Abstract
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.
1. Introduction
When the bifurcation problem of a system 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 one-parameter bifurcations would be violated, the two-parameter bifurcations can also happen [1–3]. 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 can lead to 1 : 3 resonance. One can find more information in [1–3] 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 [7–11]. 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, 8–11, 14–22].
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 , , , and are positive parameters. Model (2) can describe the transitions between the above five dynamical behaviors, that is, quiescence, spiking, bursting, irregular spiking, and irregular bursting. More related works can be found in [4, 7, 9–13, 17–21, 23–26].
Applying the forward Euler method to model (2), we obtain the following discrete-time Hindmarsh-Rose system: where 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.
2. Local Dynamics for Fixed Points of Map (3)
The fixed points of map (3) satisfy the following equations:
So is the root of the following equation:
Using the Cardan formula (see [28]), we get the following results (see also [8, 21]).
Lemma 1. (1) If , then map (3) has a unique fixed point , where .
(2) If , then map (3) has two fixed points and , where .
(3) If , then map (3) has three different fixed points, , where .
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 and , then we have .
Here, we present the bifurcation set of 1 : 3 resonance as follows: It is obvious to find that and from the bifurcation set. Hence, the 1 : 3 resonance only can occur at , , , and . In the following, we present our discussions for . The similar arguments can be undertaken at the fixed points , , and .
3. 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 [1–3]. Here, the step sizes and are considered as bifurcation parameters to present bifurcation analysis at the fixed point .
We discuss the 1 : 3 resonance of map (3) at when the parameters vary in a small neighborhood of . Taking parameters arbitrarily from , we consider map (3) with , which is described by The eigenvalues of map (11) at the fixed point are .
Now, we consider a perturbation of map (11) as follows: where which are small perturbation parameters.
Let and . Then we transform the fixed point 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 and their corresponding eigenvector as follows:
Here, we also introduce the adjoint eigenvector , satisfying which is normalized according to where means the standard scalar product in .
Now any vector can be represented in the form From the above equation, we have Since where , we get which implies that Using (19), (21), and (24), we have From (21) and (25), we get After calculation, we can choose as and , respectively.
By (26), map (15) can be transformed into the complex form where
Here, we denote by , with . And would be denoted by , with in the introduced transformation.
Now, we introduce the following transformation to annihilate some second order terms: where coefficients with 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 By setting then we have , and can be simplified in the following. Hence, the transformation (30) is defined and
To further simplify map (32), we introduce the following transformation: After using (36) and its inverse transformation, map (32) is changed into the following form: where By setting then we have . Hence, the transformation (36) is defined. Using transformation (36), map (32) finally becomes the following normal form of the bifurcation with 1 : 3 resonance: where
If , , a similar argument as in Lemma 9.13 in [2] can be obtained.
Theorem 2. Let . If and , then map (3) has the the following complex dynamical behaviors: (a)there is a Neimark-Sacker bifurcation curve at the trivial fixed point of map (40);(b)there is a saddle cycle of period-three corresponding to the saddle fixed point of map (40);(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.
4. 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 , , , , and in map (3). We know that map (3) has a fixed point . The eigenvalues of the corresponding Jacobian matrix are . After calculating, we further have . Therefore, from Theorem 2, we see that fixed point is a 1 : 3 resonance point.
Figures 1(a) and 1(b) show the 2-dimensional bifurcation diagrams when and , respectively, and varies in a neighborhood of . Figure 1(c) shows the 3-dimensional bifurcation diagram when , vary in a neighborhood of . 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 . Here, 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 are computed and plotted in Figure 2. We easily see that there are the positive Lyapunov exponents and negative Lyapunov exponents. It means that map (3) has chaotic and periodic behaviors near the 1 : 3 resonance. The 3-dimensional maximum Lyapunov exponents are given in Figure 2(c).
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Figures 3(a)–3(o) show the phase portraits of map (3) near for different and . Furthermore, as varies around and , from Figures 3(a)–3(g), we can see that period-3 orbits and period-6 orbits, eventually leading to chaos when decreases to 0.11325. This is the classical route to chaos. Besides, as increases and from Figures 3(h)–3(j), we can observe that there exists a fixed point connecting to three saddles, chaos, and more new complex phenomena in certain regions near . Here, in Figure 3(k), the different colours are chosen to demonstrate the different orbits near the 1 : 3 resonance, which shows the homoclinic structure near a 1 : 3 resonance. Furthermore, both Smale horseshoes and an infinite number of long-period orbits occur. Finally, from Figures 3(m)–3(o), the phase portraits show that Neimark-Sacker bifurcation occurs.
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5. 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 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the editor and the referees for their valuable suggestions and comments which led to the improvement of the paper.