Article 1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

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.


Introduction
When the bifurcation problem of a system   →  (, ) ,  ∈ R  ,  ∈ R  , is studied by normal form method, researchers are often likely to compute the following two conditions: nondegeneracy conditions and transversality conditions, in which the derivatives of (, ) with respect to the variables  and the parameters  are involved, respectively.If some of the nondegeneracy and transversality conditions for the oneparameter bifurcations would be violated, the two-parameter bifurcations can also happen [1][2][3].In this case, one can obtain cusp, generalized flip, and Chenciner bifurcation [2].In the other case, extra multipliers can approach the unit circle for discrete dynamical system, thus changing the dimension of the center manifold   .There are eleven kinds of two-parameter bifurcations for discrete dynamical system listed by Kuznetsov (see Section 4 in [2]).
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  0 = 2/3 can lead to 1 : 3 resonance.One can find more information in [1][2][3] and references cited therein.
In 1982, Hindmarsh and Rose [4] described a two-variable model of the action potential which is a modification of Fitzhugh's B.v.P model in [5] and explained how the close proximity of the nullclines can be exploited to give a qualitative explanation for burst generation.The Hindmarsh-Rose model is known to reproduce all dynamical behaviors, such as quiescence, spiking, bursting, irregular spiking, and irregular bursting [4,6].Bifurcation analysis is examined once more in the past, with respect to one or two bifurcation parameters [7][8][9][10][11].Local bifurcations and global bifurcations are also analysed and these bifurcation phenomena can be used to explain the transitions between the dynamical behaviors.For example, the transition between spiking and bursting in the model can be understood by homoclinic bifurcations [12,13].More information on bifurcation can be found in [1,4,[8][9][10][11][14][15][16][17][18][19][20][21][22].
Recently, X. Liu and S. Liu [8] discussed the codimension-2 bifurcations of the following two-dimensional Hindmarsh-Rose model: where  represents the membrane,  is recovery variable, and Applying the forward Euler method to model (2), we obtain the following discrete-time Hindmarsh-Rose system: where  > 0 is the step size.In [21,27], we proved that map (3) possesses flip bifurcation, Neimark-Sacker bifurcation, and 1 : 1, 1 : 2, and 1 : 4 resonance.The aim of this paper is to prove that this discrete model possesses the 1 : 3 resonance.The method we used is based on the normal form method and bifurcation theory of discrete dynamical system (see Kuznetsov, Sections 4 and 9 in [2]).This paper is organized as follows.In Section 2, we present the existence and local stability of fixed points for map (3).In Section 3, we show that there exist some values of parameters such that map (3) undergoes 1 : 3 resonance.In Section 4, we present numerical simulations, which not only illustrate our results with the theoretical analysis but also exhibit complex dynamical behaviors.Finally, a brief discussion is given in Section 5.

Lemma 1. (1) If 27𝑎
The stability of these fixed points can be found in [21].In this paper, we focus on the existence and bifurcation analysis of 1 : 3 resonance.Here, we would like to give the bifurcation set of 1 : 3 resonance.
The Jacobian matrix of map (3) at the fixed point ( * ,  * ) is given by and the corresponding characteristic equation of the Jacobian matrix ( * ,  * ) can be written as where It is easy to get that two eigenvalues of ( * ,  * ) are Further, if  = −3/ and  2 = 3, then we have Here, we present the bifurcation set of 1 : 3 resonance as follows: It is obvious to find that  < 0 and  > 0 from the bifurcation set.Hence, the 1 :

1 : 3 Resonance
In this section, we show that there exist some values of parameters such that map (3) undergoes 1 : 3 resonance by using bifurcation theory [1][2][3].Here, the step sizes  and  are considered as bifurcation parameters to present bifurcation analysis at the fixed point  11 ( 11 ,  11 ).
Let  =  −  11 and V =  −  11 .Then we transform the fixed point  11 ( 11 ,  11 ) to the origin and map (12) becomes where Map ( 13) can be denoted as where In the following, we will present our analysis in the critical case.It is easy to find the eigenvalues of ( 0 ,  0 ) and their corresponding eigenvector ( 0 ,  0 ) ∈ C 2 as follows: Here, we also introduce the adjoint eigenvector which is normalized according to where ⟨⋅, ⋅⟩ means the standard scalar product in C 2 : ⟨, ⟩ =  1  1 +  2  2 .Now any vector  = (, V)  ∈ R 2 can be represented in the form From the above equation, we have Since where  = ( √ 3 − 1)/2, we get which implies that Using ( 19), (21), and ( 24), we have From ( 21) and ( 25), we get After calculation, we can choose (27) as ( 0 ,  0 ) and ( 0 ,  0 ), respectively.By (26), map (15) can be transformed into the complex form where Here, we denote   ( 0 ,  0 ) by   , with  +  = 2, 3.And ℎ  ( 0 ,  0 ) would be denoted by ℎ  , with  +  = 2, 3 in the introduced transformation.Now, we introduce the following transformation to annihilate some second order terms: where coefficients ℎ  with  +  = 2 will be confirmed in the following, and we can obtain Thus, using (30) and its inverse transformation, map ( 28) is changed into the following form: where  where (41) )( 0 ,  0 ), a similar argument as in Lemma 9.13 in [2] can be obtained.(c) 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.
Remark 3. 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.

Numerical Simulations
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 (3).Take parameters  = 2,  = 3,  = 0.8,  =  0 = 0.12428002, and  =  0 = 1.6771238 in map (3).We know that map (3) has a fixed point (1.1176267,−0.95523687).The eigenvalues of the corresponding Jacobian matrix ()  From Figure 1(c), we can observe the relations between 1 : 3 resonance and Neimark-Sacker bifurcation.In fact, the 1 : 3 resonance is the degenerate case of Neimark-Sacker bifurcation when  0 = 2/3.Here,  ± 0 is the eigenvalues of the Jacobian matrix (6).Moreover, the flip bifurcation occurs after the Neimark-Sacker bifurcation and 1 : 3 resonance.The Lyapunov exponents corresponding to the bifurcation diagram in Figure 1  the phase portraits show that Neimark-Sacker bifurcation occurs.

Conclusion
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 Z 3 symmetric system, which are not discussed in this paper.From the presented phase portraits, some symmetric phenomena can be observed.The homoclinic loop can explain the transition between spiking and bursting.