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Complex dynamics in a two-dimensional noninvertible map
Chaos, Solitons and Fractals 39 (2009) 1798–1810 www.elsevier.com/locate/chaos
Complex dynamics in a two-dimensional noninvertible map Yinghui Gao
*
Department of Mathematics, Beihang University, Beijing 100083, PR China Key Laboratory of Mathematics, Informatics and Behavioral Semantics (Beihang University and Peking University), Ministry of Education, China Accepted 18 June 2007
Abstract A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction If one-dimensional noninvertible maps have given rise to a lot of publications, the study of two-dimensional noninvertible maps remained a long time in an underdeveloped state. It is only since these last years that the interest for this subject has increased. One reason of this situation is due to the fact that more and more mathematical models of dynamical process are related to such Maps [9]. In [9], the authors considered the two-dimensional noninvertible map investigated in this paper, and they described the construction of chaotic areas for this map. But they did not prove the existence of chaos theoretically. Here we will do it. In addition, we will study this kind of map in another point of view in this paper. As we have seen, many non-linear systems have parameters which appear in the defining systems of equations. As the parameter is changed, changes may occur in the qualitative structure of the orbits for certain parameter values [2]. A central problem in non-linear dynamics is that of discovering how the properties of orbits change and evolve as a parameter of a dynamical system is changed [13]. So, we mainly discuss the changing properties of the map. Among them, bifurcations and chaos are very important phenomena which many excellent researchers [8,10–12,15] apply themselves to study. Here, we firstly study the stability of the fixed points, then analyze the bifurcations of the system using center manifold theorem and bifurcation theory [2,14]. We find that there exist pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation. *
Address: Department of Mathematics, Beihang University, Beijing 100083, PR China. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.051
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In 1975, Li and Yorke [4] proved that period three implies chaos on the interval. From then on, many mathematicians began to devote their research to further exploring the complex mechanics of the non-linear transformations, especially in high-dimensional systems. Among them, the result, given by Marotto [6], on the criterion for chaos existence in high-dimensional systems is, of course standout and surprising. A snap-back repeller, just like period three in non-linear transformations, is regarded as an inducement to produce chaos in high-dimensional systems. Thus, this criterion is widely used in proving the existence of chaos in many non-linear dynamical systems [5]. Many years after this work first appeared, it was brought to Marotto’s attention that there is a minor technical flaw in the reasoning he used in some of his arguments. A fixed point z is referred to as repelling under f if all eigenvalues of Df(z) exceed 1 in magnitude. But z is expanding only if kf(x) f(y)k > skx yk, where s > 1, for all x,y sufficiently close to z with x 5 y. Although all expanding fixed points are repelling, the converse is not true. As a result, Marotto’s original definition of a snap-back repeller and proof that the existence of such a point implies chaos are in error. But Marotto quickly realized that the flaw is of a minor technical nature. However, during the past decade or so several papers have appeared that first overstate the severity of the error, and then propose correct but profoundly altered and weakened versions of his theorem. Marotto believed they were mistaken and gave a better version of Marotto theorem [7]. Here, we rigorously prove that the two-dimensional noninvertible map possesses chaotic phenomenon using the better version. This paper is organized as follows. In Section 2, we describe first the two-dimensional noninvertible map, then analyze two basic features of the map including the location and stability of its fixed points. And in this section, we show that there exist pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation using center manifold theorem and bifurcation theory. In Section 3, it is rigorously proven that the two-dimensional noninvertible map possesses a snap-back repeller. The results of numerical simulations and the computations of Lyapunov exponents are presented in Section 4 to verify the theoretical analysis and display the complex and interesting dynamics.
2. Existence and stability of fixed points and bifurcations 2.1. Existence and stability of fixed points Consider a two-dimensional noninvertible map [1,3,9] which takes the form ( x # ax þ y; y # bx þ x3 ;
ð1Þ
where b < 0. The fixed points of map (1) satisfy the following equations: ( ða 1Þx þ y ¼ 0; x3 þ bx y ¼ 0: By a simple analysis, it is easy to obtain the following proposition. Proposition 1 (1) If 1 a b 6 0, then map (1) has a unique fixed point O(0, 0). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) If 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b > 0, then map (1) has three fixed points: ð0; 0Þ; 1 a b; ð1 aÞ 1 a b , 1 a b; ð1 aÞ 1 a bÞ. We now investigate the linear stability of the fixed points for (1). The Jacobian matrix J of map (1) evaluated at the fixed point (x0, y0) is given by a 1 J¼ 3x20 þ b 0 and the characteristic equation of the Jacobian matrix J can be written as k2 ak ð3x20 þ bÞ: A simple calculation shows the stability of fixed points as the following.
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Proposition 2. The fixed point (0, 0) of map (1) is stable if one of the following conditions is satisfied: (i) (ii) (iii) (iv)
a2 + 4b < 0, a2 + 4b = 0, a2 + 4b > 0, a2 + 4b > 0,
1 < b < 0; |a| < 2; 0 < a < 2, 1 a b > 0; 2 < a < 0, 1 + a b > 0;
and if 1 a b > 0, then the two fixed points always unstable.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a b; ð1 aÞ 1 a b , 1 a b; ð1 aÞ 1 a b are
2.2. Bifurcations In analysis of bifurcations b is as bifurcation parameter. Let ~b ¼ b ð1 aÞ, we consider the parameter ~b as a new and dependent variable, then map (1) becomes: 8 > < x # ax þ y y #ð1 aÞx þ x~b þ x3 : > :~ ~ b#b 1 0 0 1 0 1 1 u x 1 0 C B1 a @ A @ A Let T ¼ @ 1 1 0 A and use the translation y~ ¼ T v , then map (2) becomes l b 0 0 0 1 1 10 1 0 0 1 f ðu; v; lÞ u 1 0 0 u C CB C B B C B @ v A#@ 0 a 1 0 A@ v A þ @ gðu; v; lÞ A; 0 l 0 0 1 l
ð2Þ
where 3 1 a u 1 a u þv l þv ; a2 1a a2 1a 3 1 u 1 u gðu; v; lÞ ¼ þv lþ þv : a2 1a a2 1a f ðu; v; lÞ ¼
By center manifold theory, we know that the stability of (u, v) = (0, 0) near l = 0 can be determined by studying a oneparameter family of equations on a center manifold, which can be represented as follows. Wc(0) = {(u, v, l) 2 R3|v = h(u, l),h(0, 0) = 0,Dh(0, 0) = 0}, for u and l sufficiently small. We assume a center manifold of the form hðu; lÞ ¼ a1 u2 þ a2 ul þ a3 l2 þ a4 u3 þ Oððjuj þ jljÞ3 Þ: The center manifold must satisfy N ðhðu; lÞÞ ¼ hðu þ f ðu; hðu; lÞ; lÞÞ ða 1Þhðu; lÞ gðu; hðu; lÞ; lÞ ¼ 0: The map restricted to the center manifold is given by u#f~ ðu; lÞ ¼ u
1 1 u3 þ Oððjuj þ jljÞ3 Þ: ul a2 ða 2Þð1 aÞ2
2~ 3~ ~ ~ o2 f~ 1 6 Since f~ ð0; 0Þ ¼ 0; oouf ð0; 0Þ ¼ 1; oolf ð0; 0Þ ¼ 0; oouf2 ð0; 0Þ ¼ 0; ouol ð0; 0Þ ¼ a2 –0ða–2Þ; oouf3 ð0; 0Þ ¼ ða2Þð1aÞ 2 –0 ða–1; 2Þ, the fixed point (u, l) = (0, 0) is a pitchfork bifurcation point for map (2). The number of fixed points is chan3~ o2 f~ 1 ð0; 0Þ ¼ ð1aÞ ged at (u(l), v(l)) = (0, 0) as l = 0, and oouf3 ð0; 0Þ= ouol 2 < 0, so there are three fixed points for l < 0 and one fixed point for l > 0. From the above analysis, we have the theorem.
Theorem 1. If a 5 1,2, then map (1) undergoes a pitchfork bifurcation at (0, 0) for b = 1 a. Moreover, there are three fixed points for b < 1 a and one fixed point for b > 1 a. We next consider the flip bifurcation of map (1). The characteristic equation associated with the linearization of map (1) about the fixed point (0, 0) is given by k2 ak b ¼ 0:
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Correspondingly, the eigenvalues of the fixed point (0, 0) are k1 = 1, k2 = 1 + a for b = 1 + a(a < 1). The condition |k2| = (1 + a) 5 1 leads to a 5 2. Let ~b ¼ b ð1 þ aÞ, then map (1) becomes 8 > < x # ax þ y; ð3Þ y # ð1 þ aÞx þ x~b þ x3 ; > :~ ~ 1 0 0 1 0 1 b # b: 1 u x 1 0 C B 1þa @ y A ¼ T @ v A, map (3) can be and use the translation We construct an invertible matrix T ¼ @ 1 A 1 0 ~ l b written as 0 0 1 1 1 0 0 1 0 1 0 f ðu; v; lÞ u 1 0 0 u C CB C B B C B ð4Þ 1 þ a 0 A@ v A þ @ gðu; v; lÞ A; @ v A#@ 0 0
l where
0
1
0
l
1þa u 1þa u 3 v lþ ðv Þ; aþ2 1þa aþ2 1þa 3 1 u 1 u gðu; v; lÞ ¼ : v lþ v aþ2 1þa aþ2 1þa f ðu; v; lÞ ¼
We again apply the center manifold theorem to determine the nature of the bifurcation of the fixed point (u, v) = (0, 0) at l = 0. There exists a center manifold for (4) which can be represented as follows: W c ð0Þ ¼ fðu; v; lÞ 2 R3 jv ¼ hðu; lÞ; hð0; 0Þ ¼ 0; Dhð0; 0Þ ¼ 0g; We assume h(u, l) = b1u2 + b2ul + b3l2 + b4 u3 + O((|u| + |l|)3). Equating terms of like powers to zero gives b1 ¼ 0;
b2 ¼
1 ða þ 2Þ2 ða þ 1Þ
;
b3 ¼ 0;
b4 ¼
1 ða þ 2Þ2 ða þ 1Þ3
:
Thus, the map restricted to the center manifold is given by f^ : u# u
1 1 u3 þ Oððjuj þ jljÞ3 Þ; ul aþ2 ða þ 2Þða þ 1Þ2
the second iteration of f^ is given by f^ 2 : u#u þ We have of^ a1 ¼ ol 0 1 a2 ¼ @ 2
2 2 u3 þ Oððjuj þ jljÞ3 Þ: ul þ aþ2 ða þ 2Þða þ 1Þ2
! o2 f^ o2 f^ þ2 ou2 ouol
ð0;0Þ
o2 f^ ou2
!2
¼
!1 1 o3 f^ A þ 3 ou3
2 –0; aþ2 ¼
ð0;0Þ
2 ða þ 2Þða þ 1Þ2
–0:
of^ of^ 2 o2 f^ 2 2 ð0; 0Þ ¼ 0; ð0; 0Þ ¼ f^ ð0; 0Þ ¼ 0; ð0; 0Þ ¼ 1; –0; ol ouol ou aþ2 , o3 f^ 2 o2 f^ 2 6 3 ð0; 0Þ ð0; 0Þ ¼ < 0: ou ouol ð1 þ aÞ2
o3 f^ 2 12 ð0; 0Þ ¼ –0; ou3 ða þ 2Þða þ 1Þ2
Summarize the above analysis into the following theorem. Theorem 2. Map (1) undergoes a flip bifurcation at (0, 0) for b = 1 + a if a < 1 and a 5 2. Moreover, the period two points lie on left side of b = 1 + a, and if a < 2 (resp. 2 < a < 1) the period two points are stable (resp. unstable).
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pffiffiffi pffiffiffi pffiffiffi pffiffiffi Let ~x ¼ x a; ~y ¼ y ð1 aÞ a; ~b ¼ b ð1 2aÞ a > 12 , we transform the fixed point ð a; ð1 aÞ aÞ of map (1) to the origin and take ~b as a new dependent variable, then map (1) becomes 10 1 0 0 1 0 1 ~x 0 ~x a 1þa 0 pffiffiffi CB C B B C B C 0 A@ ~y A þ @ ~xb~ þ 3 a~x2 þ ~x3 A: ð5Þ @ ~y A#@ 1 0 p ffiffi ffi b~ 0 a 1 b~ 0 Let 0
1 B1 þ a B B T ¼B B1 @
1 1 1 pffiffiffi C 2 a C C 1 aC 1 pffiffiffi C 2 aA
0
00 11 0 1 ~x u and use the translation @ ~y A ¼ T @ v A, then map (5) becomes ~b l 1 10 1 0 0 1 0 f ðu; v; lÞ u 1 0 0 u C CB C B B C B 1 þ a 0 A@ v A þ @ gðu; v; lÞ A; @ v A#@ 0 0 l 0 0 1 l where 2 3 # pffiffiffi u 1 u 1 u 1 ; v pffiffiffi l l þ 3 a v pffiffiffi l þ v pffiffiffi l 1þa 2 a 1þa 2 a 1þa 2 a " # 2 3 pffiffiffi 1 u 1 u 1 u 1 : gðu; v; lÞ ¼ v pffiffiffi l l þ 3 a v pffiffiffi l þ v pffiffiffi l aþ2 1þa 2 a 1þa 2 a 1þa 2 a 1þa f ðu; v; lÞ ¼ aþ2
"
We again apply the center manifold theorem to determine the nature of the bifurcation of the fixed point (u, v) = (0, 0) at l = 0. Using the same method, we can get the map restricted to the center manifold pffiffiffi 2 3 a 1þa a 16 f : u# u þ u3 þ Oððjuj þ jljÞ3 Þ; ul þ u2 þ pffiffiffi l2 2þa ð1 þ aÞð2 þ aÞ 4 að2 þ aÞ ð1 þ aÞ2 ð2 þ aÞ2 4 16 f 2 : u#u u3 þ Oððjuj þ jljÞ3 Þ; ul 2þa ð1 þ aÞ2 ð2 þ aÞ a1 ¼
4 16 –0; –0; a2 ¼ 2þa ð1 þ aÞ2 ð2 þ aÞ
of 2 o2 f 2 4 ð0; 0Þ ¼ 0; ð0; 0Þ ¼ –0; ol ou ol 2þa , o3 f 2 96 o3 f 2 o2 f 2 24 ð0; 0Þ ¼ –0; ð0; 0Þ ¼ ð0; 0Þ < 0: 2 3 ou3 ou ou ol ð1 þ aÞ ð2 þ aÞ ð1 þ aÞ2 pffiffiffi pffiffiffi Thus, map (1) undergoes a flip bifurcation at ð a; ð1 aÞ aÞ. Similarly, we can prove that pffiffiffi pffiffiffi ð a; ð1 aÞ aÞ is also a flip bifurcation point. pffiffiffi pffiffiffi Theorem 3. Map (1) undergoes a flip bifurcation at ð a; ð1 aÞ aÞ for b ¼ 1 2a a > 12 . Moreover, the period two points lie on left side of b = 1 2a and they are stable. Finally, we give the conditions of existence of Naimark–Sacker bifurcation by using the theorem in [2], here b is still as bifurcation parameter. We will prove that there exists Naimark– Sacker bifurcation at (0, 0) for b = 1. f ð0; 0Þ ¼ 0;
of ð0; 0Þ ¼ 1; ou
The characteristic equation associated with the linearization of map (1) at the fixed point (0, 0) is given by k2 ak b ¼ 0;
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
pffiffiffiffiffiffiffiffiffi
a2 þ4b
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the eigenvalues of the characteristic equation are k; k ¼ , the eigenvalues k; k are complex conjugate for b < a4 . 2 For b = 1, the eigenvalues of the matrix associated with the linearized map (1) at the fixed point (0, 0) are complex conjugate with modulus 1. pffiffiffiffiffiffiffi djkðbÞj Under the conditions a2 < 4 and b = 1, there are jkj ¼ b ¼ 1 and d ¼ ¼ 12 –0. In addition, a 5 0, 1 db 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b¼1 2 4a a A and use the translation x ¼ T u , map (1) leads to kn(1) 5 1,n = 1, 2, 3, 4. Let T ¼ @ 2 2 y v 0 1 becomes pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 4 a2 a C f ðu; vÞ u u B 2 2 ffi Cþ B pffiffiffiffiffiffiffiffiffiffiffiffi ; ð6Þ # A gðu; vÞ v v @ 4 a2 a a
2
2
2
where !3 pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a2 a a f ðu; vÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi uþ v ; 2 2 4 a2 !3 pffiffiffiffiffiffiffiffiffiffiffiffiffi a 4 a2 uþ v : gðu; vÞ ¼ 2 2 Notice that (6) is exactly in the form on the manifold, in which the coefficient a is given by
ð1 2kÞk2 1 n11 n20 n11 j2 jn02 j2 þ Reð kn21 Þ–0; a ¼ Re 1k 2 where 1 n20 ¼ ½ðfuu fvv þ 2guv Þ þ iðguu gvv 2f uv Þ ¼ 0; 8 1 n11 ¼ ½ðfuu þ fvv Þ þ iðguu þ gvv Þ ¼ 0; 4 1 n02 ¼ ½ðfuu fvv 2guv Þ þ iðguu gvv þ 2f uv Þ ¼ 0; 8 and 1 ½ðfuuu þ fuvv þ guuv þ gvvv Þ þ iðguuu þ guvv fuuv fvvv Þ 16 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 3a4 3a2 4 a2 3ð4 a2 Þ 4 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ : ¼ þ 2 4 16 4 4 a2
n21 ¼
Thus 3 a ¼ Reðkn21 Þ ¼ < 0: 8 From the above analysis, we have the theorem. Theorem 4. If a2 < 4,a 5 0,1, then map (1) undergoes a Naimark–Sacker bifurcation at the fixed point (0, 0) for b = 1. Moreover, an attracting invariant closed curve bifurcates from the fixed point (0, 0) for b < 1. 3. Existence of Marotto chaos In this section, we rigorously prove that map (1) possesses chaotic behavior in the sense of Marotto’s definition. We first present Marotto chaos definition and theorem which are quoted from [6,7]. For any map f:Rn ! Rn, and any positive integer k. Let f k represent the composition of f with itself k, times. For a differentiable function f, let Df(z) denote the Jacobian matrix of f evaluated at the point z 2 Rn, and |Df(z)| its determinant. Let Br(z) denote the closed ball in Rn of radius r centered at the point z and B0r ðzÞ its interior. Also let kzk be the usual Euclidean norm of z in Rn.
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Definition 1. [7] Suppose z is a fixed point of f with all eigenvalues of Df(z) exceeding 1 in magnitude and suppose there exists a point x0 5 z in a repelling neighborhood Br(z) of z such that xM = z and |Df(xk)| 5 0 for 1 < k < M, where xk = f k(x0). Then z is called a snap-back repeller of f. Repelling neighborhood means that for any x in such a Br(z), the pre-image points f k(x) remain within the local unstable manifold for all k P 0 (although not necessarily within Br(z)), and f k(x) ! z as k ! 1. Marotto Theorem [6]. If f possesses a snap-back repeller, then the map f is chaotic in the sense of Marotto. That is, there exist (i) a positive integer N such that for each integer p P N, f has a point of period p; (ii) a ‘‘scrambled set’’ of f, i.e. an uncountable set S containing no periodic points of f such that: (a) f(S) S, (b) for every x,y 2 S with x 5 y lim sup kf k ðxÞ f k ðyÞk > 0; k!1
(c) for every x 2 S and any periodic point y of f lim sup kf k ðxÞ f k ðyÞk > 0; k!1
(iii) an uncountable subset S0 of S such that for every x,y 2 S0: lim inf kf k ðxÞ f k ðyÞk ¼ 0: k!1 Now we theoretically give the conditions of existence of chaotic phenomena for map (1) in the sense of Marotto’s definition of chaos. We first give the conditions such that p the fixed point (0, 0) is a snap-back repeller. The eigenvalues associated with ffiffiffiffiffiffiffiffiffi a a2 þ4b the fixed point (0, 0) are given by k ¼ . According to Definition 1, we begin to find a neighborhood Br(O) of 2 (0, 0) in which the norms of conjugate complex eigenvalues exceed 1 for all x,y 2 Br(O). Let s1(x) = a2 + 4(3x2 + b) = 12x2 + a2 + 4b. It is easy to see that ifa2 + 4b < 0 then the equation s1(x) = 0 has two qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 þ4bÞ ða2 þ4bÞ ða2 þ4bÞ ða2 þ4bÞ real roots x1 ¼ ; x2 ¼ . And s1(x) < 0 for x 2 I 1 ¼ ; . 12 12 12 12 2 + ffiffiffiffiffiffiffiffiffiffi b) ffi 1 = 3x2 b 1. Under the condition b< ffiffiffiffiffiffiffiffiffiffi 1 theffiequation s2(x) = 0 has two real roots Letqsffiffiffiffiffiffiffiffiffiffi 2(x) ffi= (3x q qffiffiffiffiffiffiffiffiffiffi ffi q ðbþ1Þ ðbþ1Þ ðbþ1Þ ðbþ1Þ ; x ¼ . And s2 (x) > 0 for all x 2 I 2 ¼ ; . xþ ¼ 3 3 3 3
Lemma 1. (i) If a2 < 4 then I0 = I1 \ I2 = I2, (ii) If a2 > 4 then I0 = I1 \ I2 = I1. Moreover, if one of the above two conditions holds, then (0, 0) is a repelling fixed point of (1) in U0 = {(x, y)|x 2 I0,y 2 R}. Due to the definition of snap-back repeller, we need to find one point z 2 Br(O). such that z 5 O,fM(z) = O,|DfM(z)| 5 0 for some positive integer M(M 5 1). In fact, we have ax þ y ¼ X ; ð7Þ bx þ x3 ¼ Y ; and
aX þ Y ¼ 0; bX þ X 3 ¼ 0:
ð8Þ
A f 2 map has been constructed to map the point z(x, y) to the fixed point (0, 0) after two iteration if there are solutions different from (0, 0) for Eqs. (7) and (8). By the calculation, the solutions different from (0, 0) for (8) are obtained as follows: pffiffiffiffiffiffiffi X ¼ b; pffiffiffiffiffiffiffi ð9Þ Y ¼ a b; for b < 1.
pffiffiffiffiffiffiffi Substituting Y þ ¼ a b into (7) and solving x, we have p ffiffiffiffiffiffi ffi x3 þ bx þ a b ¼ 0:
ð10Þ
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We next need to find one real root x* 2 I0 of (10). Case (i) Under the conditions a > 0 and a2 < 4 we have I0 = I2, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2b þ 1 ðb þ 1Þ pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b ¼ þb þ a b > 0; 3 3 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2b 1 ðb þ 1Þ pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b ¼ þb þ a b < 0 for ðb þ 1Þð2b 1Þ2 < a2 b: 3 3 3 3 Thus, (10) has a real nonzero root x* 2 I0. Case (ii) Under the conditions a < 0 and a2 < 4 we have I0 = I2, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b > 0 for ðb þ 1Þð2b 1Þ2 < a2 b; þb 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b < 0: þb 3 3 Thus, (10) has a real nonzero root x* 2 I0. Case (iii) Under the conditions a > 0 and a2 > 4 we have I0 = I1, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a2 8b ða2 þ 4bÞ pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þ a b ¼ þb þ a b > 0; 12 12 12 12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 8b a2 ða2 þ 4bÞ pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þ a b ¼ þb þ a b < 0 for 12 12 12 12 2 ða2 þ 4bÞða2 8bÞ < 123 a2 b:
Thus, (10) has a real nonzero root x* 2 I0. Case (iv) Under the conditions a < 0 and a2 > 4 we have I0 = I1, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þb þ a b > 0 for ða2 þ 4bÞða2 8bÞ2 < 123 a2 b; 12 12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þb þ a b < 0: 12 12 * Thus, (10) has a real nonzero pffiffiffiffiffiffiffi root x 2 I0. Substituting Y ¼ a b into (5) and solving x, we have pffiffiffiffiffiffiffi x3 þ bx a b ¼ 0:
Similarly, we can discuss the conditions for a real nonzero root x* 2 I0. Here, we omit the analysis process. Lemma 2. If one of the following two conditions holds, the equation x3 + bx Y = 0 has a real root x* different from 0, x* 2 I0 for b < 1 and a2 + 4b < 0. (i) a2 < 4,(b + 1)(2b 1)2 < a2b, (ii) a2 > 4,(a2 + 4b)(a2 8b)2 < 123a2b. From (7), we also obtain y* = X ax*. 2 2 Let U 0 ¼ fðx; yÞ xr1 þ yr2 6 1; jx j < r1 ; x r1 2 I 0 ; r2 ¼ jy j þ g; g is some positive constantg:jDf 2 ðzÞj ¼ jDf 2 ðx ; y Þj ¼
½3ðax þ y Þ2 þ bð3x2 þ bÞ ¼ ð3X 2 þ bÞð3x2 þ bÞ, from the above computation, we know that 3X2 + b 5 0,3x*2 + b 5 0, hence |Df 2(z)| 5 0.
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Obviously, if the conditions in Lemmas 1 and 2 are satisfied, then (0, 0) is a snap-back repeller in U 0 . Thus, we have the theorem. Theorem 5. If one of the following conditions are satisfied (i) b < 1,a2 < 4,a2 + 4b < 0,(b + 1)(2b 1)2 < a2b, (ii) b < 1,a2 > 4,a2 + 4b < 0,(a2 + 4b)(a2 8b)2 < 123a2b, then (0, 0) is a snap-back repeller of map (1), and hence map (1) is chaotic in the sense of Marotto’s definition. Example 1. For a = 0.63, b = 1.9 [9], there is the fixed point O(0, 0), and its eigenvalues are k± = 0.315 ± 1.34193i. From s1(x) = 0 and s2(x) = 0, we obtain x1 = 0.774763, x2 = 0.774763, x± = ±0.547723. Thus I1 \ I2 5 ;. We take x2 y2 6 1g. Let X = X+,Y = Y+, then (10) becomes x3 1.9x + 0.868392. We can find one point Br ðOÞ ¼ fðx; yÞ ð0:54Þ 2 þ ð1:1Þ2 z = (x, y) where x = 0.53986, y = 1.03829 such that f2(z) = O and |Df2(z)| 5 0. So, O is a snap-back repeller.
4. Numerical simulations In this section, we present the bifurcation diagrams, the maximum lyapunov exponents corresponding to bifurcation diagram, phase portraits and iteration series for system (1) to demonstrate the above theoretical analysis and show the new interesting complex dynamical behaviors by using numerical simulations. Now We give the specific values of the parameter in system (1). Let a = 0.63 and b range from 2.5 to 0, then map (1) becomes x#0:63x þ y; ð11Þ y#bx þ x3 : After simple calculation, one may discover that system (11) generates an invariant circle (quasi-period orbit) while parameter b goes through 1, which is the Naimark–Sacker bifurcation value of map (11). In fact, the Jacobian matrix pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i; kðbÞ ¼ 0:315 4b0:3969 i and it is of map (11) has a pair of complex conjugate eigenvalues: kðbÞ ¼ 0:315 þ 4b0:3969 2 2 easy to verify jkð1Þj ¼ jkð1Þj ¼ 1; kn –1; n ¼ 1; 2; 3; 4 and djkðbÞj jb¼1 ¼ 12 –0. Fig. 1 is the bifurcation diagram, db showing the output of x component with respect to the parameter b. The fixed point O of system (11) loses its stability at b = 1 on account of the norm of complex eigenvalues of its corresponding Jacobian matrix equal to 1, so there appears an attracting invariant circle when the parameter b < 1. The phase portrait of the circle for b = 1.1 is shown in Fig. 4. The maximum Lyapunov exponent is also calculated and plotted in Fig. 2 where we can easily see that the maximum Lyapunov exponents are negative for the parameter b 2 (1, 0) while the origin is stable correspondingly. For b 2 (1.72, 1) the maximum Lyapunov exponents are in the neighborhood of zero which is corresponding to
2 1.5 1
x
0.5 0
—0.5 —1 —1.5 —2 —2.5
—2
—1.5
b
—1
—0.5
0
Fig. 1. Bifurcation diagram of x output vs. b where the initial values are x0 = 0.01, y0 = 0.01.
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
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0.4
Maximum Lyapunov Exponent
0.2 0 —0.2 —0.4 —0.6 —0.8 —1 —1.2 —2.5
—2
—1.5
—1
—0.5
0
b
Fig. 2. The maximum Lyapunov exponent vs. the parameter b. The initial values are x0 = 0.01, y0 = 0.01.
1.5
1
y
0.5
0
—0.5
—1
—1.5 —1.5
—1
—0.5
0 x
0.5
1
1.5
Fig. 3. Phase portrait of map (1) for a = 0.63 and b = 1.9.
quasi-period solutions or coexistence of chaos and quasi-period solutions. For b 2 (2.25, 1.72), the maximum Lyapunov exponents are positive with a few are negative which shows that period window occurs in the chaotic region. Some phase portraits of system (11) for b 2 (2.5, 0) are plotted in Fig. 4 where we can see that there are period-34 (b = 1.75) and period-5 (b = 1.33) orbits. In addition, there are many complex dynamics: the chaotic attractor collides with unstable fixed points on its basin boundary for b 2.25, the sudden destruction of this chaotic attractor occurs, that is to say boundary crisis occurs (see Fig. 5a, the dotted lines denote the unstable fixed points). Following a boundary crisis, we have chaotic transient [13]: the orbits stay in the vicinity of a non attracting chaotic set before the 21,368th iteration, but, later, it leaves the non attracting chaotic set (see Fig. 5b–d). In addition, when a = 0.63, b = 1.9, (0, 0) is a snap-back repeller. The Marotto’s chaotic attractor is shown in Fig. 3. Next, we consider the case of b = 1.9 in which a is as the bifurcation parameter. The bifurcation diagram (Fig. 6a) shows that for values of the parameter a less than (resp. larger than) a critical transition value ac 0.3382 (resp. 0.3382) the attractor is a periodic orbit. For a slightly larger than (resp. less than) ac, we can see from the iteration series (Fig. 6b) that there are long stretches of time during which the orbit appears to be periodic and closely resembles the orbit for a < 0.3382 (resp.a > 0.3382), but this regular (approximately periodic) behavior is intermittently interrupted by a finite duration ‘burst’ in which the orbit behaves in a decidedly different manner. According to [13], this is just the phenomenology of the Poneau–Manneville intermittency transition to a chaotic attractor.
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Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810 1.5
1 0.8
1 0.6 0.4
0.5
0
0
y
y
0.2
—0.2 —0.5 —0.4 —0.6
—1
—0.8 —1.5 —2
—1.5
—1
—0.5
0 x
0.5
1
1.5
—1 —1.5
2
—1
—0.5
0 x
b = –2.152
0.5
1
1.5
b = –1.75
0.8
0.6
0.6
0.4
0.4 0.2 0.2 y
y
0 0
—0.2 —0.2 —0.4
—0.4
—0.6
—0.6
—0.8
—0.8 —1
—0.8
—0.6
—0.4
—0.2
0 x
0.2
0.4
0.6
0.8
—0.8
1
—0.6
—0.4
b = –1.5
—0.2
0 x
0.2
0.4
0.6
0.8
b = –1.33 1
0.4
0.8
0.3
0.6 0.2 0.4 0.2
0
y
y
0.1
0
—0.2
—0.1
—0.4 —0.2 —0.6 —0.3
—0.8
—0.4 —0.4
—0.3
—0.2
—0.1
0 x
0.1
0.2
0.3
0.4
—1 —1
—0.8
—0.6
—0.4
b = –1.1
—0.2
0 x
b = –0.5
Fig. 4. Phase portraits for some fixed b, a = 0.63.
0.2
0.4
0.6
0.8
1
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810 2
40
1.5
35
1809
b= —2.258, n=1:21368
30
1
25
0.5
x
x
20 0
15 —0.5 10 —1
5
—1.5
0
—2 —2.5
c
—2
—1.5
—1
b
—0.5
—5 0
0
b= —2.258, n=1:21367 3
4
2
3
1
2
x
x
1
n
1.5
2
2.5 x 10 4
x 10 4 5
d
4
0.5
0
1
—1
0
—2
b= —2.258, n=1:21370
—1 0
0.5
1
n
1.5
2
2.5 x 10 4
0
0.5
1
n
1.5
2
2.5 x 10 4
1.5
1.5
1
1
0.5
0.5
0
0
x
x
Fig. 5. Boundary crisis and chaotic transient.
—0.5
—0.5
—1
—1
—1.5
—1.5 —2
—1.5
—1
—0.5
0 a
0.5
1
1.5
2
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n
Fig. 6. Poneau–Manneville intermittency.
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Acknowledgements I thank the reviewers and the editor for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper.
References [1] Cathala JC. Multi connected chaotic areas in second order endomorphisms. Int J Syst Sci 1990;21(5):863–87. [2] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: SpringerVerlag; 1997. [3] Kawakami H, Kobayashi K. Computer experiments on chaotic solutions of x(t + 2) ax(t + 1) x2(t) = 0. Bull Fac Eng, vol. 16. Tokushima University; 1979. pp. 29–46. [4] Li TY, Yorke JA. Period three implies chaos. Am Math Mon 1975;82:985–92. [5] Lin W, Ruan J, Zhao W. On the mathematical clarification of the snap-back repeller in high-dimensional systems and chaos in a discrete neural network model. Int J Bifurcat Chaos 2002;12(5):1129–39. [6] Marotto FR. Snap-back repellers imply chaos in Rn. J Math Anal Appl 1978;63:199–223. [7] Marotto FR. On redefining a snap-back repeller. Chaos Solitons & Fractals 2005;25:25–8. [8] Mira C. Chaos and fractal properties induced by noninvertibility of models in the form of maps. Chaos Solitons & Fractals 2000;11(1–3):251–62. [9] Mira C, Gardini L, Barugola A, Cathala JC. Chaotic dynamics in two-dimensional noninvertible maps. Singapore: World Scientific; 1996. [10] El Naschie MS. Introduction to chaos, information and diffusion in quantum physics. Chaos Solitons & Fractals 1996;7(5):vii–x. [11] El Naschie MS. Elastic buckling loads and fission critical mass as an eigenvalue of a symmetry breaking bifurcation. Chaos Solitons & Fractals 2000;11(4):631–9. [12] El Naschie MS. Nonlinear classical dynamics and knot invariants. Chaos Solitons & Fractals 2000;11(4):641. [13] Ott E. Chaos in dynamical systems. Cambridge: Cambridge University Press; 1993. [14] Wiggins S. An introduction to applied nonlinear dynamics and chaos. New York: Springer-Verlag; 1990. [15] Zhang H, Holden AV. Chaotic meander of spiral waves in the FitzHugh–Nagumo system. Chaos Solitons & Fractals 1995;5(3/4): 661–70.
Complex dynamics in a two-dimensional noninvertible map Yinghui Gao
*
Department of Mathematics, Beihang University, Beijing 100083, PR China Key Laboratory of Mathematics, Informatics and Behavioral Semantics (Beihang University and Peking University), Ministry of Education, China Accepted 18 June 2007
Abstract A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction If one-dimensional noninvertible maps have given rise to a lot of publications, the study of two-dimensional noninvertible maps remained a long time in an underdeveloped state. It is only since these last years that the interest for this subject has increased. One reason of this situation is due to the fact that more and more mathematical models of dynamical process are related to such Maps [9]. In [9], the authors considered the two-dimensional noninvertible map investigated in this paper, and they described the construction of chaotic areas for this map. But they did not prove the existence of chaos theoretically. Here we will do it. In addition, we will study this kind of map in another point of view in this paper. As we have seen, many non-linear systems have parameters which appear in the defining systems of equations. As the parameter is changed, changes may occur in the qualitative structure of the orbits for certain parameter values [2]. A central problem in non-linear dynamics is that of discovering how the properties of orbits change and evolve as a parameter of a dynamical system is changed [13]. So, we mainly discuss the changing properties of the map. Among them, bifurcations and chaos are very important phenomena which many excellent researchers [8,10–12,15] apply themselves to study. Here, we firstly study the stability of the fixed points, then analyze the bifurcations of the system using center manifold theorem and bifurcation theory [2,14]. We find that there exist pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation. *
Address: Department of Mathematics, Beihang University, Beijing 100083, PR China. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.051
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In 1975, Li and Yorke [4] proved that period three implies chaos on the interval. From then on, many mathematicians began to devote their research to further exploring the complex mechanics of the non-linear transformations, especially in high-dimensional systems. Among them, the result, given by Marotto [6], on the criterion for chaos existence in high-dimensional systems is, of course standout and surprising. A snap-back repeller, just like period three in non-linear transformations, is regarded as an inducement to produce chaos in high-dimensional systems. Thus, this criterion is widely used in proving the existence of chaos in many non-linear dynamical systems [5]. Many years after this work first appeared, it was brought to Marotto’s attention that there is a minor technical flaw in the reasoning he used in some of his arguments. A fixed point z is referred to as repelling under f if all eigenvalues of Df(z) exceed 1 in magnitude. But z is expanding only if kf(x) f(y)k > skx yk, where s > 1, for all x,y sufficiently close to z with x 5 y. Although all expanding fixed points are repelling, the converse is not true. As a result, Marotto’s original definition of a snap-back repeller and proof that the existence of such a point implies chaos are in error. But Marotto quickly realized that the flaw is of a minor technical nature. However, during the past decade or so several papers have appeared that first overstate the severity of the error, and then propose correct but profoundly altered and weakened versions of his theorem. Marotto believed they were mistaken and gave a better version of Marotto theorem [7]. Here, we rigorously prove that the two-dimensional noninvertible map possesses chaotic phenomenon using the better version. This paper is organized as follows. In Section 2, we describe first the two-dimensional noninvertible map, then analyze two basic features of the map including the location and stability of its fixed points. And in this section, we show that there exist pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation using center manifold theorem and bifurcation theory. In Section 3, it is rigorously proven that the two-dimensional noninvertible map possesses a snap-back repeller. The results of numerical simulations and the computations of Lyapunov exponents are presented in Section 4 to verify the theoretical analysis and display the complex and interesting dynamics.
2. Existence and stability of fixed points and bifurcations 2.1. Existence and stability of fixed points Consider a two-dimensional noninvertible map [1,3,9] which takes the form ( x # ax þ y; y # bx þ x3 ;
ð1Þ
where b < 0. The fixed points of map (1) satisfy the following equations: ( ða 1Þx þ y ¼ 0; x3 þ bx y ¼ 0: By a simple analysis, it is easy to obtain the following proposition. Proposition 1 (1) If 1 a b 6 0, then map (1) has a unique fixed point O(0, 0). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) If 1 a p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b > 0, then map (1) has three fixed points: ð0; 0Þ; 1 a b; ð1 aÞ 1 a b , 1 a b; ð1 aÞ 1 a bÞ. We now investigate the linear stability of the fixed points for (1). The Jacobian matrix J of map (1) evaluated at the fixed point (x0, y0) is given by a 1 J¼ 3x20 þ b 0 and the characteristic equation of the Jacobian matrix J can be written as k2 ak ð3x20 þ bÞ: A simple calculation shows the stability of fixed points as the following.
1800
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
Proposition 2. The fixed point (0, 0) of map (1) is stable if one of the following conditions is satisfied: (i) (ii) (iii) (iv)
a2 + 4b < 0, a2 + 4b = 0, a2 + 4b > 0, a2 + 4b > 0,
1 < b < 0; |a| < 2; 0 < a < 2, 1 a b > 0; 2 < a < 0, 1 + a b > 0;
and if 1 a b > 0, then the two fixed points always unstable.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a b; ð1 aÞ 1 a b , 1 a b; ð1 aÞ 1 a b are
2.2. Bifurcations In analysis of bifurcations b is as bifurcation parameter. Let ~b ¼ b ð1 aÞ, we consider the parameter ~b as a new and dependent variable, then map (1) becomes: 8 > < x # ax þ y y #ð1 aÞx þ x~b þ x3 : > :~ ~ b#b 1 0 0 1 0 1 1 u x 1 0 C B1 a @ A @ A Let T ¼ @ 1 1 0 A and use the translation y~ ¼ T v , then map (2) becomes l b 0 0 0 1 1 10 1 0 0 1 f ðu; v; lÞ u 1 0 0 u C CB C B B C B @ v A#@ 0 a 1 0 A@ v A þ @ gðu; v; lÞ A; 0 l 0 0 1 l
ð2Þ
where 3 1 a u 1 a u þv l þv ; a2 1a a2 1a 3 1 u 1 u gðu; v; lÞ ¼ þv lþ þv : a2 1a a2 1a f ðu; v; lÞ ¼
By center manifold theory, we know that the stability of (u, v) = (0, 0) near l = 0 can be determined by studying a oneparameter family of equations on a center manifold, which can be represented as follows. Wc(0) = {(u, v, l) 2 R3|v = h(u, l),h(0, 0) = 0,Dh(0, 0) = 0}, for u and l sufficiently small. We assume a center manifold of the form hðu; lÞ ¼ a1 u2 þ a2 ul þ a3 l2 þ a4 u3 þ Oððjuj þ jljÞ3 Þ: The center manifold must satisfy N ðhðu; lÞÞ ¼ hðu þ f ðu; hðu; lÞ; lÞÞ ða 1Þhðu; lÞ gðu; hðu; lÞ; lÞ ¼ 0: The map restricted to the center manifold is given by u#f~ ðu; lÞ ¼ u
1 1 u3 þ Oððjuj þ jljÞ3 Þ: ul a2 ða 2Þð1 aÞ2
2~ 3~ ~ ~ o2 f~ 1 6 Since f~ ð0; 0Þ ¼ 0; oouf ð0; 0Þ ¼ 1; oolf ð0; 0Þ ¼ 0; oouf2 ð0; 0Þ ¼ 0; ouol ð0; 0Þ ¼ a2 –0ða–2Þ; oouf3 ð0; 0Þ ¼ ða2Þð1aÞ 2 –0 ða–1; 2Þ, the fixed point (u, l) = (0, 0) is a pitchfork bifurcation point for map (2). The number of fixed points is chan3~ o2 f~ 1 ð0; 0Þ ¼ ð1aÞ ged at (u(l), v(l)) = (0, 0) as l = 0, and oouf3 ð0; 0Þ= ouol 2 < 0, so there are three fixed points for l < 0 and one fixed point for l > 0. From the above analysis, we have the theorem.
Theorem 1. If a 5 1,2, then map (1) undergoes a pitchfork bifurcation at (0, 0) for b = 1 a. Moreover, there are three fixed points for b < 1 a and one fixed point for b > 1 a. We next consider the flip bifurcation of map (1). The characteristic equation associated with the linearization of map (1) about the fixed point (0, 0) is given by k2 ak b ¼ 0:
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
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Correspondingly, the eigenvalues of the fixed point (0, 0) are k1 = 1, k2 = 1 + a for b = 1 + a(a < 1). The condition |k2| = (1 + a) 5 1 leads to a 5 2. Let ~b ¼ b ð1 þ aÞ, then map (1) becomes 8 > < x # ax þ y; ð3Þ y # ð1 þ aÞx þ x~b þ x3 ; > :~ ~ 1 0 0 1 0 1 b # b: 1 u x 1 0 C B 1þa @ y A ¼ T @ v A, map (3) can be and use the translation We construct an invertible matrix T ¼ @ 1 A 1 0 ~ l b written as 0 0 1 1 1 0 0 1 0 1 0 f ðu; v; lÞ u 1 0 0 u C CB C B B C B ð4Þ 1 þ a 0 A@ v A þ @ gðu; v; lÞ A; @ v A#@ 0 0
l where
0
1
0
l
1þa u 1þa u 3 v lþ ðv Þ; aþ2 1þa aþ2 1þa 3 1 u 1 u gðu; v; lÞ ¼ : v lþ v aþ2 1þa aþ2 1þa f ðu; v; lÞ ¼
We again apply the center manifold theorem to determine the nature of the bifurcation of the fixed point (u, v) = (0, 0) at l = 0. There exists a center manifold for (4) which can be represented as follows: W c ð0Þ ¼ fðu; v; lÞ 2 R3 jv ¼ hðu; lÞ; hð0; 0Þ ¼ 0; Dhð0; 0Þ ¼ 0g; We assume h(u, l) = b1u2 + b2ul + b3l2 + b4 u3 + O((|u| + |l|)3). Equating terms of like powers to zero gives b1 ¼ 0;
b2 ¼
1 ða þ 2Þ2 ða þ 1Þ
;
b3 ¼ 0;
b4 ¼
1 ða þ 2Þ2 ða þ 1Þ3
:
Thus, the map restricted to the center manifold is given by f^ : u# u
1 1 u3 þ Oððjuj þ jljÞ3 Þ; ul aþ2 ða þ 2Þða þ 1Þ2
the second iteration of f^ is given by f^ 2 : u#u þ We have of^ a1 ¼ ol 0 1 a2 ¼ @ 2
2 2 u3 þ Oððjuj þ jljÞ3 Þ: ul þ aþ2 ða þ 2Þða þ 1Þ2
! o2 f^ o2 f^ þ2 ou2 ouol
ð0;0Þ
o2 f^ ou2
!2
¼
!1 1 o3 f^ A þ 3 ou3
2 –0; aþ2 ¼
ð0;0Þ
2 ða þ 2Þða þ 1Þ2
–0:
of^ of^ 2 o2 f^ 2 2 ð0; 0Þ ¼ 0; ð0; 0Þ ¼ f^ ð0; 0Þ ¼ 0; ð0; 0Þ ¼ 1; –0; ol ouol ou aþ2 , o3 f^ 2 o2 f^ 2 6 3 ð0; 0Þ ð0; 0Þ ¼ < 0: ou ouol ð1 þ aÞ2
o3 f^ 2 12 ð0; 0Þ ¼ –0; ou3 ða þ 2Þða þ 1Þ2
Summarize the above analysis into the following theorem. Theorem 2. Map (1) undergoes a flip bifurcation at (0, 0) for b = 1 + a if a < 1 and a 5 2. Moreover, the period two points lie on left side of b = 1 + a, and if a < 2 (resp. 2 < a < 1) the period two points are stable (resp. unstable).
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Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
pffiffiffi pffiffiffi pffiffiffi pffiffiffi Let ~x ¼ x a; ~y ¼ y ð1 aÞ a; ~b ¼ b ð1 2aÞ a > 12 , we transform the fixed point ð a; ð1 aÞ aÞ of map (1) to the origin and take ~b as a new dependent variable, then map (1) becomes 10 1 0 0 1 0 1 ~x 0 ~x a 1þa 0 pffiffiffi CB C B B C B C 0 A@ ~y A þ @ ~xb~ þ 3 a~x2 þ ~x3 A: ð5Þ @ ~y A#@ 1 0 p ffiffi ffi b~ 0 a 1 b~ 0 Let 0
1 B1 þ a B B T ¼B B1 @
1 1 1 pffiffiffi C 2 a C C 1 aC 1 pffiffiffi C 2 aA
0
00 11 0 1 ~x u and use the translation @ ~y A ¼ T @ v A, then map (5) becomes ~b l 1 10 1 0 0 1 0 f ðu; v; lÞ u 1 0 0 u C CB C B B C B 1 þ a 0 A@ v A þ @ gðu; v; lÞ A; @ v A#@ 0 0 l 0 0 1 l where 2 3 # pffiffiffi u 1 u 1 u 1 ; v pffiffiffi l l þ 3 a v pffiffiffi l þ v pffiffiffi l 1þa 2 a 1þa 2 a 1þa 2 a " # 2 3 pffiffiffi 1 u 1 u 1 u 1 : gðu; v; lÞ ¼ v pffiffiffi l l þ 3 a v pffiffiffi l þ v pffiffiffi l aþ2 1þa 2 a 1þa 2 a 1þa 2 a 1þa f ðu; v; lÞ ¼ aþ2
"
We again apply the center manifold theorem to determine the nature of the bifurcation of the fixed point (u, v) = (0, 0) at l = 0. Using the same method, we can get the map restricted to the center manifold pffiffiffi 2 3 a 1þa a 16 f : u# u þ u3 þ Oððjuj þ jljÞ3 Þ; ul þ u2 þ pffiffiffi l2 2þa ð1 þ aÞð2 þ aÞ 4 að2 þ aÞ ð1 þ aÞ2 ð2 þ aÞ2 4 16 f 2 : u#u u3 þ Oððjuj þ jljÞ3 Þ; ul 2þa ð1 þ aÞ2 ð2 þ aÞ a1 ¼
4 16 –0; –0; a2 ¼ 2þa ð1 þ aÞ2 ð2 þ aÞ
of 2 o2 f 2 4 ð0; 0Þ ¼ 0; ð0; 0Þ ¼ –0; ol ou ol 2þa , o3 f 2 96 o3 f 2 o2 f 2 24 ð0; 0Þ ¼ –0; ð0; 0Þ ¼ ð0; 0Þ < 0: 2 3 ou3 ou ou ol ð1 þ aÞ ð2 þ aÞ ð1 þ aÞ2 pffiffiffi pffiffiffi Thus, map (1) undergoes a flip bifurcation at ð a; ð1 aÞ aÞ. Similarly, we can prove that pffiffiffi pffiffiffi ð a; ð1 aÞ aÞ is also a flip bifurcation point. pffiffiffi pffiffiffi Theorem 3. Map (1) undergoes a flip bifurcation at ð a; ð1 aÞ aÞ for b ¼ 1 2a a > 12 . Moreover, the period two points lie on left side of b = 1 2a and they are stable. Finally, we give the conditions of existence of Naimark–Sacker bifurcation by using the theorem in [2], here b is still as bifurcation parameter. We will prove that there exists Naimark– Sacker bifurcation at (0, 0) for b = 1. f ð0; 0Þ ¼ 0;
of ð0; 0Þ ¼ 1; ou
The characteristic equation associated with the linearization of map (1) at the fixed point (0, 0) is given by k2 ak b ¼ 0;
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
pffiffiffiffiffiffiffiffiffi
a2 þ4b
1803
the eigenvalues of the characteristic equation are k; k ¼ , the eigenvalues k; k are complex conjugate for b < a4 . 2 For b = 1, the eigenvalues of the matrix associated with the linearized map (1) at the fixed point (0, 0) are complex conjugate with modulus 1. pffiffiffiffiffiffiffi djkðbÞj Under the conditions a2 < 4 and b = 1, there are jkj ¼ b ¼ 1 and d ¼ ¼ 12 –0. In addition, a 5 0, 1 db 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b¼1 2 4a a A and use the translation x ¼ T u , map (1) leads to kn(1) 5 1,n = 1, 2, 3, 4. Let T ¼ @ 2 2 y v 0 1 becomes pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 4 a2 a C f ðu; vÞ u u B 2 2 ffi Cþ B pffiffiffiffiffiffiffiffiffiffiffiffi ; ð6Þ # A gðu; vÞ v v @ 4 a2 a a
2
2
2
where !3 pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a2 a a f ðu; vÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi uþ v ; 2 2 4 a2 !3 pffiffiffiffiffiffiffiffiffiffiffiffiffi a 4 a2 uþ v : gðu; vÞ ¼ 2 2 Notice that (6) is exactly in the form on the manifold, in which the coefficient a is given by
ð1 2kÞk2 1 n11 n20 n11 j2 jn02 j2 þ Reð kn21 Þ–0; a ¼ Re 1k 2 where 1 n20 ¼ ½ðfuu fvv þ 2guv Þ þ iðguu gvv 2f uv Þ ¼ 0; 8 1 n11 ¼ ½ðfuu þ fvv Þ þ iðguu þ gvv Þ ¼ 0; 4 1 n02 ¼ ½ðfuu fvv 2guv Þ þ iðguu gvv þ 2f uv Þ ¼ 0; 8 and 1 ½ðfuuu þ fuvv þ guuv þ gvvv Þ þ iðguuu þ guvv fuuv fvvv Þ 16 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 3a4 3a2 4 a2 3ð4 a2 Þ 4 a2 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ : ¼ þ 2 4 16 4 4 a2
n21 ¼
Thus 3 a ¼ Reðkn21 Þ ¼ < 0: 8 From the above analysis, we have the theorem. Theorem 4. If a2 < 4,a 5 0,1, then map (1) undergoes a Naimark–Sacker bifurcation at the fixed point (0, 0) for b = 1. Moreover, an attracting invariant closed curve bifurcates from the fixed point (0, 0) for b < 1. 3. Existence of Marotto chaos In this section, we rigorously prove that map (1) possesses chaotic behavior in the sense of Marotto’s definition. We first present Marotto chaos definition and theorem which are quoted from [6,7]. For any map f:Rn ! Rn, and any positive integer k. Let f k represent the composition of f with itself k, times. For a differentiable function f, let Df(z) denote the Jacobian matrix of f evaluated at the point z 2 Rn, and |Df(z)| its determinant. Let Br(z) denote the closed ball in Rn of radius r centered at the point z and B0r ðzÞ its interior. Also let kzk be the usual Euclidean norm of z in Rn.
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Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
Definition 1. [7] Suppose z is a fixed point of f with all eigenvalues of Df(z) exceeding 1 in magnitude and suppose there exists a point x0 5 z in a repelling neighborhood Br(z) of z such that xM = z and |Df(xk)| 5 0 for 1 < k < M, where xk = f k(x0). Then z is called a snap-back repeller of f. Repelling neighborhood means that for any x in such a Br(z), the pre-image points f k(x) remain within the local unstable manifold for all k P 0 (although not necessarily within Br(z)), and f k(x) ! z as k ! 1. Marotto Theorem [6]. If f possesses a snap-back repeller, then the map f is chaotic in the sense of Marotto. That is, there exist (i) a positive integer N such that for each integer p P N, f has a point of period p; (ii) a ‘‘scrambled set’’ of f, i.e. an uncountable set S containing no periodic points of f such that: (a) f(S) S, (b) for every x,y 2 S with x 5 y lim sup kf k ðxÞ f k ðyÞk > 0; k!1
(c) for every x 2 S and any periodic point y of f lim sup kf k ðxÞ f k ðyÞk > 0; k!1
(iii) an uncountable subset S0 of S such that for every x,y 2 S0: lim inf kf k ðxÞ f k ðyÞk ¼ 0: k!1 Now we theoretically give the conditions of existence of chaotic phenomena for map (1) in the sense of Marotto’s definition of chaos. We first give the conditions such that p the fixed point (0, 0) is a snap-back repeller. The eigenvalues associated with ffiffiffiffiffiffiffiffiffi a a2 þ4b the fixed point (0, 0) are given by k ¼ . According to Definition 1, we begin to find a neighborhood Br(O) of 2 (0, 0) in which the norms of conjugate complex eigenvalues exceed 1 for all x,y 2 Br(O). Let s1(x) = a2 + 4(3x2 + b) = 12x2 + a2 + 4b. It is easy to see that ifa2 + 4b < 0 then the equation s1(x) = 0 has two qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 þ4bÞ ða2 þ4bÞ ða2 þ4bÞ ða2 þ4bÞ real roots x1 ¼ ; x2 ¼ . And s1(x) < 0 for x 2 I 1 ¼ ; . 12 12 12 12 2 + ffiffiffiffiffiffiffiffiffiffi b) ffi 1 = 3x2 b 1. Under the condition b< ffiffiffiffiffiffiffiffiffiffi 1 theffiequation s2(x) = 0 has two real roots Letqsffiffiffiffiffiffiffiffiffiffi 2(x) ffi= (3x q qffiffiffiffiffiffiffiffiffiffi ffi q ðbþ1Þ ðbþ1Þ ðbþ1Þ ðbþ1Þ ; x ¼ . And s2 (x) > 0 for all x 2 I 2 ¼ ; . xþ ¼ 3 3 3 3
Lemma 1. (i) If a2 < 4 then I0 = I1 \ I2 = I2, (ii) If a2 > 4 then I0 = I1 \ I2 = I1. Moreover, if one of the above two conditions holds, then (0, 0) is a repelling fixed point of (1) in U0 = {(x, y)|x 2 I0,y 2 R}. Due to the definition of snap-back repeller, we need to find one point z 2 Br(O). such that z 5 O,fM(z) = O,|DfM(z)| 5 0 for some positive integer M(M 5 1). In fact, we have ax þ y ¼ X ; ð7Þ bx þ x3 ¼ Y ; and
aX þ Y ¼ 0; bX þ X 3 ¼ 0:
ð8Þ
A f 2 map has been constructed to map the point z(x, y) to the fixed point (0, 0) after two iteration if there are solutions different from (0, 0) for Eqs. (7) and (8). By the calculation, the solutions different from (0, 0) for (8) are obtained as follows: pffiffiffiffiffiffiffi X ¼ b; pffiffiffiffiffiffiffi ð9Þ Y ¼ a b; for b < 1.
pffiffiffiffiffiffiffi Substituting Y þ ¼ a b into (7) and solving x, we have p ffiffiffiffiffiffi ffi x3 þ bx þ a b ¼ 0:
ð10Þ
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
1805
We next need to find one real root x* 2 I0 of (10). Case (i) Under the conditions a > 0 and a2 < 4 we have I0 = I2, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2b þ 1 ðb þ 1Þ pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b ¼ þb þ a b > 0; 3 3 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2b 1 ðb þ 1Þ pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b ¼ þb þ a b < 0 for ðb þ 1Þð2b 1Þ2 < a2 b: 3 3 3 3 Thus, (10) has a real nonzero root x* 2 I0. Case (ii) Under the conditions a < 0 and a2 < 4 we have I0 = I2, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b > 0 for ðb þ 1Þð2b 1Þ2 < a2 b; þb 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ðb þ 1Þ ðb þ 1Þ þ a b < 0: þb 3 3 Thus, (10) has a real nonzero root x* 2 I0. Case (iii) Under the conditions a > 0 and a2 > 4 we have I0 = I1, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a2 8b ða2 þ 4bÞ pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þ a b ¼ þb þ a b > 0; 12 12 12 12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 8b a2 ða2 þ 4bÞ pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þ a b ¼ þb þ a b < 0 for 12 12 12 12 2 ða2 þ 4bÞða2 8bÞ < 123 a2 b:
Thus, (10) has a real nonzero root x* 2 I0. Case (iv) Under the conditions a < 0 and a2 > 4 we have I0 = I1, then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þb þ a b > 0 for ða2 þ 4bÞða2 8bÞ2 < 123 a2 b; 12 12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffi ða2 þ 4bÞ ða2 þ 4bÞ þb þ a b < 0: 12 12 * Thus, (10) has a real nonzero pffiffiffiffiffiffiffi root x 2 I0. Substituting Y ¼ a b into (5) and solving x, we have pffiffiffiffiffiffiffi x3 þ bx a b ¼ 0:
Similarly, we can discuss the conditions for a real nonzero root x* 2 I0. Here, we omit the analysis process. Lemma 2. If one of the following two conditions holds, the equation x3 + bx Y = 0 has a real root x* different from 0, x* 2 I0 for b < 1 and a2 + 4b < 0. (i) a2 < 4,(b + 1)(2b 1)2 < a2b, (ii) a2 > 4,(a2 + 4b)(a2 8b)2 < 123a2b. From (7), we also obtain y* = X ax*. 2 2 Let U 0 ¼ fðx; yÞ xr1 þ yr2 6 1; jx j < r1 ; x r1 2 I 0 ; r2 ¼ jy j þ g; g is some positive constantg:jDf 2 ðzÞj ¼ jDf 2 ðx ; y Þj ¼
½3ðax þ y Þ2 þ bð3x2 þ bÞ ¼ ð3X 2 þ bÞð3x2 þ bÞ, from the above computation, we know that 3X2 + b 5 0,3x*2 + b 5 0, hence |Df 2(z)| 5 0.
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Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
Obviously, if the conditions in Lemmas 1 and 2 are satisfied, then (0, 0) is a snap-back repeller in U 0 . Thus, we have the theorem. Theorem 5. If one of the following conditions are satisfied (i) b < 1,a2 < 4,a2 + 4b < 0,(b + 1)(2b 1)2 < a2b, (ii) b < 1,a2 > 4,a2 + 4b < 0,(a2 + 4b)(a2 8b)2 < 123a2b, then (0, 0) is a snap-back repeller of map (1), and hence map (1) is chaotic in the sense of Marotto’s definition. Example 1. For a = 0.63, b = 1.9 [9], there is the fixed point O(0, 0), and its eigenvalues are k± = 0.315 ± 1.34193i. From s1(x) = 0 and s2(x) = 0, we obtain x1 = 0.774763, x2 = 0.774763, x± = ±0.547723. Thus I1 \ I2 5 ;. We take x2 y2 6 1g. Let X = X+,Y = Y+, then (10) becomes x3 1.9x + 0.868392. We can find one point Br ðOÞ ¼ fðx; yÞ ð0:54Þ 2 þ ð1:1Þ2 z = (x, y) where x = 0.53986, y = 1.03829 such that f2(z) = O and |Df2(z)| 5 0. So, O is a snap-back repeller.
4. Numerical simulations In this section, we present the bifurcation diagrams, the maximum lyapunov exponents corresponding to bifurcation diagram, phase portraits and iteration series for system (1) to demonstrate the above theoretical analysis and show the new interesting complex dynamical behaviors by using numerical simulations. Now We give the specific values of the parameter in system (1). Let a = 0.63 and b range from 2.5 to 0, then map (1) becomes x#0:63x þ y; ð11Þ y#bx þ x3 : After simple calculation, one may discover that system (11) generates an invariant circle (quasi-period orbit) while parameter b goes through 1, which is the Naimark–Sacker bifurcation value of map (11). In fact, the Jacobian matrix pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i; kðbÞ ¼ 0:315 4b0:3969 i and it is of map (11) has a pair of complex conjugate eigenvalues: kðbÞ ¼ 0:315 þ 4b0:3969 2 2 easy to verify jkð1Þj ¼ jkð1Þj ¼ 1; kn –1; n ¼ 1; 2; 3; 4 and djkðbÞj jb¼1 ¼ 12 –0. Fig. 1 is the bifurcation diagram, db showing the output of x component with respect to the parameter b. The fixed point O of system (11) loses its stability at b = 1 on account of the norm of complex eigenvalues of its corresponding Jacobian matrix equal to 1, so there appears an attracting invariant circle when the parameter b < 1. The phase portrait of the circle for b = 1.1 is shown in Fig. 4. The maximum Lyapunov exponent is also calculated and plotted in Fig. 2 where we can easily see that the maximum Lyapunov exponents are negative for the parameter b 2 (1, 0) while the origin is stable correspondingly. For b 2 (1.72, 1) the maximum Lyapunov exponents are in the neighborhood of zero which is corresponding to
2 1.5 1
x
0.5 0
—0.5 —1 —1.5 —2 —2.5
—2
—1.5
b
—1
—0.5
0
Fig. 1. Bifurcation diagram of x output vs. b where the initial values are x0 = 0.01, y0 = 0.01.
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810
1807
0.4
Maximum Lyapunov Exponent
0.2 0 —0.2 —0.4 —0.6 —0.8 —1 —1.2 —2.5
—2
—1.5
—1
—0.5
0
b
Fig. 2. The maximum Lyapunov exponent vs. the parameter b. The initial values are x0 = 0.01, y0 = 0.01.
1.5
1
y
0.5
0
—0.5
—1
—1.5 —1.5
—1
—0.5
0 x
0.5
1
1.5
Fig. 3. Phase portrait of map (1) for a = 0.63 and b = 1.9.
quasi-period solutions or coexistence of chaos and quasi-period solutions. For b 2 (2.25, 1.72), the maximum Lyapunov exponents are positive with a few are negative which shows that period window occurs in the chaotic region. Some phase portraits of system (11) for b 2 (2.5, 0) are plotted in Fig. 4 where we can see that there are period-34 (b = 1.75) and period-5 (b = 1.33) orbits. In addition, there are many complex dynamics: the chaotic attractor collides with unstable fixed points on its basin boundary for b 2.25, the sudden destruction of this chaotic attractor occurs, that is to say boundary crisis occurs (see Fig. 5a, the dotted lines denote the unstable fixed points). Following a boundary crisis, we have chaotic transient [13]: the orbits stay in the vicinity of a non attracting chaotic set before the 21,368th iteration, but, later, it leaves the non attracting chaotic set (see Fig. 5b–d). In addition, when a = 0.63, b = 1.9, (0, 0) is a snap-back repeller. The Marotto’s chaotic attractor is shown in Fig. 3. Next, we consider the case of b = 1.9 in which a is as the bifurcation parameter. The bifurcation diagram (Fig. 6a) shows that for values of the parameter a less than (resp. larger than) a critical transition value ac 0.3382 (resp. 0.3382) the attractor is a periodic orbit. For a slightly larger than (resp. less than) ac, we can see from the iteration series (Fig. 6b) that there are long stretches of time during which the orbit appears to be periodic and closely resembles the orbit for a < 0.3382 (resp.a > 0.3382), but this regular (approximately periodic) behavior is intermittently interrupted by a finite duration ‘burst’ in which the orbit behaves in a decidedly different manner. According to [13], this is just the phenomenology of the Poneau–Manneville intermittency transition to a chaotic attractor.
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Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810 1.5
1 0.8
1 0.6 0.4
0.5
0
0
y
y
0.2
—0.2 —0.5 —0.4 —0.6
—1
—0.8 —1.5 —2
—1.5
—1
—0.5
0 x
0.5
1
1.5
—1 —1.5
2
—1
—0.5
0 x
b = –2.152
0.5
1
1.5
b = –1.75
0.8
0.6
0.6
0.4
0.4 0.2 0.2 y
y
0 0
—0.2 —0.2 —0.4
—0.4
—0.6
—0.6
—0.8
—0.8 —1
—0.8
—0.6
—0.4
—0.2
0 x
0.2
0.4
0.6
0.8
—0.8
1
—0.6
—0.4
b = –1.5
—0.2
0 x
0.2
0.4
0.6
0.8
b = –1.33 1
0.4
0.8
0.3
0.6 0.2 0.4 0.2
0
y
y
0.1
0
—0.2
—0.1
—0.4 —0.2 —0.6 —0.3
—0.8
—0.4 —0.4
—0.3
—0.2
—0.1
0 x
0.1
0.2
0.3
0.4
—1 —1
—0.8
—0.6
—0.4
b = –1.1
—0.2
0 x
b = –0.5
Fig. 4. Phase portraits for some fixed b, a = 0.63.
0.2
0.4
0.6
0.8
1
Y. Gao / Chaos, Solitons and Fractals 39 (2009) 1798–1810 2
40
1.5
35
1809
b= —2.258, n=1:21368
30
1
25
0.5
x
x
20 0
15 —0.5 10 —1
5
—1.5
0
—2 —2.5
c
—2
—1.5
—1
b
—0.5
—5 0
0
b= —2.258, n=1:21367 3
4
2
3
1
2
x
x
1
n
1.5
2
2.5 x 10 4
x 10 4 5
d
4
0.5
0
1
—1
0
—2
b= —2.258, n=1:21370
—1 0
0.5
1
n
1.5
2
2.5 x 10 4
0
0.5
1
n
1.5
2
2.5 x 10 4
1.5
1.5
1
1
0.5
0.5
0
0
x
x
Fig. 5. Boundary crisis and chaotic transient.
—0.5
—0.5
—1
—1
—1.5
—1.5 —2
—1.5
—1
—0.5
0 a
0.5
1
1.5
2
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 n
Fig. 6. Poneau–Manneville intermittency.
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Acknowledgements I thank the reviewers and the editor for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper.
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