The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:...

In this paper, a predator-prey model with time delay and habitat complexity is investigated. By analyzing the characteristic equations, the local stability of each feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the sum of two delays as a bifurcation parameter, we show that Hopf bifurcations can oc...

Mathematically, bifurcation theory attempts to investigate the changes in the qualitative or topological structure of a studied equation models. Given paper focuses on an analytical investigation of the nonlinear behavior of an indirect filedoriented control of induction motor. In this context, steady-state responses of the motor model are discussed and an analytical study of the generic codime...

Consider a polynomial Liénard system depending on three parameters a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii). If a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the equilibrium. We investigate analytically this bifurcation in dependence on the parameters b and c and establish the existenc...

In this paper we study Arnold’s tongues in a Z2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The...

The numerical approximation of the generalized Lienard equation is considered using delay as parameter. First, the delay difference equation obtained by using Euler method is written as a map.According to the theories of bifurcation for discrete dynamical systems,the conditions to guarantee the existence of Hopf bifurcation for numerical approximation are given. The relations of Hopf bifurcatio...

UMIST The use of boundary locus plots in the identiication of bifurcation points in numerical approximation of delay diierential equations Abstract We are interested in nonlinear delay diierential equations which have a Hopf bifurcation. We assume zero is a steady state for the problem, and so a Hopf bifurcation point lies on the boundary of the region of asymptotic stability for the zero solut...

This paper illustrates transcritical and Hopf bifurcations in a realistic ac/dc power system model. These bifurcations are thoroughly analyzed based on bifurcation theory to understand and characterize their e ect on the dynamic behavior of the system, particularly on voltage stability. A new technique to trace transcritical bifurcations diagrams is proposed, based on the generic characteristic...

Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable [É. Benôıt et al., Collect. Math., 31 (1981), pp. 37–119; F. Dumortier and R. Roussarie, Mem. Amer. Math. Soc., 121 (577) (1996); W. Eckhaus, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449–494; M. Krupa and P. Szmolyan, SIAM J. Math....