In this work we study a wide class of symmetric control systems that has the Chua’s circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-thre...

We study the linear instabilities and bifurcations in Selkov model for glycolysis with diffusion. show that this has a zero wave-vector, finite frequency Hopf bifurcation, which is forward or supercritical to growing oscillatory but spatially homogeneous state saddle-node backward subcritical inhomogeneous steady pattern characterised by wavevector. further demonstrate tuning relative diffusivi...

We present two ways of regularizing a one parameter family piece-wise smooth dynamical systems undergoing codimension grazing-sliding global bifurcation periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that regularized has saddle-node Then perform hysteretic show chaotic dynamics. Our result shows that, in spite regularizations will give same dynamics sliding modes,...

In this paper we develope a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known Morsification results for nonisolated singulatities and generalize them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects t...

A transversally driven isotropic ferromagnet being under the influence of a static external and an uniaxial internal anisotropy field is studied. We consider the dissipative Landau-Lifshitz equation as the fundamental equation of motion and treat it in 1 + 1 dimensions. The stability of the spatially homogeneous magnetizations against inhomogeneous perturbations is analyzed. Subsequently the dy...

Through multiple-scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady-state pattern-forming instabilities, in the case that the wavelengths of the instabilities are nearly in the ratio 1 : 2. In the case of exact 1 : 2 resonance the amplitude equations are ODEs; here they are PDEs. We discuss the stability of spatially-periodic solu...

We present a theoretical study of unnested period-doubling islands in three-dimensional rate equations modeling a semiconductor laser subject to external optical injection. In this phenomenon successive curves of period doublings are not arranged in nicely nested islands, but intersect each other. This overall structure is globally organized by several codimension-2 bifurcations. As a consequen...

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for su...

The main purpose of this article is to demonstrate that bursting oscillations can be observed not only in the slow–fast autonomous dynamical systems with multiple scales associated with time domain, but also in the non-autonomous dynamical systems with periodic excitations when an order gap exists between the exciting frequency and the natural frequency, implying multiple scales in frequency do...

We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synap...