Study of limit cycles for some non-smooth Liénard systems

3 011 5 ( ) Journal of East China Normal University (Natural Science) No. 3 May 011 Article ID: 1000-5641(011)03-0044-10 Study of limit cycles for some non-smooth Liénard systems YANG Lu, LIU Xia, XING Ye-peng (Department of Mathematics, Shanghai Normal University, Shanghai 0034, China) Abstract: Algebraic method was used to study the Hopf cyclicity of non-smooth Liénard systems on the plain. Some new formulas for computing focus values were presented. Based on the formulas, the number of limit cycles bifurcating from some non-smooth Liénard systems was obtained. The results improve the known results. Key words: Liénard systems; non-smooth; limit cycle; Hopf cyclicity CLC number: O19 Document code: A DOI: 10.3969/j.issn.1000-5641.011.03.007 Liénard,, (, 0034) : Liénard Hopf,, Liénard Hopf,. : Liénard ; ; ; Hopf 1 Introduction and preliminary Several papers have been concerned with the bifurcation problems of non-smooth systems, see [1-7] for example. In this paper, we consider a non-smooth Liénard system of the form ẋ = p(y) F(x, a), ẏ = g(x), (1) where a R n, F(x, a) = F (x, a), x 0, F (x, a), x 0, g(x) = g (x), x 0, g (x), x 0. : 010-5 : (10971139), (10YZ7) :,,,. :,,,. :,,,,. E-mail: ypxing@shnu.edu.cn.

3, : Liénard 45 Here, F ± and g ± are all C functions and satisfy F ± (0, a) = 0, p(0) = 0, g ± (0) = 0, (g ± ) (0) = g ± 1 > 0, p (0) = p 0 > 0, (F ± x (0, a 0 )) 4p 0 g ± 1 < 0, a 0 R n. () For s > 0 small, the Poincaré return maps h ± (s) were defined in a neighborhood of the origin of general non-smooth systems in [7] and a succession function d was introduced as follows Definition 1.1 [7] If there is a k 1 such that d(s) = h (h (s)) s = V k s k O(s k1 ), V k 0 for s > 0 close to the zero, then V k is called the kth Liapunov constant or focus value. The origin is called fine or weak focus of order k if k > 1. as follows In [1] the definition of succession function of non-smooth systems is presented for s 0 Definition 1. [1] Define h (h (s)) s, for s > 0, d(s) = 0, for s = 0, h (h (s)) s, for s < 0. The function d(s) is called a succession function or a displacement function. Let ( g ) x x 1 g (x), x > 0, G(x) = g(u)du = ( 0 g ) x 1 g (x), x 0, where g ± (0) = 0. Using the above definitions, for system (1), the authors obtained the following lemmas and theorems in [1] Lemma 1.1 [1] and only if Let (1) satisfy (). Then the origin is a fine or weak focus for a = a 0 if g1 F x (0, a 0) g 1 F x (0, a 0) = 0. (3) Lemma 1. [1] Let () and (3) hold. Suppose formally for 0 < x 1 F(α(x), a) F(x, a) = F (α(x), a) F (x, a) = i 1 B i (a)x i, (4) where α(x) = g 1 xo(x ) satisfies G(α(x)) G(x) for 0 < x 1. Then we have formally g 1 d(r 0, a) = i 1 d i (a)r i 0 for a a 0 small, where d 1 (a) = B 1 N 1 (a), d i(a) = B i N i (a) O( B 1, B 1,, B i 1 ),

46 ( ) 011 with N i C and N i (a 0) > 0 for i 1. Theorem 1.1 [1] Suppose () and (3) hold. If there exists k 1 such that B j (a 0 ) = 0, j = 1,, k, B k1 (a 0 ) < 0(> 0), then the origin is a stable (unstable) focus of order k 1 of system (1) for a = a 0 and there are at most k limit cycles near the origin for all a a 0 small and k limit cycles can appear if rank (B 1,, B k ) (a 1,, a n ) a=a 0 = k. Corollary 1.1 [1] Let () and (3) hold. If there exists k 1 such that F (α(x), a) F (x, a) when B j1 = 0, j = 0,, k (5) for all a R m, then the origin is a focus of order at most k 1 of system (1) unless it is a center. further Theorem 1. [1] Suppose () and (3) are satisfied. Let (5) hold for some k 1. If B j1 (a 0 ) = 0, j = 0,, k, rank (B 1,, B k1 ) a=a0 = k 1, (6) (a 1,, a n ) for some a 0 R m, then system (1) has Hopf cyclicity k at the origin for a a 0 small. Theorem 1.3 [1] Let () and (3) hold. Suppose there exists k 1 such that (5) holds for all a R m and (6) holds for some a 0 R m. If F is linear in a then for any constant N > a 0, system (1) has Hopf cyclicity k for all a N. For general system (1), let G G(x) = x G 3 x3 G 4 x4 G 5 x5, x > 0 G x G 3 x3 G 4 x4 G 5 x5, x 0, F(x, a) = F 1 x F x F 3 x3 F 4 x4 F 5 x5, x > 0 F 1 x F x F 3 x3 F 4 x4 F 5 x5, x 0. Suppose α(x) = α 1 x α x α 3 x 3 α 4 x 4 α 5 x 5 for 0 < x 1. By inserting α(x) into the equation G(α(x)) = G(x), it was obtained in [1] G α 1 =, α = G 3 α3 1G 3 G α 1 G, α 3 = G 4 G 4 α4 1 3G 3 α 1α G α α 1 G, α 4 = G 5 G 5 α5 1 4G 4 α3 1 α 3G 3 (α 1 α 3 α 1 α ) G α α 3 α 1 G, α 5 = G 6 G 6 α6 1 5G 5 α4 1 α G 4 (4α3 1 α 3 6α 1 α ) G 3 (3α 1 α 4 α 3 6α 1α α 3 ) α 1 G G (α α 4 α 3) α 1 G, (7)

3, : Liénard 47 α 6 = G 7 G 7 α7 1 6G 6 α5 1 α 5G 5 (α3 1 α α4 1 α 3) 4G 4 (3α 1 α α 3 α 1 α 3 α3 1 α 4) α 1 G 3G 3 (α 1α 5 α 1 α α 4 α 1 α 3 α α 3 ) G (α 3α 4 α α 5 ) α 1 G. Then by using (4), the following formulas were obtained in [1] B 1 (a) = α 1 F1 F 1, B (a) = α F1 α 1F F, B 3 (a) = α 3 F1 α 1α F α3 1F3 F 3, B 4 (a) = α 4 F1 (α 1α 3 α )F 3α 1 α F3 α4 1 F 4 F 4, B 5 (a) = α 5 F1 (α 1α 4 α α 3 )F 3(α 1 α 3 α 1 α )F 3 4α3 1 α F4 α5 1 F 5 F 5, B 6 (a) = α 6 F1 (α 1α 5 α α 4 α 3 )F (3α 1 α 4 6α 1 α α 3 α 3 )F 3 (6α 1α 4α 3 1α 3 )F4 5α4 1α F5 α6 1F6 F 6. (8) Applying Theorem 1.3 to the following system, ẋ = y F n (x, a), ẏ = g(x), (9) where F n (x, a) = n a i xi, x > 0, n a i xi, x 0, i=1 i=1 g(x) = x g x, x > 0, x g x, x 0, the authors in [1] proved that for n = 1, or 3 system (9) respectively has Hopf cyclicity 1, 3, 5 at the origin. In this paper we first give a formula for computing B i, i = 7, 8, 9, 10, and then apply Theorems 1.1-1.3 to study the Hopf bifurcation of some non-smooth Liénard systems. In particular, we found the Hopf cyclicity for system (9) in the case of n = 3 can be 4, which improve the result in [1], see Theorem. below. Main results and proofs For general system (1) we have Theorem.1 Let (7) and (8) hold. We can further obtain α 7 = [G 8 G 8 α8 1 7G 7 α6 1α G 6 (6α5 1α 3 15α 4 1α ) 5G 5 (α4 1α 4 4α 3 1α α 3 α 1α 3 ) G 4 (1α 1α α 4 6α 1α 3 4α 3 1α 5 1α 1 α α 3 α 4 ) 3G 3 (α 1α α 5 α 1 α 3 α 4 α 1α 6 α α 3 α α 4 ) G (α α 6 α 3 α 5 α 4 )]/(α 1G ),

48 ( ) 011 α 8 = [G 9 G 9 α9 1 8G 8 α7 1α 7G 7 (α6 1α 3 3α 5 1α ) G 6 (6α5 1α 4 30α 4 1α α 3 0α 3 1α 3 ) 5G 5 (α4 1α 5 4α 3 1α α 4 α 3 1α 3 6α 1α α 3 α 1 α 4 ) 4G 4 (3α 1 α 3α 4 3α 1 α α 5 3α α 1 α 3 α3 1 α 6 α 3 α 3 3α α 1α 4 ) G 3 (3α 1 α 7 3 α 1 α 4 6 α 1α 3 α 5 6 α 1 α α 6 α 3 3 6 α α 3 α 4 3 α α 5) G (α α 7 α 3 α 6 α 4 α 5 )]/(α 1 G ), α 9 = [ G 10 α10 1 G 10 9α8 1α G 9 4(7α6 1α α 7 1α 3 )G 8 7(α6 1α 4 5α 4 1α 3 6α 5 1α 3 α )G 7 (6α 5 1α 5 15α 4 1α 3 15α 4 α 1 30α 4 1α 4 α 60α 3 1α α 3 )G 6 (5α 6α 4 1 30α 1α α 3 0α 1 α 3 α 3 0α 3 1α α 5 0α 3 1α 3 α 4 30α 1α α 4 α 5 )G 5 (4α 3 α 4 4α 3 3α 1 4α 7 α 3 1 6α α 3 6α 1α 4 1α 1 α α 5 1α 1α 3 α 5 1α 6 α 1 α 4α 1 α α 4 α 3 )G 4 3(α 3 α 4 α α 6 α 1 α 8 α α 4 α 1α α 7 α 1 α 4 α 5 α 1 α 3 α 6 α α 3 α 5 )G 3 (α 3 α 7 α 4 α 6 α α 8 α 5 )G ]/( α 1G ), α 10 = [ G 11 α11 1 G 11 10α9 1α G 10 9(α8 1α 3 4α 7 1α )G 9 8(α7 1α 4 7α 5 1α 3 7α 6 1α α 3 )G 8 7(α 6 1α 5 5α 3 1α 4 3α 5 1α 3 15α 4 1α 3 α 6α 5 1α 4 α )G 7 6(α5 α 1 α 6 α 5 1 5α 4 1α 3 α 4 10α 1α 3 α 3 10α 3 1α 3α 10α 3 1α α 4 5α 4 1α 5 α )G 6 5(α4 α 3 α 7 α 4 1 α 3 3α 1 α 3 1 α 4 6α 1α α 3 6α 1 α α 5 4α 3 1 α 3α 5 4α 1 α 3 α 4 4α 6 α 3 1 α 1α 1 α α 4 α 3 )G 5 4(α3 3 α α 3 α 5 α 8 α 3 1 3α 7α 1 α 3α 6 α 1 α 3 3α 6 α 1 α 3α α 4α 3 3α 1 α α 4 3α 1α 3 α 4 3α 1 α 4α 5 6α 1 α α 3 α 5 )G 4 3(α 1α 5 α 3α 4 and α α 7 α 3α 5 α 1α 9 α α 4 α 5 α 1 α α 8 α 1 α 3 α 7 α 1 α 4 α 6 α α 3 α 6 )G 3 (α 4 α 7 α 5 α 6 α 3 α 8 α α 9 )G ]/( α 1G ) B 7 (a) = α 7 F 1 (α 1α 6 α α 5 α 3 α 4 )F 3(α 1 α 5 α 1 α α 4 α 1 α 3 α α 3)F 3 4(α 3 1α 4 3α 1α α 3 α 1 α 3 )F 4 5(α4 1α 3 α 3 1α )F 5 6α5 1α F 6 α7 1F 7 F 7, B 8 (a) = α 8 F 1 (α 1α 7 α α 6 α 3 α 5 α 4)F 3(α 1α 6 α 1 α α 5 α 1 α 3 α 4 α α 4 α α 3)F 3 (4α3 1α 5 1α 1α α 4 6α 1α 3 1α α 1 α 3 α 4 )F 4 5(α4 1α 4 4α 3 1 α α 3 α 1 α3 )F 5 3(α5 1 α 3 5α 4 1 α )F 6 7α6 1 α F 7 α8 1 F 8 F 8

3, : Liénard 49 B 9 (a) = α 9 F 1 (α α 7 α 4 α 5 α 1 α 8 α 3 α 6 )F (3α α 5 3α 1 α 4 3α 1α 7 6α 1 α α 6 6α 1 α 3 α 5 6α α 3 α 4 α 3 3)F 3 4(α3 α 3 α 6 α 3 1 3α 1α α 5 3α 1 α α 3 3α 1 α α 4 3α 1 α 3α 4 )F 4 5(α4 α 1 α 4 1 α 5 α 3 1 α 3 4α3 1 α α 4 6α 1 α α 3)F 5 (0α 3 1 α3 6α5 1 α 4 30α 4 1 α 3α )F 6 7(α6 1 α 3 3α 5 1 α )F 7 8α7 1 α F 8 α 9 1 F 9 F 9, B 10 (a) = α 10 F 1 (α 3α 7 α 1 α 9 α 4 α 6 α α 8 α 5 )F 3(α 1 α 8 α 3 α 4 α α 4 α α 6 α 1 α 4 α 5 α α 3 α 5 α 1 α α 7 α 1 α 3 α 6 )F 3 (4α 7α 3 1 4α3 α 4 6α α 3 4α 1α 3 3 6α 1 α 4 1α 1 α 3α 5 1α 6 α 1 α 1α 1 α α 5 4α 1 α α 3 α 4 )F 4 Then (5α 6 α 4 1 0α 3 1α 3 α 4 0α 1 α 3 α 3 30α 1α α 4 0α 3 1α α 5 30α 1α α 3 α 5 )F 5 3(5α 4 α 1 5α 4 1α 3 α 5 1α 5 10α 4 1α 4 α 0α 3 1α α 3 )F 6 [mm] 7(5α 4 1α 3 α 6 1α 4 6α 5 1α 3 α )F 7 (8α7 1α 3 8α 6 1α )F 8 9α8 1α F 9 α 10 1 F 10 F 10. Proof Let α(x) = α 1 x α x α 3 x 3 α 4 x 4 α 5 x 5, xα(x) < 0. G(α(x)) = G (α(x)) G 3 (α(x))3 G 4 (α(x))4 G 5 (α(x))5 G 6 (α(x))6 G 7 (α(x))7 G 8 (α(x))8, G(x) = G x G 3 x3 G 4 x4 G 5 x5 G 6 x6 G 7 x7 G 8 x8. Note that G(α(x)) = G(x). Comparing the coefficients of x 8 in the both sides, we obtain G 8 = G (α 1α 7 α α 6 α 3 α 5 α 4 ) G 3 (6α 1α α 5 6α 1 α 3 α 4 3α 1 α 6 3α α 3 3α α 4) G 4 (1α 1 α α 4 6α 1 α 3 4α3 1 α 5 1α 1 α α 3 α 4 ) G 5 (5α4 1 α 4 0α 3 1 α α 3 10α 1 α3 ) G 6 (6α5 1 α 3 15α 4 1 α ) 7G 7 α6 1 α G 8 α8 1. Then α 7 can be solved from the above. The coefficients α 8, α 9, α 10 can be obtained in the same way. By F (α(x), a) F (x, a) = F1 (α(x)) F (α(x)) F3 (α(x))3 F4 (α(x))4 F5 (α(x))5 F6 (α(x))6 F7 (α(x))7 F8 (α(x))8 (F 1 x F x F 3 x3 F 4 x4 F 5 x5 F 6 x6 F 7 x7 F 8 x8 ),

50 ( ) 011 and using (4), by combining the coefficients of x i, i = 7, 8, 9, 10 respectively, we can obtain the expressions of B i (a), i = 7, 8, 9, 10. The proof is completed. where Let Theorem. Consider the following system F n (x, a) = ẋ = y F n (x, a), ẏ = g(x), (10) n a i xi, x > 0, n a i xi, x 0, i=1 i=1 g(x) = x g x, x > 0, x g x, x 0. M 3 = 7 97 (g )6 7 34 (g )5 g 7 486 (g )4 (g ) 7 486 (g )3 (g )3 7 34 (g ) (g )4 7 97 (g )(g )5, M 4 = 77 19683 (g )5 (g )6 11 4374 (g )3 (g )8 11 4374 (g )9 (g ) 44 59049 (g )10 g 77 19683 (g )7 (g )4 11 118098 (g )11 44 59049 (g ) (g )9 88 19683 (g )4 (g )7 11 118098 (g )(g )10 88 19683 (g )8 (g )3, M 5 = 5115 41348516 (g )9 (g )9 505 95851 (g )14 (g 3935 )4 103311304 (g )4 (g )14 505 95851 (g )5 (g 715 715 )13 86497043 (g )18 86497043 g (g )17 3575 754990144 (g )17 g 455455 0664608 (g )6 (g )1 3535 103311304 (g )7 (g )11 3575 754990144 (g )16 ( ) 1045 344373768 (g )11 (g 995 )7 103311304 (g )16 (g ) 995 103311304 (g )3 (g )15 3535 103311304 (g )1 (g 3935 )6 103311304 (g )15 (g )3 455455 0664608 (g )13 (g )5 5115 41348516 (g )10 (g )8 1045 344373768 (g )8 (g )10. Then system (10) has Hopf cyclicity n at the origin if M n 0 for n = 3, 4 and 5 respectively. Proof In fact, for (10) we have G = G = 1, G 3 = g 3, G 3 = g 3, G± j By (7), (8) and Theorem.1, we can obtain = 0, j 4. α 1 = 1, α = g g, 3 α 3 = 9 g g 5 18 (g ) 1 18 (g ), α 4 = 5 18 (g ) g 8 7 (g )3 1 54 (g )3, α 5 = 77 16 (g )4 3 81 (g )3 g 5 108 (g ) (g ) 5 648 (g )4,

3, : Liénard 51 α 6 = 11 43 (g )5 7 1944 (g )5 385 648 (g )4 g 3 43 (g )3 (g ) 5 97 (g ) (g )3, α 7 = 431 3888 (g )6 7 3888 (g )6 4 43 (g )5 g 385 196 (g )4 (g ) 5 3888 (g ) (g )4, α 8 = 448 79 (g )5 (g ) 17017 11664 (g )6 (g ) 11 11664 (g )7 640 79 (g )7 385 11664 (g )4 (g )3 5 11664 (g ) (g )5, α 9 = 385 139968 (g )4 (g )4 85085 69984 (g )6 (g ) 35 0995 (g ) (g )6 510 187 (g )7 g 896 6561 (g )5 (g )3 143 79936 (g )8 106347 839808 (g )8, and α 10 = 85085 0995 (g )6 (g )3 5 69984 (g ) (g )7 77 139968 (g )4 (g )5 510 187 (g )7 (g ) 106347 79936 (g )8 g 36608 19683 (g )9 715 51944 (g )9, B 1 (a) = a 1 a 1, B (a) = α a 1 a a, B 3 (a) = α 3 a 1 α a a 3 a 3, B 4 (a) = α 4 a 1 ( α 3 α )a 3α a 3 a 4 a 4, B 5 (a) = α 5 a 1 ( α 4 α α 3 )a 3(α 3 α )a 3 4α a 4 a 5 a 5, B 6 (a) = α 6 a 1 ( α 5 α α 4 α 3)a (3α 4 6α α 3 α 3 )a 3 (6α 4α 3 )a 4 5α a 5, B 7 (a) = α 7 a 1 ( α 6 α α 5 α 3 α 4 )a 3(α 5 α α 4 α 3 α α 3 )a 3 4( α 4 3α α 3 α 3 )a 4 5(α 3 α )a 5, B 8 (a) = α 8 a 1 ( α 7 α α 6 α 3 α 5 α 4)a 3(α 6 α α 5 α 3 α 4 α α 4 α α 3)a 3 ( 4α 5 1α α 4 6α 3 1α α 3 α 4 )a 4 5(α3 α 4 4α α 3 )a 5, B 9 (a) = α 9 a 1 (α 3α 6 α 4 α 5 α α 7 α 8 )a (α3 3 6 α α 3 α 4 3 α α 5 3 α 7 3 α 4 6 α α 6 6 α 3 α 5 )a 3 4(3 α α 5 α 3 α 3 3 α 3 α 4 3 α α 4 3 α α 3 α 6 )a 4 5( α 3 α 5 α 4 4 α α 4 6 α α 3 )a 5, B 10 (a) =α 10 a 1 ( α 4α 6 α 5 α α 8 α 3 α 7 α 9 )a 3( α α 7 α 4 α 5 α α 4 α α 3 α 5 α 3 α 6 α 3 α 4 α α 6 α 8 )a 3 (α3 α 4 α 7 1α α 3 α 4 3α α 3 3α 4 6α α 6 6α 3 α 5 α 3 3 6α α 5)a 4 (5α 6 0α α 5 30α α 4 0α 3 α 3 0α 3 α 4 30α α 3 α5 )a 5. We take g, g as constants. For n = 3, take a = (a 1, a 1, a, a, a 3, a 3 ) R6, we have det (B 1, B, B 3, B 4, B 5, B 6 ) (a 1, a 1, a, a, a 3, a 3 ) = 0, and det (B 1, B, B 3, B 4, B 5 ) (a 1, a 1, a, a, a 3 ) = M 3.

5 ( ) 011 Then by Solving the equations B i (a) = 0, i = 1,, 3, 4, 5, we have and (11), we have a 1 = 0, a 1 = 0, a = 3 a 3, a = 3 a 3, a 3 = g a 3. (11) G (α(x)) = G (x) 1 α (x) g 3 α3 (x) = 1 x g 3 x3, F (α(x)) F (x) = a 1 α(x) a α (x) a 3 α3 (x) a 1 x a x a 3 x3 = a 3 3 g ( 1 α (x) g 3 α3 (x) 1 x g 3 x3 ) = 0. Hence, for i = 1,, 3, 4, 5, B i (a) = 0 implies F(α(x)) = F(x). Taking a 0 = ( 0, 0, 3 g the conclusion for n = 3 follows from Theorem 1.3. For n = 4, we have ) a 3, 3 a 3, g a 3, a 3, det (B 1, B, B 3, B 4, B 5, B 6, B 7, B 8 ) (a 1, a 1, a, a, a 3, a 3, a 4, a 4 ) = 0, Solve B i (a) = 0, i = 1,,, 7, we obtain det (B 1, B, B 3, B 4, B 5, B 6, B 7 ) (a 1, a 1, a, a, a 3, a 4, a 4 ) = M 4. a 1 = a 1 = a 4 = a 4 = 0, a = a = 3 a 3, a 3 = g a 3, which follows F (α(x)) = F (x) from G (α(x)) = G (x) using the discussion for the case n = 3. Thus, when B i (a) = 0, i = 1,,, 7, one has F(α(x)) = F(x). The conclusion follows for n = 4 by taking For n = 5, we have a 0 = ( 0, 0, 3 g ) a 3, 3 a 3, g a 3, a 3, 0, 0. det (B 1, B, B 3, B 4, B 5, B 6, B 7, B 8, B 9, B 10 ) (a 1, a 1, a, a, a 3, a 3, a 4, a 4, a 5, a 5 ) = 0, det (B 1, B, B 3, B 4, B 5, B 6, B 7, B 8, B 9 ) (a 1, a 1, a, a, a 3, a 4, a 4, a 5, a 5 ) = M 5. Solve B i (a) = 0, i = 1,,, 9, we obtain a 1 = a 1 = a 4 = a 4 = a 5 = a 5 = 0, a = a = 3 a 3, a 3 = g a 3.

3, : Liénard 53 As before, when B i (a) = 0, i = 1,,, 9, one has F(α(x)) = F(x). The conclusion follows for n = 5 by taking This ends the proof. a 0 = ( 0, 0, 3 g a 3, 3 a 3, g a 3, a 3 )., 0, 0, 0, 0 [ References ] [ 1 ] HAN Mao-an, LIU Xia. Hopf bifurcation for non-smooth Liénard systems[j]. Int J Bifurcation and Chaos, 009, 19(7): 401-415. [ ] COLL B, GASULL A, PROHENS R. Limit cycles for non smooth differential equations via schwarzian derivative [J]. J Diff Eqs, 1996, 13: 03-1. [ 3 ] LEINE R I, VAN CAMPEN D H. Bifurcation phenomena in non-smooth dynamical systems[j]. European Journal of Mechanics, 006, 5: 595-616. [ 4 ] LEINE R I. Bifurcation of equilibria in non-smooth continuous systems[j] Physica D, 006, 3: 11-137. [ 5 ] ZOU Y, KÜPPER T, BEYN W J. Generalized Hopf bifurcation for palnar Filippov systems continuous at the origin[j]. J. Nonlinear Sci, 006, 16: 159-177. [ 6 ] GASULL A, TORREGROSA J. Center-focus problem for discontinuous planar differ ential equations[j]. Int J Bifurcation and Chaos, 003, 13: 1755-1765. [ 7 ] KUNZE M. Non-Smooth Dynamical Systems[M]. Berlin: Springer-Verlag, 000. ( 0 ) [ ] [ 1 ] MERTON R C. Option pricing when underlying stock returns are discontinuous[j]. Journal of Financial Economics, 1976(3): 15-144. [ ] NAIK V, LEE M. General equilibrium prcing of options on the market portfolio with discontinuous returns[j]. Review of Financial Studies, 1990(3): 493-51. [ 3 ] AHN C M, CHO D C, PARK K. The pricing of foreign currency options under jump-diffusion processes[j]. Journal of Futures Markets, 007, 7: 669-695. [ 4 ] KOU S G. A jump diffusion model for option pricing[j]. Management Science, 00, 48: 1086-1101. [ 5 ] KOU S G., WANG H. Option pricing under a double exponential jump diffusion model[j]. Management Science, 004, 50: 1178-119. [ 6 ] BIGER N, HULL J. The valuation of currency options[j]. Financial Manage, 1983(1): 4-8. [ 7 ] COX J C, INGERSOLL J E, ROSS S A. A theory of the term structure of interest rates[j]. Ecomometrica, 1985, 53: 385-407. [ 8 ] AMIN K I, Jarrow R. Pricing foreign currency options under stochastic interest rates [J]. Journal of International Money and Finance, 1991(10): 310-39. [ 9 ] HULL J. Options, Futures, and Other Derivatives[M] 5th ed. New York: Prentice Hall International Inc, 003. [10] BLENMAN L P, Clark S P. Power exchange options[j]. Finance Research Letters, 005 (): 97-106. [11] JAIMUNGAL S, Wang T. Catastrophe options with stochastic interest rates and compound Poisson losses.[j]. Insurance: Mathematics and Economics, 006, 38: 469-483. [1] SHREVE. Stochatic Calculus for Finance II: Continuous time models[m]. New York: Springer, 000.