JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/896478 896478 Research Article 1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model Li Bo He Zhimin Teng Zhidong School of Mathematics and Statistics Central South University Changsha Hunan 410083 China csu.edu.cn 2014 17122014 2014 10 07 2014 30 11 2014 17 12 2014 2014 Copyright © 2014 Bo Li and Zhimin He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (1)xfx,α,xRn,αRk, 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 f(x,α) with respect to the variables x and the parameters α 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 . In this case, one can obtain cusp, generalized flip, and Chenciner bifurcation . In the other case, extra multipliers can approach the unit circle for discrete dynamical system, thus changing the dimension of the center manifold Wc. There are eleven kinds of two-parameter bifurcations for discrete dynamical system listed by Kuznetsov (see Section 4 in ).

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 θ0=2π/3 can lead to 1 : 3 resonance. One can find more information in  and references cited therein.

In 1982, Hindmarsh and Rose  described a two-variable model of the action potential which is a modification of Fitzhugh’s B.v.P model in  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 . 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, 811, 1422].

Recently, X. Liu and S. Liu  discussed the codimension-2 bifurcations of the following two-dimensional Hindmarsh-Rose model: (2)dxdt=y-ax3+bx2,dydt=-c-dx2-y, where x represents the membrane, y is recovery variable, and a, b, c, and d 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, 913, 1721, 2326].

Applying the forward Euler method to model (2), we obtain the following discrete-time Hindmarsh-Rose system: (3)xyx+δ(y-ax3+bx2)y-δ(c+dx2+y), where δ>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 ).

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 (<xref ref-type="disp-formula" rid="EEq2">3</xref>)

The fixed points of map (3) satisfy the following equations: (4)y-ax3+bx2=0,c+dx2+y=0.

So x* is the root of the following equation: (5)ax3+d-bx2+c=0,y*=-c-dx*2.

Using the Cardan formula (see ), we get the following results (see also [8, 21]).

Lemma 1.

(1) If 27a2c-4b-d3>0, then map (3) has a unique fixed point E11(x11,y11), where x11<min{0,2(b-d)/3a}.

(2) If 27a2c-4(b-d)3=0, then map (3) has two fixed points E21(x21,y21) and E22(x22,y22), where x21<0<x22=2(b-d)/3a.

(3) If 27a2c-4(b-d)3<0, then map (3) has three different fixed points, E3ix3i,y3i(i=1,2,3), where x31<0<x32<2(b-d)/3a<x33.

The stability of these fixed points can be found in . 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 (x*,y*) is given by (6)Jx*,y*=1+δ(-3ax*2+2bx*)δ-2δdx*1-δ, and the corresponding characteristic equation of the Jacobian matrix J(x*,y*) can be written as (7)F(λ):=λ2-2+Gδλ+1+Gδ+Hδ2=0, where (8)G=-3ax*2+2bx*-1,H=3ax*2-2bx*+2dx*.

It is easy to get that two eigenvalues of J(x*,y*) are (9)λ1,2=1+δ2(G±G2-4H). Further, if δ=-3/G and G2=3H, then we have λ1,2=±3i-1/2.

Here, we present the bifurcation set of 1 : 3 resonance as follows: (10)F==2G+32+324x*,δ,a,b,c,d>0(a,b,c,d,δ):lllllllδ=-3G,d=2G+32+324x*,δ,a,b,c,d>0. It is obvious to find that G<0 and H>0 from the bifurcation set. Hence, the 1 : 3 resonance only can occur at E11(x11,y11), E21(x21,y21), E31(x31,y31), and E33(x33,y33). In the following, we present our discussions for E11(x11,y11). The similar arguments can be undertaken at the fixed points E21(x21,y21), E31(x31,y31), and E33(x33,y33).

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 . Here, the step sizes δ and d are considered as bifurcation parameters to present bifurcation analysis at the fixed point E11(x11,y11).

We discuss the 1 : 3 resonance of map (3) at E11(x11,y11) when the parameters vary in a small neighborhood of F. Taking parameters (a,b,c,d0,δ0) arbitrarily from F, we consider map (3) with (a,b,c,d0,δ0), which is described by (11)xyx+δ0(y-ax3+bx2)y-δ0(c+d0x2+y). The eigenvalues of map (11) at the fixed point E11(x11,y11) are λ1,2=±3i-1/2.

Now, we consider a perturbation of map (11) as follows: (12)xyx+δ(y-ax3+bx2)y-δ(c+dx2+y), where |δ-δ0|,|d-d0|1 which are small perturbation parameters.

Let u=x-x11 and v=y-y11. Then we transform the fixed point E11(x11,y11) to the origin and map (12) becomes (13)uv(1+δa11)u+δv+δa13u2-δau3δda21u+(1-δ)v-δdu2, where (14)a11=-3ax112+2bx11,a13=-3ax11+b,a21=-2x11.

Map (13) can be denoted as (15)uvA(δ,d)uv+F(u,v,δ,d), where (16)Aδ,d=1+δa11δδa211-δ,F(u,v,δ,d)=δa13u2-δu3-δdu2.

In the following, we will present our analysis in the critical case. It is easy to find the eigenvalues of A(δ0,d0) and their corresponding eigenvector q(δ0,d0)C2 as follows: (17)A(δ0,d0)q(δ0,d0)=3i-12q(δ0,d0).

Here, we also introduce the adjoint eigenvector p(δ0,d0)C2, satisfying (18)Aδ0,d0Tp(δ0,d0)=-3i+12p(δ0,d0), which is normalized according to (19)pδ0,d0,qδ0,d0=1, where ·,· means the standard scalar product in C2:p,q=p¯1q1+p¯2q2.

Now any vector X=(u,v)TR2 can be represented in the form (20)X=zqδ0,d0+z-qδ0,d0¯,zC. From the above equation, we have (21)pδ0,d0,X=pδ0,d0,zqδ0,d0+z-qδ0,d0¯=zpδ0,d0,qδ0,d0+z-pδ0,d0,qδ0,d0¯. Since (22)pδ0,d0,qδ0,d0¯=pδ0,d0,1λ¯Aqδ0,d0¯=1λ¯ATpδ0,d0,qδ0,d0¯=λλ¯pδ0,d0,qδ0,d0¯, where λ=3i-1/2, we get (23)1-λλ¯pδ0,d0,qδ0,d0¯=0, which implies that (24)pδ0,d0,qδ0,d0¯=0. Using (19), (21), and (24), we have (25)z=pδ0,d0,X. From (21) and (25), we get (26)zn+1=pδ0,d0,Xn+1=pδ0,d0,Aδ0,d0Xn+FXn,δ0,d0=pδ0,d0,Aδ0,d0znqδ0,d0+zn¯  qδ0,d0¯llllll+Fznqδ0,d0+zn¯  qδ0,d0¯,δ0,d0=znqδ0,d0+zn¯  qδ0,d0¯,δ0,d0pδ0,d0,znAδ0,d0qδ0,d0+zn¯A  q(δ0,d0)¯llllll+Fznqδ0,d0+zn¯  qδ0,d0¯,δ0,d0=3i-12zn+znqδ0,d0+zn¯  qδ0,d0¯,δ0,d0pδ0,d0,lllllllllllllllllllllllllFznqδ0,d0+zn¯  qδ0,d0¯,δ0,d0. After calculation, we can choose (27)δ0,3i-32-δ0a11T,3i(3+2δ0a11)+36δ0,3i3T as q(δ0,d0) and p(δ0,d0), respectively.

By (26), map (15) can be transformed into the complex form (28)z3i-12z+2k+l31k!l!gklδ0,d0zkz-l, where (29)g20=g11=g02=δ02a13+33δ02i2δ0d-3a13-2δ0a11a13,g30=g21=g12=g03=-3δ03a+3δ03ai3+2δ0a11.

Here, we denote gkl(δ0,d0) by gkl, with k+l=2,3. And hkl(δ0,d0) would be denoted by hkl, with k+l=2,3 in the introduced transformation.

Now, we introduce the following transformation to annihilate some second order terms: (30)z=ω+12h20ω2+h11ωω¯+12h20ω¯2, where coefficients hkl with k+l=2 will be confirmed in the following, and we can obtain (31)ω=z-12h20z2-h11zz¯-12h20z¯2+12(h202+h11h¯02)z3+32h20h11+h112+h022z2z¯+12h11h¯20+h112+h02h¯11+12h20h02zz¯2+12h02h¯20+h11h02z¯3+Oz4. Thus, using (30) and its inverse transformation, map (28) is changed into the following form: (32)ω3i-12ω+2k+l31k!l!ϱklωkω¯l+O(ω4), where (33)ϱ20=g20+3h20i,ϱ11=2g11+3i-3h11,ϱ02=g02,ϱ30=32(3-3i)h20g20+3g11h¯02lllllllll-32(3i-1)h11g¯02+32(3i+3)h202+g30,ϱ21=12(5+3i)h11g20+(2-3i)h20g11lllllllll+2h¯11g11+(1-3i)h11g¯11+g02h¯02lllllllll+12(1+3i)h02g¯02+g21lllllllll+12(53i-3)h11h20+(3i-3)h¯11h11,ϱ12=g20h02+12(1-3i)h11g¯20lllllllll+(h¯20+(3i+3)h11)g11+(3i+1)h02g¯11+g12lllllllll+12(1-3i)h20+2h¯11g02lllllllll-12(3+3i)h11h¯20-(3+3i)h112-23ih02h¯11,ϱ03=32(1+3i)h02g¯20+3g11h02+3h¯20g02lllllllll+32(1+3i)h11g02+g03+32(3-3i)h02h¯20. By setting (34)h20=3i3g20,h11=3+3i6g11,h02=0, then we have ϱ20=ϱ11=0, ϱ02=g02 and ϱ30,ϱ21,ϱ12,ϱ03 can be simplified in the following. Hence, the transformation (30) is defined and (35)ϱ30=3-3i2g11g¯02+3ig202+g30,ϱ21=3+23i3g20g11+3-3i3g112+g21,ϱ12=3+3i6g20g02+3-3i3g¯11g02lllllllll+3+3i3g112-3i3g¯20g11+g12,ϱ03=3ig11g02-3ig¯20g02+g03.

To further simplify map (32), we introduce the following transformation: (36)ω=ζ+16h30ζ3+12h12ζζ¯2+12h21ζ2ζ¯+16h03ζ¯3. After using (36) and its inverse transformation, map (32) is changed into the following form: (37)ζ3i-12ζ+g022ζ¯2+k+l=31k!l!ϱ~klζkζ¯l+O(ζ4), where (38)ϱ~30=ϱ30+3i-32h30,ϱ~21=ϱ21,ϱ~12=ϱ12+3ih12,ϱ~03=ϱ03+3i-32h03. By setting (39)h30=3+3i6ϱ30,h21=0,h12=3i3ϱ12,h03=3+3i6ϱ03, then we have ϱ~30=ϱ~21=ϱ~03=0. Hence, the transformation (36) is defined. Using transformation (36), map (32) finally becomes the following normal form of the bifurcation with 1 : 3 resonance: (40)ζ3i-12ζ+B~δ0,d0ζ¯2+C~(δ0,d0)ζζ2+Oζ4, where (41)B(δ0,d0)=g022,C(δ0,d0)=g20g11(3+23i)6+(3-3i)g1126+g212.

If B1(δ0,d0)=-3/2(3i+1)B(δ0,d0), C1(δ0,d0)=-3Bδ0,d02-3(1+3i)/2C(δ0,d0), a similar argument as in Lemma  9.13 in  can be obtained.

Theorem 2.

Let (δ0,d0)F. If B1(δ0,d0)0 and Re(C1(δ0,d0))0, then map (3) has the the following complex dynamical behaviors:

there is a Neimark-Sacker bifurcation curve at the trivial fixed point E0 of map (40);

there is a saddle cycle of period-three corresponding to the saddle fixed point Ek  (k=1,2,3) of map (40);

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 a=2, b=3, c=0.8, d=d0=0.12428002, and δ=δ0=1.6771238 in map (3). We know that map (3) has a fixed point E(1.1176267,-0.95523687). The eigenvalues of the corresponding Jacobian matrix J(E) are λ1,2=±0.8660254347i-0.5000000165±3i-1/2. After calculating, we further have Re(C1(δ0,d0))=-284.66342920. Therefore, from Theorem 2, we see that fixed point E is a 1 : 3 resonance point.

Figures 1(a) and 1(b) show the 2-dimensional bifurcation diagrams when d=0.12428002 and d=0.14, respectively, and δ varies in a neighborhood of δ0=1.6771238. Figure 1(c) shows the 3-dimensional bifurcation diagram when δ,  d vary in a neighborhood of (δ0,d0)=(1.6771238,0.12428002). 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 θ0=2π/3. Here, e±iθ0 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).

1 : 3 resonance bifurcation diagram at E with a=2, b=3, c=0.8, d=0.12428002, and δ=1.6771238. (a) In (δ,x) plane with d=0.12428002. (b) In (δ,x) plane with d=0.14. (c) In (δ,d,x) plane.

Maximum Lyapunov exponents of map (3) near 1 : 3 resonance E as δ and d vary. (a) and (b) are maximum Lyapunov exponents corresponding to (a) and (b) in Figure 1. (c) and (d) are 3-dimensional maximum Lyapunov exponents in [1.5,1.82]×[0.124,0.14] and 2-dimensional projection onto (δ,d).

Figures 3(a)3(o) show the phase portraits of map (3) near E for different δ and d. Furthermore, as d varies around d=d0=0.12428002 and δ0=1.6771238, from Figures 3(a)3(g), we can see that period-3 orbits and period-6 orbits, eventually leading to chaos when d decreases to 0.11325. This is the classical route to chaos. Besides, as δ increases and d=d0=0.12428002 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 E. 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.

Phase portraits corresponding to Figure 1 for different δ and d. (a) δ=1.6771238, d=0.13; (b) δ=1.6771238, d=0.125; (c) δ=1.6771238, d=0.12457; (d) δ=1.6771238, d=0.12; (e) δ=1.6771238, d=0.115; (f) δ=1.6771238, d=0.1134; (g) δ=1.6771238, d=0.09; (h) δ=1.685, d=0.12428002; (i) δ=1.78, d=0.12428002; (j) δ=1.83, d=0.12428002; (k) δ=1.75, d=0.14; (l) δ=1.9, d=0.14; (m) δ=1.76, d=0.15; (n) δ=1.77, d=0.15; (o) δ=1.83, d=0.15.

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 Z3 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.

Guckenheimer J. Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983 Berlin, Germany Springer 10.1007/978-1-4612-1140-2 MR709768 Kuznetsov Y. A. Elements of Applied Bifurcation Theory 2004 3rd New York, NY, USA Springer 10.1007/978-1-4757-3978-7 MR2071006 Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos 2003 2 2nd New York, NY, USA Springer MR2004534 Hindmarsh J. L. Rose R. M. A model of the nerve impulse using two first-order differential equations Nature 1982 296 5853 162 164 2-s2.0-0020049721 10.1038/296162a0 FitzHugh R. Impulses and physiological state in theoretical models of nerve membrane Biophysical Journal 1961 1 445 467 10.1016/s0006-34956186902-6 Nagumo J. Arimoto S. Yoshizawa S. An active pulse transmission line simulating nerve axon Proceedings of the IRE 1962 50 2061 2070 10.1109/jrproc.1962.288235 Gonzalez-Miranda J. M. Complex bifurcation structures in the Hindmarsh-Rose neuron model International Journal of Bifurcation and Chaos 2007 17 9 3071 3083 10.1142/s0218127407018877 MR2372289 2-s2.0-36749092272 Liu X. Liu S. Codimension-two bifurcation analysis in two-dimensional Hindmarsh-Rose model Nonlinear Dynamics 2012 67 1 847 857 10.1007/s11071-011-0030-6 MR2869243 2-s2.0-82255186539 Storace M. Linaro D. de Lange E. The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations Chaos 2008 18 3 033128 10.1063/1.2975967 MR2464307 2-s2.0-54749122455 Nikolov S. An alternative bifurcation analysis of the Rose-Hindmarsh model Chaos, Solitons and Fractals 2005 23 5 1643 1649 10.1016/j.chaos.2004.06.080 MR2101579 2-s2.0-9544252187 Tsuji S. Ueta T. Kawakami H. Fujii H. Aihara K. Bifurcations in two-dimensional Hindmarsh-Rose type model International Journal of Bifurcation and Chaos 2007 17 3 985 998 10.1142/s0218127407017707 MR2324992 2-s2.0-34249105813 Linaro D. Champneys A. Desroches M. Storace M. Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose burster SIAM Journal on Applied Dynamical Systems 2012 11 3 939 962 10.1137/110848931 MR3022055 2-s2.0-84866434344 Shilnikov A. Kolomiets M. Methods of the qualitative theory for the Hindmarsh-Rose model: a case study. A tutorial International Journal of Bifurcation and Chaos 2008 18 8 2141 2168 10.1142/s0218127408021634 MR2463856 2-s2.0-54949130175 Chen Q. Teng Z. Wang L. Jiang H. The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems 2013 71 1-2 55 73 10.1007/s11071-012-0641-6 MR3010563 2-s2.0-84871999135 He Z. Lai X. Bifurcation and chaotic behavior of a discrete-time predator-prey system Nonlinear Analysis. Real World Applications 2011 12 1 403 417 10.1016/j.nonrwa.2010.06.026 MR2729029 2-s2.0-77958015050 He Z. Qiu J. Neimark-Sacker bifurcation of a third-order rational difference equation Journal of Difference Equations and Applications 2013 19 9 1513 1522 10.1080/10236198.2013.764998 MR3173501 2-s2.0-84883655458 Hindmarsh J. L. Rose R. M. A model of neuronal bursting using three coupled first order differential equations Proceedings of the Royal Society of London Series B: Biological sciences 1984 221 1222 87 102 10.1098/rspb.1984.0024 2-s2.0-0021768729 Hodgkin A. L. Huxley A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve The Journal of Physiology 1952 117 4 500 544 2-s2.0-35649001607 Innocenti G. Morelli A. Genesio R. Torcini A. Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos Chaos 2007 17 4 043128 10.1063/1.2818153 MR2380043 2-s2.0-37649016614 Jing Z. Chang Y. Guo B. Bifurcation and chaos in discrete FitzHugh-Nagumo system Chaos, Solitons & Fractals 2004 21 3 701 720 10.1016/j.chaos.2003.12.043 MR2043746 2-s2.0-1342306868 Li B. He Z. Bifurcations and chaos in a two-dimensional discrete Hindmarsh-Rose model Nonlinear Dynamics 2014 76 1 697 715 10.1007/s11071-013-1161-8 MR3189203 2-s2.0-84899087686 Liu X. Xiao D. Complex dynamic behaviors of a discrete-time predator-prey system Chaos, Solitons & Fractals 2007 32 1 80 94 10.1016/j.chaos.2005.10.081 MR2271103 2-s2.0-33748554203 Chen S.-S. Cheng C.-Y. Lin Y.-R. Application of a two-dimensional Hindmarsh-Rose type model for bifurcation analysis International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2013 23 3 1350055 10.1142/s0218127413500557 MR3047970 2-s2.0-84876204360 Ma S. Feng Z. Fold-Hopf bifurcations of the Rose-Hindmarsh model with time delay International Journal of Bifurcation and Chaos 2011 21 2 437 452 10.1142/s0218127411028490 MR2793962 2-s2.0-79953851483 Terman D. The transition from bursting to continuous spiking in excitable membrane models Journal of Nonlinear Science 1992 2 2 135 182 10.1007/BF02429854 MR1169590 2-s2.0-34249839599 Wang X.-J. Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle Physica D: Nonlinear Phenomena 1993 62 1–4 263 274 10.1016/0167-27899390286-a MR1207426 Li B. He Z. M. 1 : 2 and 1 : 4 resonances in a two-dimensional discrete Hindmarsh-Rose model Nonlinear Dynamics 2014 10.1007/s11071-014-1696-3 Polyanin A. D. Chernoutsan A. I. A Concise Handbook of Mathematics, Physics, and Engineering Science 2011 New York, NY, USA CRC Press