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Hopf Bifurcation and Stability of Periodic Solutions for Van Der Pol

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Hopf Bifurcation and Stability of Periodic Solutions for Van Der Pol Nonlinear Analysis 62 (2005) 141–165 www.elsevier.com/locate/na Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delayଁ WenwuYu, Jinde Cao ∗ Department of Mathematics, Southeast University, Nanjing 210096, China Received 5 January 2005; accepted 9 March 2005 Abstract In this paper, the van der Pol equation with a time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem. © 2005 Published by Elsevier Ltd. Keywords: Van der Pol equation; Time delay; Hopf bifurcation; Periodic solutions 1. Introduction The well-known van der Pol equation, which describes the oscillations is the second-order nonlinear damped system governed by x(t)˙ = y(t) − f (x(t)), y(t)˙ =−x(t). (1.1) This model is considered as one of the most intensely studied system in nonlinear system [9,12,23] and has served as a basic model in physics, electronics, biology, neurology and ଁ This work was jointly supported by the National Natural Science Foundation of China under Grant 60373067, the 973 Program of China under Grant 2003CB317004, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053, Qing-Lan Engineering Project of Jiangsu Province, China. ∗ Corresponding author. Tel.: +86 25 83792315; fax: +86 25 83792316. E-mail address: [email protected] (J. Cao). 0362-546X/$ - see front matter © 2005 Published by Elsevier Ltd. doi:10.1016/j.na.2005.03.017 142 W. Yu, J. Cao / Nonlinear Analysis 62 (2005) 141–165 so on. Many efforts have been made to find its approximate solutions [1,10,24].Aswe know, in ordinary differential equation, a non-constant periodic solution can arise a Hopf bifurcation. This occurs when a eigenvalue crosses the imaginary axis from left to right as a real parameter in the equation passes through a critical value [15,21].In[10], the authors proposed a class of relaxation algorithms for finding the periodic steady-state solution of a van der Pol oscillation. In [1], the periodic solution of van der Pol equation is given in the form of a series converging for all values of the damping parameter. Recently, dynamical systems with time delays have been found in neural networks [2–8,11,18,20]. It is worth noting that the dynamical characteristics (including stable, unstable, oscillatory, and chaotic behavior) of neural networks with time delays have become a subject of intense research activities ([2–8,11,18,20] and the references cited therein), and neural networks involving persistent oscillations such as limit cycle may be applied to pattern recognition and asso- ciative memory. Thus it is of great interest to understand the mechanism of neural networks that cause and sustain such periodic activities. However, neural networks are complex and large-scale nonlinear dynamical systems. For simplicity, many researchers have directed their attention to the study of simple systems. This is still useful since the complexity found may be carried over to large networks. In [22], Murakaimi introduced a discrete time delay into the van der Pol equation (1.1) and the following delayed differential equation was obtained: x(t)˙ = y(t − ) − f(x(t − )), y(t)˙ =−x(t − ). (1.2) And the author discussed in detail the existence of periodic solution by using the center man- ifold approaches. However, the stability of bifurcating periodic solution was not discussed. Clearly, if =0, (1.2) can reduce to (1.1). In [17], Liao studied Hopf bifurcation and stability of periodic solutions for van der Pol equation with distributed delay. Therefore, dynamical analysis of time-delay systems is an important topic in many fields [13,14,16,19,25]. In this paper, we will consider the van der Pol equation with a discrete delay, and study the existence of a Hopf bifurcation and the stability of bifurcating periodic solutions of Eq. (1.2). The obtained results find that both of them depend on the parameters a and the delay . The organization of this paper is as follows: In Section 2, we will discuss the stability of the trivial solutions and the existence of Hopf bifurcation. In Section 3, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions will be given by using the normal form method and center manifold theorem introduced by Hassard et al. [14]. In Section 4, numerical simulations aimed at justifying the theoretical analysis will be reported. 2. Stability of trivial solutions and existence of Hopf bifurcation In this section, we consider the following delayed differential equation: x(t)˙ = y(t − ) − f(x(t − )), y(t)˙ =−x(t − ), (2.1) W. Yu, J. Cao / Nonlinear Analysis 62 (2005) 141–165 143 where f(x)= ax + bx2, a and b are positive and real. The linear equation of (2.1) are as follows: x(t)˙ =−ax(t − ) + y(t − ), y(t)˙ =−x(t − ). (2.2) Clearly, the point (0, 0) is the unique equilibria of (2.2). The characteristic equation of the linearized system (2.2) is + ae− −e− det = 0. (2.3) e− By simple calculation, we obtain the following characteristic equation: − − 2 + ae + e 2 = 0. (2.4) Lemma 2.1. (i) If a>2 holds, then (2.4) has a pair of purely imaginary roots ±iw0l when = nl (l = 1, 2; n = 0, 1, 2,...), where √ a ± a2 − 4 w l = , (2.5) 0 2 2n + /2 nl = (l = 1, 2; n = 0, 1, 2,...). (2.6) w0l (ii) If 0 <a2 holds, then (2.4) has a pair of purely imaginary roots ±iw0 when = j (j = 0, 1, 2,...), where w0 = 1, (2.7) a j = arcsin + 2j. (2.8) 2 Proof. Suppose that = iw, with w>0, is a root of Eq. (2.4), then we obtain −w2 + aiw(cos(w) − i sin(w)) + (cos(2w) − i sin(2w)) = 0. Separating the real and imaginary parts, we have w2 − aw sin(w) − cos(2w) = 0, (2.9) aw cos(w) − sin(2w) = 0. Which can be rewritten as the following two equations: w2 − aw sin(w) − cos(2w) = 0, (2.10) cos(w) = 0, 144 W. Yu, J. Cao / Nonlinear Analysis 62 (2005) 141–165 or w2 − aw sin(w) − cos(2w) = 0, aw (2.11) sin(w) = . 2 Next we discuss it in two cases: Case I: From cos(w)=0, it follows that cos(2w)=−1. If sin(w)=−1, then substituting 2 this into the first equation of (2.10), we have w + aw + 1 = 0. Suppose w1 and w2 are the two roots of this equation. Clearly, w1 + w2 =−a<0,w1w2 = 1 > 0. The equation have two negative roots. Since we suppose w>0, we choose sin(w) = 1. Substituting this into the first equation of (2.10), we obtain w2 − aw + 1 = 0. (2.12) √ 2 For the case a>2, we have two roots of Eq. (2.12): w0l =a ± a − 4/2, thus nl =2n+ /2/w0l (l = 1, 2; n = 0, 1, 2,...). The proof is completed. Case II: From sin(w) = aw/2, we have cos(2w) = 1 − 2 sin2(w) = 1 − a2w2/2. Substituting these into the first equation of (2.11), from 0 <a2, we obtain w0 = 1 and j = arcsin a/2 + 2j. The proof is completed. Denote () = () + iw(), is the root of Eq. (2.4) satisfying (nl) = 0,w(nl) = w0l, where nl is defined by (2.6). Lemma 2.2. If a>2 holds, then we have d(nl) > 0. (2.13) d Proof. Taking the derivative of with respect to in (2.4), we have − − − − 2 + a e + a(−)e + a(− )e − 2( + )e 2 = 0, it follows that: d() a2e− + 2e−2 = . (2.14) d 2 + ae− − ae− − 2e−2 W. Yu, J. Cao / Nonlinear Analysis 62 (2005) 141–165 145 For the sake of simplicity, we denote w0l and nl by w, , respectively, then −aw2 −iw + w −2iw d = e 2i e d 2iw + ae−iw − aiwe−iw − 2e−2iw = −aw2[cos(w)−i sin(w)]+2iw[ cos(2w)−i sin(2w)] 2iw+a[cos(w)−i sin(w)]−aiw[cos(w)−i sin(w)]−2[cos(2w)−i sin(2w)] = [−aw2 cos(w)+2w sin(2w)]+i[aw2 sin(w)+2w cos(2w)] [a cos(w)−aw sin(w)−2 cos(2w)]+i[2w−a sin(w)−aw cos(w)+2 sin(2w)] . Let Q =[a cos(w) − aw sin(w) − 2 cos(2w)]2 +[2w − a sin(w) − aw cos(w) + 2 sin(2w)]2. d Q Re =[−aw2 cos(w) + 2w sin(2w)] d ×[a cos(w) − aw sin(w) − 2 cos(2w)] +[aw2 sin(w) + 2w cos(2w)] ×[2w − a sin(w) − aw cos(w) + 2 sin(2w)] (2.15) Q d = (aw2 − w)( w − a) Re 2 2 d =nl = w(2aw2 − a2w − 4w + 2a) = w(2a2w − 2a − a2w − 4w + 2a) = w2(a2 − 4)>0, this completes the proof. Denote () = () + iw(), is the root of Eq. (2.4) satisfying (j ) = 0,w(j ) = w0, where j is defined by (2.8). Lemma 2.3. If 0 <a2 holds, then d(j ) > 0. (2.16) d 146 W. Yu, J. Cao / Nonlinear Analysis 62 (2005) 141–165 Proof. If 0 <a<2, from (2.15) we have Q d =[−a + ( )][a − a2/ − ( − a2)/ ] Re cos 2 sin 2 cos 2 2 1 2 d =j +[a2/2 + 2 − a2][2 − a2/2 − a cos + 2 sin(2)] = a cos (a cos + a2/2 − 2) + (2 − a2/2)(2 − a2/2 + a cos ) = a2 cos2 + a cos (a2/2 − 2) + (2 − a2/2)a cos + (2 − a2/2)2 = a2(1 − a2/4) + (4 + a4/4 − 2a2) = a2 − a4/4 + 4 + a4/4 − 2a2 = 4 − a2 > 0.
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