We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines-exact bifurcation cu...

An algorithm to detect homoclinic twist bifurcation points in Z2 symmetric autonomous systems of ordinary differential equations in R4 along curves of symmetric homoclinic orbits to hyperbolic equilibria has been developed.We show convergence of numerical approximations to homoclinic twist bifurcation points in such systems. A test function is defined on the homoclinic solutions, which has a re...

Abstract Analytical determination of the bifurcation thresholds is of important interest for practical electrical machine design and analysis control. This paper presents mathematical investigation of the qualitative behavior of indirect filed oriented control induction motor based on bifurcation theory. In this context, steady-state responses analysis of the motor model is discussed and an ana...

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and non-impacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcatio...

Abstract. In many applications of practical interest, for example, in control theory, economics, electronics and neural networks, the dynamics of the system under consideration can be modelled by an endomorphism, which is a discrete smooth map that does not have a uniquely defined inverse; one also speaks simply of a noninvertible map. In contrast to the better known case of a dynamical system ...

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR is evoked by inhibitory stimulation instead of excitatory stimulation, presents a challenge the threshold concept. In this paper, 7 codimension-2 degenerate bifurcations 1...

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a twoparameter fr...

This work analyzes the O(2) symmetry breaking bifurcations in systems with anZ2×O(2) symmetry group—whereZ2 and O(2) are, respectively, spatio-temporal and spatial symmetries—that are responsible for the transitions from two-dimensional to three-dimensional hydrodynamic states. This symmetry group describes, for example, two-dimensional time-periodic flows past bodies which have reflection symm...

The equation ż = ez + eiφz|z|2 + bz̄3 models a map near a Hopf bifurcation with 1:4 resonance. It is a conjecture by V. I. Arnol’d that this equation contains all versal unfoldings of Z4-equivariant planar vector fields. We study its bifurcations at ∞ and show that the singularities of codimension two unfold versally in a neighborhood. We give an unfolding of the codimension-three singularity fo...

This paper is concerned with the analysis of two-parameter bifurcation phenomena in the buck power converter. It is shown that the complex dynamics of the converter can be unfolded by considering higher codimension bifurcation points in two-parameter space. Specifically, standard smooth bifurcations are shown to merge with discontinuity-induced bifurcation (DIB) curves, giving rise to intricate...