We analyze the second order diierential equation describing a damped harmonic oscillator with nonlinear feedback depending on both the state and the derivative of the state at some time in the past. The characteristic equation for the linear stability of the equilibrium is completely solved, and the stability region is illustrated in a parameter space consisting of the time delay and the streng...

Two dynamical models that have been proposed to describe transitions between low- and high-confinement states in confined plasmas are analyzed using singularity theory and stability theory. It is shown that the stationary-state bifurcation sets have qualitative properties identical to standard normal forms for the pitchfork and transcritical bifurcations. The analysis yields the codimension of ...

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens’ clocks, is studied using the codimension-three Elphick–Huygens equivariant normal form presented here....

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equili...

Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation sc...

An SIRS epidemic model with a modified nonlinear incidence rate is studied, which describes that the infectivity strong at first as emergence of new disease or reemergence an old disease, but then psychological effect will weaken infectivity. Lastly, goes to saturation state result crowding effect. The nonlinearity functional form infection modified, more reasonable biologically. We analyze sta...

Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation sc...

The Hopf bifurcation in the presence of 0(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classific...

Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a onedimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when t...

<p style='text-indent:20px;'>Based on the three-dimensional endocrine neuron model, a four-dimensional model was constructed by introducing magnetic flux variable and induced current according to law of electromagnetic induction. Firstly, codimension-one bifurcation Interspike Intervals (ISIs) analysis were applied study structure with respect external stimuli parameter <inline-formula...