Controllable Hopf bifurcations of codimension 1 and 2 in nonlinear control systems

The purpose of this work, given a nonlinear control system, is to design a four-parameter family of static state feedback such that the corresponding closed-loop control system exhibits controllable Hopf bifurcations of codimension 1 and 2. More precisely, the scalar law designed by us permits the control of the stability of the equilibrium points and the orientation and stability of the periodic orbits.     

Hopf bifurcations in a Ricardo-Malthus model

Brander and Taylor presented a simple and basic framework for discussing the problem on human population and renewable natural resources in the year 1998, and D’Alessandro recently extended this work mainly by introducing a nonlinear term into the model, if seeing from the mathematical point of view. A limit cycle in this new model was reported by the author via numerically simulated drawing. In this paper, we show that this limit cycle actually is a bifurcating limit cycle of a one-parameter Hopf bifurcation.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.     

Stability of singular Hopf bifurcations

A stability formula is given for the singular Hopf bifurcation arising in singularly perturbed systems of the form , in this paper. The derivation of the formula is based on a reduction technique and on an existing stability formula for Hopf bifurcation.     

Hopf bifurcation in symmetric configuration of predator-prey-mutualist systems

In this paper we apply the equivariant degree method to study Hopf bifurcations in a system of differential equations describing a symmetric predator-prey-mutualist model with diffusive migration between interacting communities. A topological classification (according to symmetry types), of symmetric Hopf bifurcation in configurations of populations with D₈, D₁₂, A₄ and S₄ symmetries, is presented with estimation on minimal number of bifurcating branches of periodic solutions.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

Dynamic transcritical bifurcations in a class of slow-fast predator-prey models

We study the stability loss delay phenomenon in a dynamic transcritical bifurcation in a class of three-dimensional prey and predator systems. The dynamics of the predator is assumed to be slow comparatively to the dynamics of the prey. As an application, a well-known model considered by Clark will be discussed.     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov–Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slow-fast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up.     

Slow-fast Bogdanov-Takens bifurcations

In this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of Bogdanov-Takens (BT) bifurcations. We introduce, among other precise definitions, the notion of slow-fast BT-bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits. Based on geometric singular perturbation theory, including blow-up, we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane.     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in Rⁿ${\mathbb{R}}^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or C^∞$C^{\infty }$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.     

Subharmonic bifurcation from infinity

We are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods 2πn) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points ±αi with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods 2πn exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors

First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates. After we study the cyclicity of a class of nilpotent foci in their analytic normal form.     

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.     

Algorithmic derivation of isochronicity conditions

This paper is concerned with obtaining the conditions under which the origin is an isochronous centre for certain planar polynomial systems. The necessity of these conditions is proved using a computer implementation of an algorithm first proposed by I.I. Pleshkan. Their sufficiency is then proved using various methods.     

Global qualitative analysis of a quartic ecological model

In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.