# bifurcations

## Simple and complex dynamics in the model of evolution of two populations coupled by migration with non-overlapping generations

Purpose is to study the mechanisms leading to genetic divergence (stable genetic differences between two adjacent populations). We considered the following classical model situation. Populations are panmictic with Mendelian rules of inheritance. The action of natural selection (differences in fitness) on each of population is the same and is determined by the genotypes of only one diallel locus. We assume that adjacent generations do not overlap and genetic transformations can be described by a discrete time model.

## Cycles-canards and torus-canards in a weakly inhomogeneous ensemble of FitzHugh–Nagumo neurons with excitatory synaptic couplings

The purpose of this work is to study the dynamics of a weakly inhomogeneous ensemble of three FitzHugh–Nagumo neurons with excitatory synaptic couplings. To single out main types of canard solutions of the system and obtain the regions in parameter space the solutions exist in. Methods. In this paper the dynamics of autonomous systems are studied by using methods based on geometric singular perturbation theory. To study the dynamics of non-autonomous systems we develop an approximate approach and use numerical methods such as obtaining of Poincare maps. Results.

## Equations with the Fermi–Pasta–Ulam and dislocations nonlinearity

Issue. The class of Fermi–Pasta–Ulam equations and equations describing dislocations are investigated. Being a bright representative of integrable equations, they are of interest both in theoretical constructions and in applied research. Investigation methods. In the present work, a model combining these two equations is considered, and local dynamic properties of solutions are investigated. An important feature of the model is the fact that the inﬁnite set of characteristic numbers of the equation linearized at zero consists of purely imaginary values.

## Bifurcations of one-parameter families of steady state regimes in model of a filtrational convection

Results of numerical investigation of bifurcations of one-parameter families of steady state regimes in a planar filtrational convection problem are presented. Galerkin’s method is applied for approximation of partial differential equations. As a result of the cosymmetry existence there are curves of equilibria with the hidden parameter. The algorithm of calculation of such curves is described. This algorithm can be applied to analyze systems with nonisolated sets of equilibria.

## Subharmonic resonance in a system of two dissipative coupled van der Pol oscillators with external force

The problem of the excitation of two coupled oscillators is discussed in the case of the simple subharmonic resonance between the external force and eigen-frequencies of the oscillators. The corresponded phase equation is obtained. We showed that the form of the synchronization tongue and transformation of the region of the two-, three-frequency tori by varying the parameter of the coupling between the oscillators is significantly different from the case of the main resonance.

## Dynamics of two nonlinearly coupled nonidentical Lang–Kobayshi oscillators

One-parameter study of system of two nonlinearly coupled nonidentical Lang– Kobayshi oscillators is presented. The time delay influence on oscillation regimes in the system is studied. The posibility of periodic and quasiperiodic oscillations is shown. Variation of delay time leads to bifurcations and an alternation of periodic and quasiperiodic oscillations. Quasiperiodic oscillations are excited as a result of Neimark–Sacker bifurcation.

## New type of bifurcations in the modified Rayleigh–Benard convection problem

The original Rayleigh–Benard convection is a standard example of the system where bifurcations occur with changing of a control parameter. In this paper we consider the modified Rayleigh–Benard convection problem including radiative effects as well as gas sources on a surface. Such formulation leads to the identification of new type of bifurcations in the problem besides the well-known Benard cells.

## Changes of the parameter plane of driven auto-oscillatory system caused by delayed modulation of the parameter

The driven auto-oscillatory system with the delayed modulation of driving amplitude was investigated. It was shown that synchronous regime destructs in different ways at small and large modulation amplitudes. The changes in the «driving amplitude–driving frequency» plane were revealed.

## Nonlinear dynamics of a ring of three phase systems

Nonlinear dynamics of the ensemble consisting of three phase-locked generators, which are coupled in a ring, is discovered. By force of computational modeling, which is based on the theory of oscillations, the regimes of the generators collective behavior is examined; the districts of synchronous and quasi-synchronous regimes are distinguished in the parameter space; the restructuring of the dynamics behavior on the boards of the distinguished districts is analyzed.

## Bifurcations of three and fourdimensional maps: universal properties

The approach, in which the picture of bifurcations of discrete maps is considered in the space of invariants of perturbation matrix (Jacobi matrix), is extended to the case of three and four dimensions. In those cases the structure of surfaces, lines and points for bifurcations, that is universal for all maps, is revealed. We present the examples of maps, whose parameters are governed directly by invariants of the Jacobian matrix.