1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

When the bifurcation problem of a system

In this paper, we focus on 1 : 3 resonance, which is less discussed in the existing papers. When Neimark-Sacker bifurcation is considered, the case that

In 1982, Hindmarsh and Rose [

Recently, X. Liu and S. Liu [

Applying the forward Euler method to model (

This paper is organized as follows. In Section

The fixed points of map (

So

Using the Cardan formula (see [

(1) If

(2) If

(3) If

The stability of these fixed points can be found in [

The Jacobian matrix of map (

It is easy to get that two eigenvalues of

Here, we present the bifurcation set of 1 : 3 resonance as follows:

In this section, we show that there exist some values of parameters such that map (

We discuss the 1 : 3 resonance of map (

Now, we consider a perturbation of map (

Let

Map (

In the following, we will present our analysis in the critical case. It is easy to find the eigenvalues of

Here, we also introduce the adjoint eigenvector

Now any vector

By (

Here, we denote

Now, we introduce the following transformation to annihilate some second order terms:

To further simplify map (

If

Let

there is a Neimark-Sacker bifurcation curve at the trivial fixed point

there is a saddle cycle of period-three corresponding to the saddle fixed point

there is a homoclinic structure formed by the stable and unstable invariant manifolds of the period-three cycle intersecting transversally in an exponentially narrow parameter region.

Here, the intersection of these manifolds, which form a homoclinic tangency, implies the existence of Smale horseshoes and therefore an infinite number of long-period orbits appear. It illustrates a route from period-3 to chaos.

In this section, the 2-dimensional and 3-dimensional bifurcation diagrams show that the 1 : 3 resonance is the degenerate case of Neimark-Sacker bifurcation. So there exists a closed invariant circle near the 1 : 3 resonance. Here, we provide the following case to illustrate the dynamic behaviors of map (

Take parameters

Figures

1 : 3 resonance bifurcation diagram at

Maximum Lyapunov exponents of map (

Figures

Phase portraits corresponding to Figure

In this paper, we investigated the 1 : 3 resonance of a discrete Hindmarsh-Rose model. Here, we examined the fixed points of the model in detail and showed that the model exhibits the 1 : 3 resonance. Furthermore, near 1 : 3 resonance point, the Neimark-Sacker bifurcation and the homoclinic bifurcation can occur. The onset of 1 : 3 resonance means that, in some cases, there is a region such that the model will have an invariant circle from three-saddle cycle.

Here, we want to note that 1 : 3 resonance involves the bifurcations of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the editor and the referees for their valuable suggestions and comments which led to the improvement of the paper.