Polyrhythmic multifrequency synchronization in coupled oscillators with exactly solvable attractors

While stable polyrhythmic multifrequency [Formula: see text] dynamics has traditionally been an important element in music performance, recently, this type of dynamics has been discovered in the human brain in terms of elementary temporal neural activity patterns. In this context, the canonical-dissipative systems framework is a promising modeling approach due to its two key features to bridge the gap between classical mechanics and life sciences, on the one hand, and to provide analytical or semi-analytical solutions, on the other hand. Within this framework, a family of testbed models is constructed that exhibit [Formula: see text] multifrequency limit cycle attractors describing two components oscillating with frequencies at [Formula: see text] ratios and stable polyrhythmic phase relationships. The attractors are super-integrable due to the existence of third invariants of motion for all [Formula: see text] ratios. Strikingly, all [Formula: see text] attractors models satisfy the same generic bifurcation diagram. The study generalizes earlier work on super-integrable systems, on the one hand, and canonical-dissipative limit cycle oscillators, on the other hand. Explicit worked-out models for 1[Formula: see text]:[Formula: see text]4 and 2[Formula: see text]:[Formula: see text]3 frequency ratios are presented.

Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

Piecewise-Linear Map for Studying Border Collision Phenomena in DC/AC Converters

Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border points in 1D). Obviously, the multitude of control strategies applied in the modern power electronics leads to different maps belonging to this class of models. However, in this paper we show that several models can be studied using the same piecewise-linear approximation, so that the bifurcation phenomena which can be observed in this approximation are generic for many models. Based on the results obtained before for piecewise-smooth models with different kinds of nonlinearities resulting from the corresponding control strategies, in the present paper we discuss the generic bifurcation patterns in the underlying piecewise-linear map.

On damping in the vicinity of critical points

The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated. Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added. Oscillators with saddle–node, pitchfork and transcritical bifurcations are shown analytically to exhibit several interesting characteristics in the free decay response near a bifurcation. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. A transcritical bifurcation was selected because it may be used to represent generic bifurcation behaviour. It is shown that the damping ratio can be used to predict a change in the stability with respect to changing system parameters.

THE EXISTENCE AND CLASSIFICATION OF SYNCHRONY-BREAKING BIFURCATIONS IN REGULAR HOMOGENEOUS NETWORKS USING LATTICE STRUCTURES

For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.