The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorem...

In this paper, we consider controlling and anti-controlling Hopf bifurcations in discrete maps using feedback controller of polynomial functions. It is shown that such a polynomial feedback controller is easy to be implemented, which not only preserves the system s equilibrium solutions, but also keeps the dimension of the system unchanged. Examples are used to show, with this type of controlle...

We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use ...

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh-Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: 1) the number and stability of the equilibrium points are ...

Transients in ecology are extremely important since they determine how equilibria approached. The debate on the dynamic stability of ecosystems has been largely focused equilibrium states. However, constantly changing due to climate conditions or perturbations driven by crisis anthropogenic actions (habitat destruction, deforestation, defaunation), it is study dynamics can proceed till equilibr...

We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find that the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We deriv...

Our previous work has shown that AFP could adjust synaptic conductance of RA network with its firing pattern changed from bursting to tonic firing, eventually leading song degradation. However, it is difficult explore the dynamic due high dimensionality and complexity. To address this issue, we use mean field theory proposed by Wilten Nicola reduce a system switching ordinary differential equat...

Bursting behavior in neurons is a recurrent transition between a quiescent state and repetitive spiking. When the transition to repetitive spiking occurs via a subcritical Andronov–Hopf bifurcation and the transition to the quiescent state occurs via double limit cycle bifurcation, the burster is said to be of subcritical elliptic type. When the fast subsystem is near a Bautin (generalized Hopf...