Bifurcation Of Periodic Solutions Research Articles

Classification of bifurcations for nonlinear dynamical problems with constraints

Bifurcation of periodic solutions widely existed in nonlinear dynamical systems is a kind of constrained one in intrinsic quality because its amplitude is always non-negative. Classification of the bifurcations with the type of constraint was discussed. All its six types of transition sets are derived, in which three types are newly found and a method is proposed for analyzing the constrained bifurcation.

Discontinuous fold bifurcations in mechanical systems

This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions.

The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations

The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations x n+1 = F( n, x n , x n−1 ,…, x n− k ), where k is a nonnegative integer, are studied. A method of analysis of bifurcation of periodic solutions in the nonautonomous difference equations are presented. We apply this result on the rational difference equation x n+1 =( a n + b n x n )/( x n−1 ).

Bifurcations of synchronized responses in synaptically coupled Bonhöffer-van der Pol neurons.

The Bonhöffer-van der Pol (BvdP) equation is considered as an important model for studying dynamics in a single neuron. In this paper, we investigate bifurcations of periodic solutions in model equations of four and five BvdP neurons coupled through the characteristics of synaptic transmissions with a time delay. The model can be considered as a dynamical system whose solution includes jumps depending on a condition related to the behavior of the trajectory. Although the solution is discontinuous, we can define the Poincaré map as a synthesis of successive submaps, and give its derivatives for obtaining periodic points and their bifurcations. Using our proposed numerical method, we clarify mechanisms of bifurcations among synchronized oscillations with phase-locking patterns by analyzing periodic solutions observed in the coupling system and its subsystems. Moreover, we show that a global behavior of chaotic itinerancy or a phenomenon of chaotic transitions among several quasiattracting states can be observed in higher-dimensional systems of the synaptically four and five coupled neurons.

Bifurcations of periodics from homoclinics in singular O.D.E.: applications to discretizations of travelling waves of P.D.E.

Bifurcations of periodic solutions from homoclinic ones are investigated for certain singularly perturbed systems of autonomous ordinary differential equations in $\mathbb R^4$. Results are applied to discretization of travelling waves of certain p.d.e.

Hopf bifurcation near a double singular point with Z2-symmetry and X0-breaking

This paper deals with nonlinear equations f( x, λ, α)=0 and the corresponding ODEs x t = f( x, λ, α) satisfying f(0, λ, α)=0 and a Z 2-symmetry. In particular, we are interested in Hopf points, which indicate the bifurcation of periodic solutions of x t = f( x, λ, α) from (steady-state) solutions of f( x, λ, α)=0. It is shown that under suitable nondegeneracy conditions, there bifurcate two paths of Hopf points from a double singular point, where x=0 and f x (0, λ, α) has a double zero eigenvalue with one eigenvector symmetric and one anti-symmetric. This result gives a new example of finding Hopf points through local singular points. Our main tools for analysis are some extended systems, which also provide easily implemented algorithms for the numerical computation of the bifurcating Hopf points. A supporting numerical example for a Brusselator model is also presented.

Bifurcation of periodic solution in a three-unit neural network with delay

A system of three-unit networks with no self-connection is investigated, the general formula for bifurcation direction of Hopf bifurcation is calculated, and the estimation formula of the period for periodic solution is given.

BIFURCATIONS IN SYNAPTICALLY COUPLED BVP NEURONS

We investigate bifurcations of periodic solutions in model equations of two and three Bonhöffer–van der Pol (BVP) neurons coupled through the characteristics of synaptic transmissions with a time delay. Bifurcations of the coupled BVP neurons are compared with bifurcations of synaptically coupled Hodgkin–Huxley neurons. We obtained a parameter set of the BVP system, such that the two systems are qualitatively very similar from a bifurcational point of view. This study is a base for the analysis of synaptically coupled neurons with a large number of coupling strategies.

Bifurcations of periodic solutions and chaos in Josephson system

The Josephson equation is investigated in detail: the existence and bifurcationsfor harmonic and subharmonic solutions under small perturbations areobtained by using second-order averaging method and subharmonic Melnikov function,and the criterion of existence for chaos is proved by Melnikov analysis; thebifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency$\omega$ effects to the forms of chaotic behaviors are given by numerical simulations.

Global Bifurcations of Periodic solutions of the Hill Lunar Problem

We describe global bifurcations of non-stationary periodic solutions of the Hill Lunar Problem. Especially we are interested in description of closed connected sets (continua) of non-stationary periodic solutions which bifurcate from stationary ones. Such continua of solutions of the Hill Lunar Problem are not admissible in H12π \ Λ(H). For the Regularized Hill Lunar Problem we prove that these families are unbounded in H12π. As the main tool we use degree theory for SO(2)-equivariant orthogonal maps defined by S.M. Rybicki.

Emergence of oscillations in a model of weakly coupled two Bonhoeffer–van der Pol equations

Bifurcations of periodic solutions in a model of weakly coupled two Bonhoeffer–van der Pol equations are studied. The model realizes a half-center model with reciprocal inhibition, a typical model used in the field of neural motor control to account for the generation of alternating rhythmic bursts observed in motoneurons and spinal neural networks. Several oscillatory solutions such as in-phase, anti-phase as well as out-of-phase solutions emerge from the model's equilibrium as one of the parameters of the model changes. Among the variety of bifurcations exhibited by the model, we analyze Hopf bifurcations, by which several periodic solutions emerge, and illustrate generation mechanisms of alternating oscillations in the model.

Examples of bifurcation of periodic solutions to variational inequalities in $$\mathbb{R}^\kappa $$

A bifurcation problem for variational inequalities $$U(t) \in K$$ , $$(\dot U(t) - B_\lambda U(t) - G(\lambda ,U(t)),{\text{ }}Z - U(t)) \geqslant 0{\text{ for all }}Z \in K,{\text{ a}}{\text{.a}}{\text{. }}t \geqslant {\text{0}}$$ is studied, where K is a closed convex cone in $$\mathbb{R}^\kappa ,\kappa \geqslant 3,B_\lambda $$ , κ ≥ 3, B λ is a κ × κ matrix, G is a small perturbation, λ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

Hopf bifurcation in models for pertussis epidemiology

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two of the models. Periods of about four years are found for epidemiologically reasonable parameter values in two of these models.

Bifurcation of periodic solutions in symmetric models of suspension bridges

We consider a nonlinear model for time-periodic oscillations of a suspension bridge. Under some additional restrictive assumptions we describe our model by a standard bifurcation scheme which allows us to use global bifurcation theorems and make some new conclusions.

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

Variational inequalities $$\begin{gathered} U(t) \in K, \hfill (\dot U(t) - B_\lambda U(t) - G(\lambda ,U(t)),\;\;Z - U(t)) \geqslant 0 for all Z \in K, a .a . t \in (0,T) \hfill \end{gathered} $$ are studied, where K is a closed convex cone in $${\mathbb{R}}^{\kappa } $$ , κ ≥ 3, B λ is a κ × κ matrix, G is a small perturbation, λ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some λ0 for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some λI ≠ λ0. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at λ0 constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.

Synchronization and stable phase-locking in a network of neurons with memory

We consider a network of three identical neurons whose dynamics is governed by the Hopfield's model with delay to account for the finite switching speed of amplifiers (neurons). We show that in a certain region of the space of (α, β), where α and β are the normalized parameters measuring, respectively, the synaptic strength of self-connection and neighbourhood-interaction, each solution of the network is convergent to the set of synchronous states in the phase space, and this synchronization is independent of the size of the delay. We also obtain a surface, as the graph of a continuous function of τ = τ( α, β) (the normalized delay) in some region of (α, β), where Hopf bifurcation of periodic solutions takes place. We describe a continuous curve on such a surface where the system undergoes mode-interaction and we describe the change of patterns from stable synchronous periodic solutions to the coexistence of two stable phase-locked oscillations and several unstable mirror-reflecting waves and standing waves.

A METHOD TO CALCULATE BIFURCATIONS IN SYNAPTICALLY COUPLED HODGKIN–HUXLEY EQUATIONS

We propose a numerical method for calculating bifurcations of periodic solutions observed in a model equation of Hodgkin–Huxley neurons coupled by excitatory synapses with a time delay. To illustrate the validity of the method, bifurcations in two-coupled Hodgkin–Huxley equations with variation of a coupling coefficient and time delay are studied.

Chaos in spacecraft attitude motion in Earth's magnetic field.

The rotational motion of a satellite with a magnetic stabilization system is discussed. The motion is described by a nonautonomous differential equation, with the magnetic moment of the satellite as a parameter. The global phase portrait of the problem is investigated in a wide range of magnetic-parameter values, using a numerical realization of the method of Poincare point maps. New periodic solutions of the problem are found, and an analysis is carried out of the evolution of the phase portrait and the bifurcation of periodic solutions with varying magnetic-parameter values. The values of the magnetic parameter that must be avoided in the design of the satellite magnetic stabilization system are discussed. (c) 1999 American Institute of Physics.

Bifurcation of periodic solutions of the Navier-Stokes equations in a thin domain

Aim of this paper is to provide conditions in order to guarantee that the periodic solutions in time and in the space variables of the Navier-Stokes equations bifurcate. Specifically, we study this problem when the considered state domain has one dimension which is small with respect to the others which we let to tend to zero. The thinness of the domain represents the bifurcation parameter in our situation.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.