Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.     

Bifurcations and transition states

Bifurcations of reaction channels are related to valley-ridge inflection points and it is examined what happens when these do not coincide with transition states. Under such conditions there result bifurcating regions. There exist a number of different prototypes for such regions which are discussed explicitly on the basis of the pertinent Taylor expansions. When bifurcations occur close enough to transition states then there result bifurcating transition regions. An example for a bifurcating transition region is exhibited which is obtained from a quantum mechanical ab initio calculation for the ring opening of cyclopropylidene to aliene. In general there exist no orthogonal trajectory patterns which could serve as simplified models for channel bifurcations.Operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences     

Existence of periodic orbits and shift-invariant curve sequences near multiple homoclinics简

In this paper, the complicated dynamics is studied near a double homoclinic loops with bellows configuration for general systems. For the non-twisted multiple homoclinics, the existence of periodic orbit with the specified route and the existence of shift-invariant curve sequences defined on the cross sections of multiple homoclinics corresponding to any specified one-side infinite sequences are given. In addition, the existence regions are also located.     

Solitary wave solutions and kink wave solutions for a generalized KDV-mKDV equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized KDV-mKDV equation. Under some parameter conditions, their explicit expressions are obtained.     

Quasi-periodic bifurcations and “amplitude death” in low-dimensional ensemble of van der Pol oscillators

The dynamics of the four dissipatively coupled van der Pol oscillators is considered. Lyapunov chart is presented in the parameter plane. Its arrangement is discussed. We discuss the bifurcations of tori in the system at large frequency detuning of the oscillators. Here are quasi-periodic saddle-node, Hopf and Neimark–Sacker bifurcations. The effect of increase of the threshold for the “amplitude death” regime and the possibilities of complete and partial broadband synchronization are revealed.     

Mathematics of Philosophy or Philosophy of Mathematics?

This article examines recent attempts to gain insight into philosophical paradoxes through using NDS models employing iterated difference equations and resulting phase portraits and escape time diagrams. The temporal nature of such models is contrasted with an alternative approach based on the a-temporal and non-dynamical construct of a lattice. Finally, there is a discussion of how such strategies for understanding paradox transcend the realm of empirical research and enter territory in the philosophy of mathematics.     

Pseudo-billiards

A new class of Hamiltonian dynamical systems with two degrees of freedom and kinetic energy of the form T = c₁|p₁| + c₂|p₂| (called “pseudo-billiards”) is studied. For any kind of interaction, the canonical equations can always be integrated on sequential time intervals; i.e. in principle all the trajectories can be found explicitly.

Depending on the potential, a dynamical system of this class can either be completely integrable or behave just as a usual non-integrable Hamiltonian system with two degrees of freedom: in its phase space there exist invariant tori, stochastic layers, domains of global chaos, etc. Pseudo-billiard models of both the types are considered.

If a potential of a pseudo-billiard system has critical points (equilibria), then trajectories close to these points (“loops”) can exist; they can be treated as images of self-localized objects with finite duration. Such a model (with quartic potential) is also studied.