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Phan Van Long Em - Volume 2 - Issue 1-2020, p.98-109. A study of fixed points and hopf bifurcation of hindmarshrose model by Phan Van Long Em (An Giang university, Vietnam) Article Info: Received 10 Sep. 2019, Accepted 20 Oct. 2019, Available online 15 Feb. 2020 Corresponding author: pvlem@agu.edu.vn (Phan Van Long Em PhD) https://doi.org/10.37550/tdmu.EJS/2020.01.002 ABSTRACT In this article, a class of Hindmarsh-Rose model is studied. First, all necessary conditions for the parameters of system are found in order to have one stable fixed point which presents the resting state for this famous model. After that, using the Hopf’s theorem proofs analytically the existence of a Hopf bifurcation, which is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. Moreover, with the suitable assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point. Keywords: Hindmarsh-Rose model, fixed point, Hopf bifurcation, limit cycle 1. Introduction In the beginning of 1980s, Hindmarsh J.L. and Rose R.M. studied a model called Hindmarsh-Rose model, to expose part of the inner working mechanism of the HodgkinHuxley equations, a famous model in study of neurophysiology since 1952. The Hindmarsh-Rose model was introduced as a dimensional reduction of the well-known Hodgkin-Huxley model (Hodgkin A. L., and Huxley A. F., 1952; Nagumo J., et al., 1962; 98 Thu Dau Mot University Journal of Science - Volume 2 - Issue 1-2020 Izhikevich E. M ., 2007; Ermentrout G. B., and Terman D. H ., 2009 ; Keener J. P., and Sney J., 2009 ; Murray J. D., 2010 ). It is constituted by two equations in two variables u and v . The first one is the fast variable called excitatory representing the transmembrane voltage. The second variable is the slow recovery variable describing the time dependence of several physical quantities, such as the electrical conductance of the ion currents across the membrane. The Hindmarsh-Rose equations (HR) are given by  du 3 2  dt  u  f (u, v)  v  au  bu  I ,   dv  v  g (u, v)  c  du 2  v,  dt (1) where u corresponds to the membrane potential, v corresponds to the slow flux ions through the membrane, I corresponds to the applied extern current, and a, b, c, d are parameters. Here, I , a, b, c, d are real numbers. The paper is organized as follows. In section 2, a study of fixed point is investigated and all necessary conditions for the parameters of Hindmarsh-Rose model are found in order to have a stable focus. In section 3, the system undergoes subcritical Hopf bifurcation is shown. And finally, conclusions are drawn in Section 4. 2. A study of fixed points Equilibria or stability are tools to study the dynamic of fixed points. In mathematics, a fixed point of a function is an element of the function's domain that is mapped to itself by the function. This paper focuses on the fixed points of the system (1) given by the resolution of the following system v  au 3  bu 2  I  0  f (u, v)  0    2 v  c  du  g (u, v)  0  It implies that au3  (d  b)u 2  c  I  0. (2) Let   d b cI and    . The equation (2) can be written a a u3   u 2    0. To solve this equation, let's use the Cardan's formula after the following variables changes: 99 Phan Van Long Em - Volume 2 - Issue 1-2020, p.98-109. u   d b ( d  b) 2 2(d  b)3 c  I ,p , q   , then  3  p  q  0. 2 3 3a 3a 27a a Let now   4 p3  27q 2 . If   0 , then the equation (2) admits only one root and hence the system (1) admits a unique fixed point. Now, if   0 , then the system (1) admits two fixed points, and finally if   0 , the system (1) admits three fixed points (see Figure 1). The Jacobian matrix of the system (1) is written as the following:  f (u, v)  u A(u )    g (u, v)   u f (u, v)  2 v   3au  2bu 1   . g (u, v)   2du 1  v  Let (u*, v*) be one fixed point of (1), we have Det ( A(u*)   I 2 )   2  Tr ( A(u*))  Det ( A(u*)), where Tr ( A(u))  3au 2  6u  1 and Det ( A(u))  3au 2  4u. The reduced discriminant of Tr ( A(u)) is  '  b2  3a. If b2  3a , then Tr ( A(u)) admits two real roots given by b  b2  3a b  D b  b2  3a b  D and uTr 2  with D  b2  3a . uTr1    3a 3a 3a 3a Two roots of Det ( A(u)) is ( d  b)  ( d  b) 2 (d  b)  (d  b) 2 d b and uDet 2  uDet1   2  0. 3a 3a 3a The nature of fixed points is rapported in Table 1. TABLE 1: Stability of fixed point If b2  3a, then Tr ( A(u))  0 for all values of u and in this case, the fixed point is only stable focus or stable node. Morever, in this study, the model is needed to generate the 100 Thu Dau Mot University Journal of Science - Volume 2 - Issue 1-2020 potential actions, it is necessary for the existence of a limit cycle. In the other word, it is need to have an unstable focus or a center. So the condition b2  3a is chosen to be in the region IV of Table 1. The infimum and superimum in the region IV are given by bD bD and M  L . 3a 3a To observe the behavior of the system (1) like Figure 1, we fix the values of parameters as the following a  1, b  3, c  1, d  5, I  0. Then, the system (1) becomes  du 3 2  dt  v  u  3u (3)   dv  1  5u 2  v  dt The system (3) has three fixed points: A  (1.618033989, 12.090169948), B  (1, 4), C  (0.618033989, 0.909830058). In Figure 1(a), we simulated two nullclines, u  0 in red and v  0 in green. The intersection point of these two nullclines is three fixed points A, B, C and one orbit of (3) is represented in blue and it is a limit cycle. Figure 1: Numerical results obtained for two nullclines u  0 in green and v  0 in blue. The intersection points are fixed points A, B and C. The red curve is the limit cycle. At the point A, we get Det ( A)  1.381966013 and Tr ( A)  18.562305903, so A is a stable node. At the point B , we get Det ( B)  1 and Tr ( B)  10 , hence B is a 101 Phan Van Long Em - Volume 2 - Issue 1-2020, p.98-109. saddle. At the point C , we get Det (C)  3.618033991 and Tr (C)  1.562305899, so C is a instable focus. 3. existence and direction of hopf bifurcation This section focuses on the existence and the direction of Hopf bifurcation, which corresponds to the passage of a fixed point to a limit cycle under the effect of variation of a parameter. Recall the Hopf's theorem (Dang-Vu Huyen, and Delcarte C., 2000). Theorem 1. Consider the system of two ordinary differential equations  u  f (u, v, a) (4)  v  g (u, v, a) Let (u*, v*) a fixed point of the system (4) for all a . If the Jacobian matrix of the system (4) at (u*, v*) admits two conjugate complex eigenvalues, 1,2 (a)   (a)  iw(a) and there is a certain value a  ac such that  (ac )  0, w(ac )  0 and  (a) (ac )  0. a Then, a Hopf bifurcation survives when the value of bifurcation parameter a passes by ac and (u*, v*, ac ) is a point of Hopf bifurcation. Moreover, let c1 in order that c1     2 F  2G  2 F  2 F  2G  2G 1    16w(ac )  u 2 u 2 u 2 uv u 2 uv  G  G  F  F  F  G  F  F  G  G  2  2    , 2 2   3 v uv v uv v v   u uv 2 u 2v v 3  2 2 2 2 2 2 3 3 3 (5) 3 where F and G are given by the method of Hassard, Kazarinoff and Wan (Dang-Vu Huyen, and Delcarte C., 2000). We can distinguish different cases TABLE 2: Stability of the fixed points according to Hopf bifurcation  (ac )  0 a c1  0 c1  0 a  ac stable equilibrium stable equilibrium and no periodic orbit and unstable periodic orbit a  ac unstable equilibrium unstable equilibrium and stable periodic orbit and no periodic orbit 102 Thu Dau Mot University Journal of Science - Volume 2 - Issue 1-2020  (ac )  0 a a  ac unstable equilibrium unstable equilibrium and stable periodic orbit and periodic orbit a  ac stable equilibrium stable equilibrium and no periodic orbit and unstable periodic orbit Now this theorem is applied to the Hindmarsh-Rose model in which a represents the bifurcation parameter  du 3 2  dt  v  au  3u  I (6)  dv 2   1  5u  v  dt Let (u*, v*) a fixed point of the system (6). Let u  u1  u * and v  v1  v * , then  3 2 u1  f (u1 , v1 , a)  (v1  v*)  a(u1  u*)  3(u1  u*)  I  2  v1  g (u1 , v1 , a)  1  5(u1  u*)  (v1  v*) With a development of the functions f and g at the neighborhood of (0,0, a) , the above systems become f f  u1  u1 u (0, 0, a)  v1 v (0, 0, a)  F (u1 , v1 , a)  1 1  v  u g (0, 0, a)  v g (0, 0, a)  G(u , v , a) 1 1 1  1 1 u1 v1 where F (u1 , v1 , a) and G(u1 , v1 , a) are the nonlinear terms, then  2 u1  (3au * 6u*)u1  v1  F (u1 , v1 , a)   v1  10u * u1  v1  G (u1 , v1 , a) with F (u1 , v1 , a)  au13  (3au * 3)u12 and G(u1 , v1 , a)  5u12 . Now, (0,0, a) is a fixed point of the system. The Jacobian matrix is given by  3au *2 6u * 1  A . 1  10u * 103 Phan Van Long Em - Volume 2 - Issue 1-2020, p.98-109. The characteristic polynomial Det ( A   I 2 )   2  (3au *2 6u * 1)  3au *2 4u *. Let P(a)  Tr ( A) and Q(a)  Det ( A) . We get  2  P(a)  Q(a)  0. Hence, the Jacobian matrix admits a pair of conjugate complex eigenvalues if 1 Det ( A)  Tr ( A) 2 and the above equation has the following roots 4 1,2   (a)  iw(a), with  (a)   3au *2 6u * 1 and w(a)  3au *2 4u *  (a)2 . 2 Moreover, the value ac of a , for which the real part of these eigenvalues is null, is given by the equations P(ac )  0 and Q(ac )  0 , then ac  6u * 1 4 1 and ac   u*  . 2 3u * 3u * 10 Moreover,  3u *2 (ac )   . a 2 Thus,  (ac )  0, w(ac )  0 and  ( I ) (ac )  0 , then ac is a bifurcation Hopf value of a the parameter a. In the following, the direction and the stability of Hopf bifurcation are investigated. To do this, let’s determine an eigenvector v1 associated with the eigenvalue 1 , obtained by resolving the system (1  i 10u * 1)u  v  0 u   (A  1 I 2 )    0   10u * u  1  i 10u * 1 v  0 v     A solution of this system is an eigenvector associated with 1 given by 1   V1   .  1  i 10 u *  1   The base change matrix is given by 104 Thu Dau Mot University Journal of Science - Volume 2 - Issue 1-2020 1 P   Re(V1 )  Im(V1 )     1  . 10u * 1  0 Then  10u * 1 0  1  . 10u * 1  1 1 P 1  Now let the variable change u  u1   u2   u2  1  1     P       P  .  v1   v2   v2   v1  Hence        u2   P 1  u1   P 1 AP  u2   P 1  F (u2 , v2 , a)  .        G(u , v , a)   v2   2 2   v2   v1    (a) w(a)  Let A '(a)  P 1 AP    . Then, for a  ac , it implies that  w(a)  (a)    w(ac )  u2   w(ac )v2  F (u2 , v2 , ac )  0 A '(ac )    0    w(ac ) v2  w(ac )u2  G (u2 , v2 , ac ) with  F (u2 , v2 , ac )   F (u2 , v2 , ac )  1 .    P   G ( u , v , a )  G(u2 , v2 , ac )  2 2 c   Then  F (u2 , v2 , a)  au23  (3au * 3)u22  1  au23  (3au * 2)u22   G(u2 , v2 , a)  10u * 1  Let c1 be given by the equation (5). The functions F and G depend only on u2 , the coefficient c1 is given by c1   1 2 F  2G 3 F (0, 0, a ) (0, 0, a )  (0, 0, ac ). c c 16w(ac ) u22 u22 u23 105 Phan Van Long Em - Volume 2 - Issue 1-2020, p.98-109. At the point (u2 , v2 )  (0,0) and for a  ac , it implies that w(ac )  10u * 1 , and 1 4 .  3acu * 3 3acu * 2  16 10u * 1 10u * 1 3  6ac  3ac2u *2  acu * 2  .  4(10u * 1) c1  6ac  Theorem 1 permits to deduce the direction and the stability of Hopf bifurcation from  the signs of (ac ) and c1 . Now we apply this theorem in fixing all parameters values a except the bifurcation parmaeter a. Let I  0, the system (6) becomes  du 3 2  dt  v  au  3u   dv  1  5u 2  v  dt (7) 2 1 The fixed points are given by resolving the equation u 3  u 2   0. a a Let u   2 42 16 1 , p   2 ,q   , 3 3a 3a 27a a then  3  p  q  0. Let now   4 p3  27q 2 . We choose arbitrarily one condition over a, in order to have only a fixed point, it means   4 2 4 2   0  a   ;  ;   .    3 3 3 3   With those values of a, we get u *(a)   9a 27a  32  27 3a  16 3 2 2  2 3  27a  32  27 3a  16 3   42 3 . 3.2 3 a  9a 27a  32  27 3a  16 3   22 3  9a  1 1 3 6 2 2 1 3 2 2 1 3 32 3 a 9a 27a  32  27 3a  16 3 1 1 3 6 1 1 3 6 2 2 106 2 1 3 3 1 3 Thu Dau Mot University Journal of Science - Volume 2 - Issue 1-2020 Then, ac  6u *(a)  1 . Moreover, ac is solution of the equation 3u *(a)2 6u *(a)  1  a  0. 3u *(a)2 (8) Figure 2: (a) The resolution of the equation (8) gives two solutions over  10;10 , corresponding to the intersections with the abscisses axis. (b) We are interested in the case where a  0, so ac   2.55165;2.5517 The graphic resolution of the equation (8) gives two solutions over  10;10 (see Figure 2(a)). Here, we are interested in the case where a  0, so ac   2.55165;2.5517 (see Figure 2(b)). With these values of ac , we get u*  0.54  2 1 1 , 3acu *2 4u*  4.392187794   3acu *2 6u * 1  1.526.10 5. 10 4 Moreover, c1  6ac  So, we 3 3ac2u *2  acu * 2   15.632152  0.  4(10u * 1) have c1  0,  (ac )  0. a u*, v*, ac    0.54, 0.46, ac  2.551655 From Theorem 1, is a Hopf bifurcation point. Moreover, for a  ac , the fixed point is unstable with a stable periodic orbit; while for a  ac , the fixed point is stable without periodic orbit (see Figure 3). Figure 3(a) shows the phase portrait in the plane (u, v) of the system (7) with a  2.54, and a stable limit cycle for a value a  2.54  ac . Figure 3(b) presents the time series corresponding to (t , u ) . Figure 3(c) shows the phase portrait in the plane (u, v) of the system (7) with a  2.57, and a focus stable for a value a  2.57  I c . Figure 3(d) presents the time series corresponding to (t , u). 107