcation, can be derived as a direct consequence of the analytical properties of this formal series.

The one to two-hole transition for cantori

Abstract The gaps in a cantorus come in orbits, which we call “holes”. In the space of parameters ( a, b ) for the “two-harmonic” reversible area-preserving twist map family, y′=y− a 2π sin2πx− b 4π sin4πx , x ′= x + y ′ (mod1), application of the idea of the anti-integrable limit establishes that there must be one to two-hole transitions for cantori of all irrational rotation numbers. We have numerically located a curve in parameter space across which a one-hole cantorus of golden rotation number develops a second hole, and we present results on scaling behaviour of several quantities near this interesting transition.

Simple Resonance Regions of Torus Diffeomorphisms

This paper discusses resonance regions for two parameter families of diffeomorphisms and vector fields on the two dimensional torus. Resonance regions with at most two resonant periodic orbits (in the discrete case) or two equilibrium points (for flows) are studied. We establish global geometric properties of these regions with topological arguments.

Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit*

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering.The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems.