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Open Journal of Mathematical Sciences

Vol. 2 (2018), Issue 1, pp. 209 – 220
ISSN: 2523-0212 (Online) 2616-4906 (Print)
DOI: 0.30538/oms2018.0029

Flip and Hopf Bifurcations of Discrete-Time Fitzhugh-Nagumo Model

Qamar Din\(^{1}\), Sadaf Khaliq
Department of Mathematics, The University of Poonch Rawalakot, Pakistan. (Q.D & S.K)

\(^{1}\)Corresponding Author: qamar.sms@gmail.com

Copyright © 2018 Qamar Din, Sadaf Khaliq. 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.
Received: February 24, 2018 – Accepted: August 10, 2018 – Published: August 14, 2018

Abstract

In this paper, dynamics of a two-dimensional Fitzhugh-Nagumo model is discussed. The discrete-time model is obtained with the implementation of forward Euler’s scheme. We present the parametric conditions for local asymptotic stability of steady-states. It is shown that the two-dimensional discrete-time model undergoes period-doubling bifurcation and Neimark-Sacker bifurcation at its positive steady-state. Furthermore, in order to illustrate theoretical discussion some interesting numerical examples are presented.

Keywords:

Fitzhugh-Nagumo model, Flip bifurcation, Hopf bifurcation.

1. Introduction

In 1961 FitzHugh and Nagumo [1] presented the following two-dimensional model:
\begin{equation}\label{eq0} \left( \begin{array}{c} \frac{dx}{dt} \\ \frac{dy}{dt}\\ \end{array} \right)=\left( \begin{array}{c} c_1\left(x+y-\frac{x^3}{3}\right) \\ \frac{1}{c_1}\left(x-a_1+b_1y\right)\\ \end{array} \right), \end{equation}
(1)
where \(a_1\), \(b_1\) and \(c_1\) are positive constants. Using the forward Euler method to system (1), we get the discrete-time model as follows:
\begin{equation}\label{eq1} \left( \begin{array}{c} x \\ y \\ \end{array} \right)\to\left( \begin{array}{c} x+h c_1\left(x+y-\frac{x^3}{3}\right) \\ y-\frac{h}{c_1}\left(x-a_1+b_1y\right)\\ \end{array} \right), \end{equation}
(2)
where \(h>0\) is step size. For further biological relevance and dynamical analysis of some models that are very close to system (2), we refer to [2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ], and references are therein. We investigate the existence of equilibria for (2) and local asymptotic stability of these steady-states by implementing linearized stability analysis techniques. Also, Neimark-Sacker bifurcation and period-doubling bifurcation are discussed.

2. Existence of equilibria and stability

The steady-states of (2) satisfy the following system of algebraic equations:
\begin{equation}\label{as} x=x+h c_1\left(x+y-\frac{x^3}{3}\right),\ y=y-\frac{h}{c_1}\left(x-a_1+b_1*y\right). \end{equation}
(3)
From (3), it is quite simple to obtain: $$ b_1 x^3+3(1-b_1 ) x-3a_1=0,\ y=\frac{a_1-x}{b_1}. $$ Next, we define the following quantity:
\begin{equation}\label{dis} \Delta:=4b_1(b_1-1)^3-9 a_1^2 b_1^2. \end{equation}
(4)
Then, it follows that
  • If \(\Delta > 0\), then system (2) has three distinct equilibrium points.
  • If \(\Delta = 0\), then system (2) has a multiple equilibrium points.
  • If \(\Delta < 0\), then system (2) has a unique positive equilibrium point.
For \(a_1=4.92\) and \(b_1=0.16\), we have \(\Delta=-5.95649< 0\) and existence for unique positive equilibrium is depicted in Figure 1.

Figure 1. For \(a_1=4.92\) and \(b_1=0.16\) existence of unique positive equilibrium for (2).

For \(a_1\in[0,50]\) and \(b_1\in[0,50]\), the region (blue) where \(\Delta< 0\) and region (red) where \(\Delta>0\) are depicted in Figure 2.

Figure 2. Regions for existence of various equilibria.

Mathematically, we have the following conditions for negativity and positivity of \(\Delta\):
  • \(\Delta< 0\) if and only if \(0< b_1\le1\), or \(b_1>1\) and \(a_1>\frac{2}{3} \sqrt{\frac{-1+3 b_1-3b_1^2+b_1^3}{b_1}}\).
  • \(\Delta>0\) if and only if \(b_1>1\) and \(a_1< \frac{2}{3} \sqrt{\frac{-1+3 b_1-3b_1^2+b_1^3}{b_1}}\).
  • \(\Delta=0\) if and only if \(b_1>1\) and \(a_1=\frac{2}{3} \sqrt{\frac{-1+3 b_1-3b_1^2+b_1^3}{b_1}}\).
Now the Jacobian matrix of (2) evaluated at arbitrary equilibrium \((x,y)\) is given by: $$ J(x,y):=\left( \begin{array}{cc} 1+\left(h-h x^2\right) c_1 & h c_1 \\ -\frac{h}{c_1} & 1-\frac{h b_1}{c_1} \end{array} \right). $$ Moreover, the characteristic polynomial of \(J(x,y)\) is given by:
\begin{equation}\label{ch} \begin{split} P(\lambda)&:=\lambda^2-\left(2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right)\lambda\\ &+1+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}. \end{split} \end{equation}
(5)

Theorem 2.1. [16] Assume that \(\Delta< 0\), then unique positive equilibrium \((x,y)\) has the following topological classification: (i) \((x,y)\) is a sink if and only if $$ \left|2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right|< 2+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}< 2. $$ (ii) \((x,y)\) is a saddle if and only if $$ \left(2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right)^2-4\left(1+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}\right)>0, $$ and $$ \left|2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right|>\left|2+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}\right|. $$ (iii) \((x,y)\) is a source if and only if $$ \left|1+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}\right|>1, $$ and $$ \left|2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right|< \left|2+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}\right|. $$ (iv) \((x,y)\) is non-hyperbolic if and only if $$ \left|2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right|=\left|2+h \left(c_1+h-c_1 x^2\right)- \frac{b_1 h \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}\right|, $$ or $$ \left(c_1+h-c_1 x^2\right)- \frac{b_1 \left(1+c_1 h \left(1-x^2\right)\right)}{c_1}=0 $$ and $$ \left|2-\frac{b_1 h}{c_1}+c_1 h \left(1-x^2\right)\right|\le2. $$

If we choose \(b_1=0.45\), \(c_1=0.15\), \(h\in[0,1]\) and \(x\in[0,10]\), then topological classification for unique positive point of system (2) is shown in Figure 3.

Figure 3. Topological classification for positive equilibrium.

3. Bifurcation analysis

Studying bifurcation analysis for discrete-time models is a topic of great interest. Recently, there are many articles have published for the investigation for period-doubling and Neimark-Sacker bifurcations in discrete-time models [17, 18, 19, 20, 21, 22, 23]. In this section, we explore the parametric conditions under which system (2) undergoes period-doubling and Neimark-Sacker bifurcations at its unique positive equilibrium point. For this, first we discuss the emergence of period-doubling bifurcation at positive equilibrium of system (2). Assume that \(P(-1)=0\), where \(P(\lambda)\) is defined in (5), then system (2) undergoes period-doubling bifurcation as \(h\) varies in a small neighborhood of \(h_0\) defined by $$ h_0:=\frac{b_1+\left(-1+x^2\right) c_1^2-\sqrt{-4 c_1^2+\left(b_1-\left(-1+x^2\right) c_1^2\right){}^2}}{\left(1+\left(-1+x^2\right) b_1\right) c_1}, $$ or $$ h_0:=\frac{b_1+\left(-1+x^2\right) c_1^2+\sqrt{-4 c_1^2+\left(b_1-\left(-1+x^2\right) c_1^2\right){}^2}}{\left(1+\left(-1+x^2\right) b_1\right) c_1}. $$ Secondly, we assume that $$ \left(b_1+\left(-1+x^2\right) c_1^2\right){}^2 \left(-4 c_1^2+\left(b_1-\left(-1+x^2\right) c_1^2\right){}^2\right)< 0. $$ Then system (2) undergoes Neimark-Sacker bifurcation as parameter \(h\) varies in a small neighborhood of \(h_1\) defined by: $$ h_1:=\frac{b_1+\left(-1+x^2\right) c_1^2}{\left(1+\left(-1+x^2\right) b_1\right) c_1}. $$ In order to verify aforementioned mathematical investigation for existence of period-doubling and Neimark-Sacker bifurcations, we choose particular parametric values for system (2) as follows:
  1. [Period-doubling bifurcation] Let \(a_1=2.6\), \(b_1=1.2\), \(c_1=1.9\) and \(h\in[0.3,0.5]\). In this case, system (2) undergoes period-doubling bifurcation as \(h\) varies in a small neighborhood of \(h_0=0.39\). Moreover, the bifurcation diagrams for period-doubling bifurcation are shown in Figure 4 and Figure 5. Moreover, maximum Lyapunov exponents (MLE) are shown in Figure 6 and a chaotic attractor is depicted in Figure 7.
  2. [Neimark-Sacker bifurcation] Taking \(a_1=2.7\), \(b_1=2.5\), \(c_1=0.95\) and \(h\in[0.65,0.72]\). Then system (2) undergoes Neimark-Sacker bifurcation as \(h\) varies in a small neighborhood of \(h_1=0.69\). The diagrams for Neimark-Sacker bifurcation are given in Figure 8 and Figure 9. Furthermore, MLE are shown in Figure 10 and phase portrait at \(h=0.69\) is depicted in Figure 11.

Figure 4. Bifurcation diagram for \(x_n\).

Figure 5. Bifurcation diagram for \(y_n\).

Figure 6. Maximum Lyapunov exponents.

Figure 7. A chaotic attractor at \(h=0.5\).

Figure 8. Bifurcation diagram for \(x_n\).

Figure 9. Bifurcation diagram for \(y_n\).

Figure 10. Maximum Lyapunov exponents.

Figure 11. Phase portrait at \(h=0.69\).

4. Conclusion

The qualitative behavior for a two-dimensional discrete-time Fitzhugh-Nagumo model is investigated. Euler's forward scheme is implemented to obtain the discrete counterpart of the continuous Fitzhugh-Nagumo model. It is investigated that discrete-time model has rich dynamical behavior as compare to its continuous counterpart. The topological classification for steady-state solutions is discussed. Furthermore, parametric conditions for the existence of period-doubling bifurcation and Neimark-Sacker bifurcation are analyzed by taking \(h\( as bifurcation parameter. At the end numerical simulations are provided to illustrate the theoretical discussion.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

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Category: OMS - Vol 2 - 2018