We analyze three codimension-two bifurcations occurring in nonsmooth systems, when a nonhyperbolic cycle (fold, flip, and Neimark–Sacker cases, in both continuous and discrete time) interacts with one of the discontinuity boundaries characterizing the system’s dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of n...

Tunnel Diodes are semiconductor devices that have many uses in physics and other fields. In this thesis, we study the effect of negative differential conductance in the current-voltage(IV) characteristic region of the tunnel diode and study the resulting bifurcation phenomenon. We first study the 3D model that has been physically studied by Steven Jones. We are able to successfully construct th...

A new bifurcation critical criterion of flip-Neimark-Sacker bifurcation is proposed for detecting or anticontrolling this type of codimension-two bifurcation of discrete systems in a general sense. The criterion is built on the properties of coefficients of characteristic equations instead of the properties of eigenvalues of Jacobian matrix of nonlinear system, which is formulated using a set o...

When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way, provided the corresponding genericity conditions. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbit...

In this paper, we study the global dynamics of a density-dependent predator-prey system with ratio-dependent functional response. The main features and challenges are that origin model is degenerate equilibrium higher order there multiple positive equilibria. Firstly, local qualitative behavior around explicitly described. Then, based on other equilibria, analysis c...

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and ...

In this paper, we show that the behavior of weakly nonlinear waves in a 2-layer model of baroclinic instability on a β-plane with varying viscosity is determined by a single, degenerate codimension three bifurcation. In the process, we show how previous studies, using the method of multiple scales to derive evolution equations for the slowly varying amplitude of the growing wave, arise as speci...

We give an explicit construction of families of Dm-equivariant polynomial vector fields in R possessing a codimension-one homoclinic cycle. The homoclinic cycle consist of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m a multiple of f...

In this paper, we show that the behaviour of weakly nonlinear waves in a 2-layer model of baroclinic instability on an f-plane with varying viscosity is determined by a single, degenerate codimension three bifurcation. In the process, we show how previous studies, using the method of multiple scales to derive evolution equations for the slowly varying amplitude of the growing wave, arise as spe...

A system of ODE’s is used to attempt an approximation of the dynamics of two delayed coupled FitzHugh-Nagumo excitable units, described by delay-differential equations. It is shown that the codimension 2 generalized Hopf bifurcation acts as the organizing center for the dynamics of ODE’s for small time-lags. Furthermore, this is used to explain important qualitative properties of the exact dyna...