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Limit cycles for extended bistable stochastic resonance system
Available online at www.sciencedirect.com
Physica A 331 (2004) 467 – 476
www.elsevier.com/locate/physa
Limit cycles for extended bistable stochastic resonance system Xue-Juan Zhanga; b , Guan-Xiang Wanga;∗ a LMAM,
School of Mathematical Sciences, Peking University, Beijing 100871, PR China College of Arts and Sciences, Zhejiang Province 312000, PR China
b ShaoXing
Received 19 May 2003; received in revised form 20 June 2003
Abstract Bistable system is a typical model in stochastic resonance (SR) researching. And it has been known that the deterministic dynamics of a noised system plays an important role in making SR phenomena occur. By embedding a non-autonomous system onto a cylinder, or extending the one-dimensional non-autonomous system as a three-dimensional autonomous system restricted on a cylinder, this paper theoretically analyzes the dynamics of the well-known periodically driven bistable system. After three limit cycles (equivalent to the periodic states of the un-extended system) are proved existent for subthreshold case, they are shown to be preserved even in the suprathreshold case with large driving frequency. Moreover, it is also proved in suprathreshold case that the three ones keep touching each other and combine into one larger globally stable limit cycle as the driving frequency decreases to some certain degree. c 2003 Elsevier B.V. All rights reserved. PACS: 05.40.−a; 05.45.−a; 02.30.Hq Keywords: Stochastic resonance; Limit cycles; Bistable system
1. Introduction As one of the most frequently used model, bistable system play an irreplaceable role in the recently developed (and still expanding) research :elds concerning stochastic resonance (SR) (e.g., see recent reviews [1–4]). Many researchers have noted that SR phenomena depend much on the dynamical properties of the corresponding deterministic system. For instance, Thorwart et al. [5], Hu et al. [6], Apostolico [7] and ∗
Corresponding author. E-mail address: [email protected] (G.-X. Wang).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.09.044
468
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Qian et al. [8] pointed out that the existence and positions of limit cycles in deterministic system are important ingredients for making SR become true when noise is introduced into the system. However, getting useful dynamics information from an non-autonomic system is not so easy as it seems even though the system is low dimensional. Taking the well-known one-dimensional bistable system x˙ = x − x3 + A cos !t
(1)
as an example, the existence, positions and the number of the stable periodic orbits of it are not trivial as it seems. The noised analog of (1) x˙ = x − x3 + A cos !(t) + D(t)
(2)
has been studied in amount of literatures e.g., Refs. [8–18]. In system (2), (t) is the Gaussian white noise satisfying (t) = 0, (t)(t ) = (t − t ), and D is the intensity of the noise. It has been shown that SR happens in (2) for non-switching case (A ¡ Ath ) and residual SR occurs for switching case (A ¿ Ath ), where Ath denotes the system bistability threshold, which is the critical value of the parameter A where one of the two stable periodic states of (1) disappears. In his monographic report [1], Jung, as traditionally done in dynamics analysis, embedded system (2) into a two-dimensional extended phase space and changed the non-autonomous system (1) into an autonomous system dx = x − x3 + A cos ! ; dt d =1 : dt
(3)
By systematically analyzing the corresponding extended Fokker–Planck operator, he gave important properties of the non-stationary stochastic process induced by (2). In another letter [19], he and his colleagues even obtained a scaling law for the unique stable periodic state of (1)(called hysteresis loop there). In our previous paper [8], interesting dynamical structures of (1) were displayed by numerical simulation, and the mechanism of SR in system (2) was explained clearly with the dynamical properties. This letter presents the mathematical analysis for the observed numerical facts. √ For the subthreshold case (A 6 A0 ), where A0 = 2 9 3 is the barrier of the potential V (x) = x4 =4 − x2 =2, we prove that there exists exactly two stable periodic orbits and another unstable one for every :xed pair of values of (A; !). For the suprathreshold case (A ¿ A0 ), we prove that the three periodic orbits still remain for suJciently large driving frequency !, and that they get touching each other to combine into one larger globally stable periodic orbit as the frequency ! decreases to some certain degree. To prove the results, instead of embedding the non-autonomous system (1) into two-dimensional Kat space R2 , we embed it into a rolled two-dimensional space, the cylinder E = R × S 1 . Let y = A cos !t; z = A sin !t, then system (1) can be equivalently
X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
469
characterized as x˙ = x − x3 + y ; y˙ = −!z ; z˙ = !y :
(4)
So the solutions to (1) are curves winding on the cylinder E = {(x; y; z)|y2 + z 2 = A2 } and the periodic orbits of (1) are now the limit cycles on the cylinder E. In traditional investigation on stochastic resonance of system (1), the eLect of the periodic driving is considered as to break the symmetry of the bistable potential. Meanwhile the driving itself alone is insuJcient to motivate the particle switching between the potential wells, it is the presence of small noise that produce a :nite probability of transition between the wells. A resonance occurs when the time scale of the noise-induced switching matches the period of the driving. In our previous paper [8], with the embedding method introduced above as well as numerical simulation, the occurrence of stochastic resonance in system (1) was clearly explained based on the dynamical structure of the extended deterministic system (4). In such a picture, the role of the periodic driving is just to modulate the relative positions of the stable limit cycles and the unstable one, and the necessary condition for SR occur is the existence of three limit cycles. For the subthreshold case where three limit cycles exist, SR occurs as long as the cycles are at suitable positions. The surprising part of the numerical result is that, even for suprathreshold case, SR still happens provided the driving is fast enough. That means the three limit cycles remaining in suprathreshold case. This paper presents the complete theoretical analysis of the limit cycles. In next section, the existence and stabilities of the three limit cycles on E are proved. The preserving of them is shown in Section 3. Lastly, in section 4, we prove the existence of one unique globally stable limit cycle on E. 2. Existence and stabilities of limit cycles for subthreshold case Now consider the extended system (4) on E. For the case of subthreshold (A 6 A0 ), the surface y = x3 − x divides the cylinder into three parts (See Fig. 1(a)) and the equations x − x3 − A = 0 and x − x3 + A = 0 have three roots, respectively. Let a be the largest root of x3 −x = A and p1 ; p2 two arbitrary values satisfying p1 ¡−a and p2 ¿ a. De:ne three ring-shape regions on the cylinder E: G−1 : (y; z; x): p1 6 x 6 √ √ √ √ − 33 ; y2 + z 2 = A2 ,G0 : (y; z; x): − 33 6 x 6 33 ; y2 + z 2 = A2 , G1 : (y; z; x): 33 6 x 6 p2 ; y2 + z 2 = A2 . Obviously, they are invariant regions of system (4). Since there is no equilibrium point in these regions, the PoincarPe–Bendixson theorem on a cylinder ensures that there is a limit cycle in every region, respectively. To obtain the stabilities of the limit cycles, we :rst prove the uniqueness of them in their corresponding invariant regions. Because of the symmetry of the system, we
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X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
x
x
y
y z
z
(a)
(b)
Fig. 1. Phase space on cylinder. (a) subthreshold case; (b) suprathreshold case.
need only prove the statement in one region. Taking G−1 for illustration, suppose that there are two limit cycles LC1 and LC2 in G−1 , which could be written as LC1 = {(x1 ; y1 ; z1 ): x1 =f1 (t); y1 =A cos !t; z1 =A sin !t}, LC2 ={(x2 ; y2 ; z2 ): x2 =f2 (t); y2 = A cos !t; z2 = A sin !t}. By Eq. (1) (or (4)), x˙ 2 (t) − x˙ 1 (t) = (x2 − x1 )[1 − (x12 + x1 x2 + x22 )] ; [x2 (t) − x1 (t)] − [x2 (0) − x1 (0)] =
0
t
(x2 − x1 )[1 − (x12 + x1 x2 + x22 )] dt :
√ Suppose x1 ¡ x2 , then (x2 − x1 )[1 − (x12 + x1 x2 + x22 )] ¿ 0 since x1 ; x2 ∈ p1 ; − 33 . As a result, [x2 (2=!) − x1 (2=!)] − [x2 (0) − x1 (0)] ¿ 0, which contradicts with the periodicity of x1 (t) and x2 (t). The uniqueness and the sign of x˙ show that the two limit cycles in G−1 and G1 are stable and the one in G0 is unstable. It should be pointed out that the three invariant regions still work for the critical do not leave their original regions at the intersecting points case √A = A0 since orbits √ − 33 ; A0 ; 0 and 33 ; −A0 ; 0 on E.
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3. Preserving of the three limit cycles in suprathreshold case with large driving frequency We prove in this section that, as long as the driving frequency ! is large enough, the three limit cycles still exist even when the driving becomes of suprathreshold (A ¿ A0 ). This fact makes sense in such a way. Although in this case there is no explicit invariant region on E (See Fig. 1(b)), one orbit starts from far below the separating curve L0 = {(x; y; z) ∈ E: y = x3 √ − x} will have no opportunity to get across the critical curve L− = (x; y; z) ∈ E: x = − 33 if the angular velocity ! is large. Thus a limit cycle still exists below L− , and the √ case for orbits starting from far above another critical curve L+ = (x; y; z) ∈ E: x = 33 is similar. Therefore, there still are three limit cycles on E and their stabilities are the same as for the subthreshold, so we say the three limit cycles are preserved. To theoretically prove the preserving of the limit cycles, we project system (4) from the cylinder E onto a new two-dimensional Kat space R2 . Let = ex y ; = ex z
(5)
then, after the time translation t = =!, system (4) becomes 1 d = − + x˙ ; d ! d 1 = + x˙ ; d !
(6)
where x˙ =
1 3 2 + 2 d x 1 3 2 + 2 − + = ln ln dt 2 A2 8 A2
2 + 2 A2
−1=2 :
Obviously, the transformation (5) is a homeomorphism, it maps any ring region on the cylinder E to a ring region on − plane, and any closed curve winding on the cylinder E to a closed curve on the − plane. For suJciently large value of !,this system can be regarded as a perturbation of the corresponding linear system d = − ; d d = : d
(7)
Denote = 1=!, f(; ) = x˙ , g(; ) = x˙ . Note that lim(; )→(0; 0) f(; ) = 0 and lim(; )→(0; 0) g(; ) = 0, the equilibrium point (0; 0) of system (7) is kept by system (6). Here we need the classical theorem of perturbation theory, which is cited as a lemma below.
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X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
Lemma. Let (r) = have
2 0
[f(; ) + g(; )] d , where = r cos ; = r sin . Then we
(1) For su8ciently small value of , the necessary condition for system (6) to have a closed orbit near the closed orbit r0 : = r0 sin ; = r0 cos of system (7) is (r0 ) = 0. (2) If r0 ¿ 0; (r0 ) = 0 and r0 is not the extreme point of (r0 ), then system (6) has a closed orbit near r0 for su8ciently small . (3) If (r0 ) = (r0 ) = · · · = (2k) (r0 ) = 0, with (2k+1) (r0 ) ¡ 0 and k is a natural number, then for su8ciently small , system (6) has a limit cycle near r0 , which is stable for ¿ 0 and unstable for ¡ 0. Elementary calculation results in r r
; (r) = 2r 2 ln − ln3 A A r r
r
(r) = 4r ln − ln3 + 2r 1 − 3 ln2 : A A A The roots of (r) = 0 are: r0 = 0 (discarded), r1 = e−1 A, r2 = A and r3 = eA. The corresponding values of (r) are: (r1 ) = −4r1 ¡ 0, (r2 ) = 2r2 ¿ 0, and (r3 ) = −4r3 ¡ 0. By the lemma, one knows that for suJciently large !, system (6) has two stable limit cycles near −1 : = r1 sin ; = r1 cos ; and 1 : = r3 sin ; = r3 cos respectively, and has one unstable limit cycle near 0 : = r2 sin ; = r2 cos . Moreover, there is no other limit cycle on the − plane since there is no other r satisfying (r) = 0. Thus, correspondingly, for suJcient large ! and suprathreshold value of A, system (4) has three limit cycles on E and system (1) still has two stable periodic states.
4. Globally stable limit cycle for suprathreshold case with small driving frequency When the driving becomes of suprathreshold (A ¿ A0 ) and the values of ! is small in system (1), Numerical simulations [8] show that the dynamics of the system is very similar to that of relaxation oscillator. So in this section, we prove the existence and global stability of the periodic state of system (1) for relatively small driving frequency, which equivalently is the unique stable limit cycle of system (4) on E. To get the existence of the global limit cycle for (4) on E, we construct an invariant region near the separating curve L0 on E. The method is to set two bourn lines L1 ; L2 on the cylinder E (See Fig. 2). L1 = P[ 1 P2 ∪ P[ 2 P3 ∪ P[ 3 P4 ∪ P[ 4 P5 ∪ P[ 5 P6 ∪ P[ 6 P7 ∪ P[ 7 P1 and L2 is the symmetric line of L1 about the origin point, L2 = Q[ 1 Q2 ∪ Q[ 2 Q3 ∪ Q[ 3 Q4 ∪ Q[ 4 Q5 ∪ Q[ 5 Q6 ∪ Q[ 6 Q7 ∪ Q[ 7 Q1 :
X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
x z
P1
473
P2
y
Q4
Q5 Q6
P3
dx/dt<0
Q7
P7 Q3 dx/dt>0
P6 P5
P4
Q2 Q1
Fig. 2. Invariant region on the cylinder.
Every part of L1 ; is described as follows: (1) P[ 1 P2 = {(x; y; z) ∈ E: x = xup ; A0 6 y 6 A} ; 3 P[ 2 P3 = {(x; y; z) ∈ E: x0 ¡ x ¡ xup ; y = x − x − 1 (x)} ; √ where x0 = 33 , 1 (x) = 2!A(a + ! − x), xup is the unique root of the equation A and a is the positive root of the equation x3 − x = A. P1 is the x3 − x − (x) = point xup ; A0 ; − A2 − A20 , P2 is the point (xup ; A; 0) and P3 is the point x0 ; −(A0 + A1 ); A2 − (A0 + A1 )2 with A1 = 2!A(a + ! − x0 ). It is easy to check that xup → a as ! → 0. P[ 3 P4 = {(x; y; z) ∈ E: − 2x0 6 x ¡ x0 ; y = 2 (x)}, where 2 (x) = −(A0 + A1 ) + √ (2) A!= 2(a + ! − x0 )(x−x0 ) and x = 2x0 is the bigger one of the two roots of equation x3 − x = A0 . P4 is the point − 2x0 ; −(A0 + A1 + A2 ); A2 − (A0 + A1 + A2 )2 with √ A2 = 3x0 A!= 2(a + ! − x0 ). 3 [ − 3 (x)}, where 3 (x) = (3) P4 P5 = {(x; y; z) ∈ E: − x1 6 x ¡ − 2x0 ; y = x − x 2!A(a + ! − x) + c with c = A0 + A + A − A and A = 2!A(a + ! + 2x0 ), and 2 3 3 1 3 −x1 is the unique root of x − x − 2!A(a + ! − x) − c = −A. Note that −x1 ¿ − a since c → 0 as ! → 0. 3 (4) P[ 5 P6 = {(x; y; z) ∈ E: x= −x1 ; −A 6 y 6 y0 }, where y0 = −x1 + x1 and P6 is the point − x1 ; y0 ; − A2 − y02 . P = {(x; y; z) ∈ E:−x1 6 x 6−x0 ; y = x3 −x}, where P7 is the point −x0 ; A0 ; (5) P[ 6 7 − A2 − A20 . (6) P[ 7 P1 is the vertical line connecting point P7 and point P1 . Thus, L1 is a closed curve on E and L1 , L2 form a closed region G on the cylinder.
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X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
To prove the region G is invariant, we just need check that all orbits starting from points on the bourn lines L1 , L2 will enter G as time increases. Also because of the symmetry of L1 to L2 , we need only prove the statement valid on L1 . And the validness is obvious on the segments P[ 6 P7 and P[ 7 P1 . 1 P2 , P[ 5 P6 , P[ The orbits of system (4) intersecting P[ P point to the inside of G since 2 3
dy
−!z
! A2 − y2
= 3 =
d x P[ x − x + y P[ 1 (x) 2 P3 2 P3 P[ 2 P3
√
√ !A !A 2 = 3x2 − 1 − ¡ ¡ 3x − 1 + 2(a + ! − x) 2(a + ! − x)
1 (x)
:
The orbits of system (4) intersecting P[ 3 P4 points to the inside of G since
dy
−!z
= 3
d x P[ x − x + y P[ 3 P4 3 P4
! A2 − y 2
=
x − x3 + (A0 + A1 ) + a!= (a + ! − x0 )(x0 − x) √
¡
P[ 3 P4
A! A! ; = A1 2(a + ! − x0 )
where we use the property x − x3 + A ¿ 0 for −x0 6 x 6 x0 . On the segment P[ 4 P5 , the situation is the same as on P[ 2 P3 since the two segments are parallel to each other. Therefore, we :nish the proof of the invariance of region G. By PoincarPe–Bendixsion theorem on cylinder, there exists a limit cycle in region G. To get the uniqueness of the limit cycle, we again project system (4) onto the plane − with translation (5), but keep the time invariable t. The changed system on plane − now is d = x˙ − ! , F(; ) ; dt d = x˙ + ! , G(; ) : dt
(8)
Since transformation (5) is a homeomorphism, the idea here is to prove that any limit cycle of system (8) is stable. So there can not be more than one limit cycle on − plane as well as on the cylinder E. Let L1 , L2 and G be the image of curves L1 , L2 and region G, respectively, under transformation (5). Suppose L! be a limit cycle of system (4) on E and its image on − plane denoted by L! , which is a limit cycle of system (8). The stability of L! on E is equivalent to that of L! on − plane.
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475
Let V (; ) be the divergence of system (8) along L! on − plane, i.e., V (; ) = 9F(; )=9 + 9G(; )=9 = 2˙x + · 9˙x=9 + · 9˙x=9. So, by the basic theory of planar dynamical systems, we only need to prove V (; ) dt ¡ 0 : (9) L!
Note that 9˙x 9˙x 9 9˙x 9 9˙x 9˙x = · + · =· +· ; 9x 9 9x 9 9x 9 9 we have
9˙x 9˙x dt 2˙x(; ) + · +· 9 9 L! 9˙x 9˙x =2 dt = dt = − dx + F (x) dt ; 9x 9x L! L! L! L!
L!
V (; ) dt =
where F(x) = x3 − x; F (x) = 3x2 − 1. Since that F (x) ¡ 0 for |x| ¡ x0 , F (x) ¿ 0 for |x| ¿ x0 and that the limit cycle L! is symmetry respect to origin point, we get F (x) dt = F (x) dt + F (x) dt L!
L! ;|x|6x0
=2 −
L! ;|x|6x0 ; z¿0
L! ; x0 6|x|6xup 2
(1 − 3x ) dt +
2
L! ; x0 6|x|6xup ; z¿0
(3x − 1) dt
:
(10)
Firstly, L! ;|x|6x0 (1−3x2 ) dt ¿ t, where t is the time spent by the orbit L! moving from x = x0 to x = −x0 between lines P3 P4 and Q7 Q1 on the cylinder E. It is already known that the maximum diLerence of y between lines P3 P4 and Q7 Q1 on E is y = A1 + A2 ∼ o(!1=2 ), so t ∼ o(!−1=2 ) since y = A cos !t. Therefore, for small values of !, there exists a constant M such that (1 − 3x2 ) dt ¿ M!−1=2 : (11) L! ;|x|6x0 ; z¿0
On the other hand, 0¡
L! ; x0 6x6xup ; z¿0
=!−1
A
A0
(3x2 − 1) dt ¡
L! ;2x0 6x6xup ; z¿0
3x2 − 1 dy ! A2 − y 2
3x2 − 1 A0 dy 6 !−1 (12x02 − 2x0 ) arcsin : 2 2 A ! A −y
(12)
Therefore, (12), (13) and (10) implies inequality (8). Thus the uniqueness of limit cycle on E is proved.
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X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
Acknowledgements This work was supported by Natural Sciences Founds of China (NSF10271009) and we thank Hong Gu and Zhu-Yu Wang for showing us some numerical results from their Master theses. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
P. Jung, Phys. Rep. 234 (1993) 175. L. Gammaitoni, P. HRanggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70 (1998) 223. S. Mitaim, B. Kosko, Proc. IEEE 86 (1998) 2152. R.D. Astumian, P. HRanggi, Phys. Today 55 (11) (2002) 33; P. Reimann, Phys. Rep. 362 (2002) 57. M. Thorwart, M. Grifoni, P. HRanggi, Ann. Phys. 293 (2001) 15. G. Hu, H. Haken, C.Z. Ning, Phys. Rev. E 47 (1993) 2321. F. Apostolico, L. Gammaitoni, F. Marchesoni, S. Santucci, Phys. Rev. E 55 (1997) 36; F. Marchesoni, F. Apostolico, L. Gammaitoni, S. Santucci, Phys. Rev. E 58 (1998) 7079. M. Qian, X.-J. Zhang, Phys. Rev. E 65 (2002) 011101-1. P. Jung, P. HRanggi, Europhy. Lett. 8 (1989) 505. P. Jung, P. HRanggi, Ber. Bunsenges. Phys. Chem. 95 (1991) 311. P. Jung, P. HRanggi, Phys. Rev. A 44 (1991) 8032. L. Gammaitoni, F. Marchesoni, E. Menichella–Saetta, S. Santucci, Phys. Rev. Lett. 62 (1989) 349. T. Zhou, F. Moss, P. Jung, Phys. Rev. A 42 (1990) 3161. M.H. Choi, R.F. Fox, P. Jung, Phys. Rev. E 57 (1998) 6335. M. Misono, T. Kohmoto, Y. Fukuda, M. Kunitomo, Phys. Rev. E 58 (1998) 5602. A. Silchenko, T. Kapitaniak, V. Anishchenko, Phys. Rev. E 59 (1999) 1593. Y. Jia, S.-N. Yu, J.-R. Li, Phys. Rev. E 62 (2000) 1869. A.L. Pankratov, M. Salerno, Phys. Rev. E 61 (2000) 1206. P. Jung, G. Gray, R. Roy, Phys. Rev. Lett. 65 (1990) 1873.
Physica A 331 (2004) 467 – 476
www.elsevier.com/locate/physa
Limit cycles for extended bistable stochastic resonance system Xue-Juan Zhanga; b , Guan-Xiang Wanga;∗ a LMAM,
School of Mathematical Sciences, Peking University, Beijing 100871, PR China College of Arts and Sciences, Zhejiang Province 312000, PR China
b ShaoXing
Received 19 May 2003; received in revised form 20 June 2003
Abstract Bistable system is a typical model in stochastic resonance (SR) researching. And it has been known that the deterministic dynamics of a noised system plays an important role in making SR phenomena occur. By embedding a non-autonomous system onto a cylinder, or extending the one-dimensional non-autonomous system as a three-dimensional autonomous system restricted on a cylinder, this paper theoretically analyzes the dynamics of the well-known periodically driven bistable system. After three limit cycles (equivalent to the periodic states of the un-extended system) are proved existent for subthreshold case, they are shown to be preserved even in the suprathreshold case with large driving frequency. Moreover, it is also proved in suprathreshold case that the three ones keep touching each other and combine into one larger globally stable limit cycle as the driving frequency decreases to some certain degree. c 2003 Elsevier B.V. All rights reserved. PACS: 05.40.−a; 05.45.−a; 02.30.Hq Keywords: Stochastic resonance; Limit cycles; Bistable system
1. Introduction As one of the most frequently used model, bistable system play an irreplaceable role in the recently developed (and still expanding) research :elds concerning stochastic resonance (SR) (e.g., see recent reviews [1–4]). Many researchers have noted that SR phenomena depend much on the dynamical properties of the corresponding deterministic system. For instance, Thorwart et al. [5], Hu et al. [6], Apostolico [7] and ∗
Corresponding author. E-mail address: [email protected] (G.-X. Wang).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.09.044
468
X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
Qian et al. [8] pointed out that the existence and positions of limit cycles in deterministic system are important ingredients for making SR become true when noise is introduced into the system. However, getting useful dynamics information from an non-autonomic system is not so easy as it seems even though the system is low dimensional. Taking the well-known one-dimensional bistable system x˙ = x − x3 + A cos !t
(1)
as an example, the existence, positions and the number of the stable periodic orbits of it are not trivial as it seems. The noised analog of (1) x˙ = x − x3 + A cos !(t) + D(t)
(2)
has been studied in amount of literatures e.g., Refs. [8–18]. In system (2), (t) is the Gaussian white noise satisfying (t) = 0, (t)(t ) = (t − t ), and D is the intensity of the noise. It has been shown that SR happens in (2) for non-switching case (A ¡ Ath ) and residual SR occurs for switching case (A ¿ Ath ), where Ath denotes the system bistability threshold, which is the critical value of the parameter A where one of the two stable periodic states of (1) disappears. In his monographic report [1], Jung, as traditionally done in dynamics analysis, embedded system (2) into a two-dimensional extended phase space and changed the non-autonomous system (1) into an autonomous system dx = x − x3 + A cos ! ; dt d =1 : dt
(3)
By systematically analyzing the corresponding extended Fokker–Planck operator, he gave important properties of the non-stationary stochastic process induced by (2). In another letter [19], he and his colleagues even obtained a scaling law for the unique stable periodic state of (1)(called hysteresis loop there). In our previous paper [8], interesting dynamical structures of (1) were displayed by numerical simulation, and the mechanism of SR in system (2) was explained clearly with the dynamical properties. This letter presents the mathematical analysis for the observed numerical facts. √ For the subthreshold case (A 6 A0 ), where A0 = 2 9 3 is the barrier of the potential V (x) = x4 =4 − x2 =2, we prove that there exists exactly two stable periodic orbits and another unstable one for every :xed pair of values of (A; !). For the suprathreshold case (A ¿ A0 ), we prove that the three periodic orbits still remain for suJciently large driving frequency !, and that they get touching each other to combine into one larger globally stable periodic orbit as the frequency ! decreases to some certain degree. To prove the results, instead of embedding the non-autonomous system (1) into two-dimensional Kat space R2 , we embed it into a rolled two-dimensional space, the cylinder E = R × S 1 . Let y = A cos !t; z = A sin !t, then system (1) can be equivalently
X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
469
characterized as x˙ = x − x3 + y ; y˙ = −!z ; z˙ = !y :
(4)
So the solutions to (1) are curves winding on the cylinder E = {(x; y; z)|y2 + z 2 = A2 } and the periodic orbits of (1) are now the limit cycles on the cylinder E. In traditional investigation on stochastic resonance of system (1), the eLect of the periodic driving is considered as to break the symmetry of the bistable potential. Meanwhile the driving itself alone is insuJcient to motivate the particle switching between the potential wells, it is the presence of small noise that produce a :nite probability of transition between the wells. A resonance occurs when the time scale of the noise-induced switching matches the period of the driving. In our previous paper [8], with the embedding method introduced above as well as numerical simulation, the occurrence of stochastic resonance in system (1) was clearly explained based on the dynamical structure of the extended deterministic system (4). In such a picture, the role of the periodic driving is just to modulate the relative positions of the stable limit cycles and the unstable one, and the necessary condition for SR occur is the existence of three limit cycles. For the subthreshold case where three limit cycles exist, SR occurs as long as the cycles are at suitable positions. The surprising part of the numerical result is that, even for suprathreshold case, SR still happens provided the driving is fast enough. That means the three limit cycles remaining in suprathreshold case. This paper presents the complete theoretical analysis of the limit cycles. In next section, the existence and stabilities of the three limit cycles on E are proved. The preserving of them is shown in Section 3. Lastly, in section 4, we prove the existence of one unique globally stable limit cycle on E. 2. Existence and stabilities of limit cycles for subthreshold case Now consider the extended system (4) on E. For the case of subthreshold (A 6 A0 ), the surface y = x3 − x divides the cylinder into three parts (See Fig. 1(a)) and the equations x − x3 − A = 0 and x − x3 + A = 0 have three roots, respectively. Let a be the largest root of x3 −x = A and p1 ; p2 two arbitrary values satisfying p1 ¡−a and p2 ¿ a. De:ne three ring-shape regions on the cylinder E: G−1 : (y; z; x): p1 6 x 6 √ √ √ √ − 33 ; y2 + z 2 = A2 ,G0 : (y; z; x): − 33 6 x 6 33 ; y2 + z 2 = A2 , G1 : (y; z; x): 33 6 x 6 p2 ; y2 + z 2 = A2 . Obviously, they are invariant regions of system (4). Since there is no equilibrium point in these regions, the PoincarPe–Bendixson theorem on a cylinder ensures that there is a limit cycle in every region, respectively. To obtain the stabilities of the limit cycles, we :rst prove the uniqueness of them in their corresponding invariant regions. Because of the symmetry of the system, we
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x
x
y
y z
z
(a)
(b)
Fig. 1. Phase space on cylinder. (a) subthreshold case; (b) suprathreshold case.
need only prove the statement in one region. Taking G−1 for illustration, suppose that there are two limit cycles LC1 and LC2 in G−1 , which could be written as LC1 = {(x1 ; y1 ; z1 ): x1 =f1 (t); y1 =A cos !t; z1 =A sin !t}, LC2 ={(x2 ; y2 ; z2 ): x2 =f2 (t); y2 = A cos !t; z2 = A sin !t}. By Eq. (1) (or (4)), x˙ 2 (t) − x˙ 1 (t) = (x2 − x1 )[1 − (x12 + x1 x2 + x22 )] ; [x2 (t) − x1 (t)] − [x2 (0) − x1 (0)] =
0
t
(x2 − x1 )[1 − (x12 + x1 x2 + x22 )] dt :
√ Suppose x1 ¡ x2 , then (x2 − x1 )[1 − (x12 + x1 x2 + x22 )] ¿ 0 since x1 ; x2 ∈ p1 ; − 33 . As a result, [x2 (2=!) − x1 (2=!)] − [x2 (0) − x1 (0)] ¿ 0, which contradicts with the periodicity of x1 (t) and x2 (t). The uniqueness and the sign of x˙ show that the two limit cycles in G−1 and G1 are stable and the one in G0 is unstable. It should be pointed out that the three invariant regions still work for the critical do not leave their original regions at the intersecting points case √A = A0 since orbits √ − 33 ; A0 ; 0 and 33 ; −A0 ; 0 on E.
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3. Preserving of the three limit cycles in suprathreshold case with large driving frequency We prove in this section that, as long as the driving frequency ! is large enough, the three limit cycles still exist even when the driving becomes of suprathreshold (A ¿ A0 ). This fact makes sense in such a way. Although in this case there is no explicit invariant region on E (See Fig. 1(b)), one orbit starts from far below the separating curve L0 = {(x; y; z) ∈ E: y = x3 √ − x} will have no opportunity to get across the critical curve L− = (x; y; z) ∈ E: x = − 33 if the angular velocity ! is large. Thus a limit cycle still exists below L− , and the √ case for orbits starting from far above another critical curve L+ = (x; y; z) ∈ E: x = 33 is similar. Therefore, there still are three limit cycles on E and their stabilities are the same as for the subthreshold, so we say the three limit cycles are preserved. To theoretically prove the preserving of the limit cycles, we project system (4) from the cylinder E onto a new two-dimensional Kat space R2 . Let = ex y ; = ex z
(5)
then, after the time translation t = =!, system (4) becomes 1 d = − + x˙ ; d ! d 1 = + x˙ ; d !
(6)
where x˙ =
1 3 2 + 2 d x 1 3 2 + 2 − + = ln ln dt 2 A2 8 A2
2 + 2 A2
−1=2 :
Obviously, the transformation (5) is a homeomorphism, it maps any ring region on the cylinder E to a ring region on − plane, and any closed curve winding on the cylinder E to a closed curve on the − plane. For suJciently large value of !,this system can be regarded as a perturbation of the corresponding linear system d = − ; d d = : d
(7)
Denote = 1=!, f(; ) = x˙ , g(; ) = x˙ . Note that lim(; )→(0; 0) f(; ) = 0 and lim(; )→(0; 0) g(; ) = 0, the equilibrium point (0; 0) of system (7) is kept by system (6). Here we need the classical theorem of perturbation theory, which is cited as a lemma below.
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Lemma. Let (r) = have
2 0
[f(; ) + g(; )] d , where = r cos ; = r sin . Then we
(1) For su8ciently small value of , the necessary condition for system (6) to have a closed orbit near the closed orbit r0 : = r0 sin ; = r0 cos of system (7) is (r0 ) = 0. (2) If r0 ¿ 0; (r0 ) = 0 and r0 is not the extreme point of (r0 ), then system (6) has a closed orbit near r0 for su8ciently small . (3) If (r0 ) = (r0 ) = · · · = (2k) (r0 ) = 0, with (2k+1) (r0 ) ¡ 0 and k is a natural number, then for su8ciently small , system (6) has a limit cycle near r0 , which is stable for ¿ 0 and unstable for ¡ 0. Elementary calculation results in r r
; (r) = 2r 2 ln − ln3 A A r r
r
(r) = 4r ln − ln3 + 2r 1 − 3 ln2 : A A A The roots of (r) = 0 are: r0 = 0 (discarded), r1 = e−1 A, r2 = A and r3 = eA. The corresponding values of (r) are: (r1 ) = −4r1 ¡ 0, (r2 ) = 2r2 ¿ 0, and (r3 ) = −4r3 ¡ 0. By the lemma, one knows that for suJciently large !, system (6) has two stable limit cycles near −1 : = r1 sin ; = r1 cos ; and 1 : = r3 sin ; = r3 cos respectively, and has one unstable limit cycle near 0 : = r2 sin ; = r2 cos . Moreover, there is no other limit cycle on the − plane since there is no other r satisfying (r) = 0. Thus, correspondingly, for suJcient large ! and suprathreshold value of A, system (4) has three limit cycles on E and system (1) still has two stable periodic states.
4. Globally stable limit cycle for suprathreshold case with small driving frequency When the driving becomes of suprathreshold (A ¿ A0 ) and the values of ! is small in system (1), Numerical simulations [8] show that the dynamics of the system is very similar to that of relaxation oscillator. So in this section, we prove the existence and global stability of the periodic state of system (1) for relatively small driving frequency, which equivalently is the unique stable limit cycle of system (4) on E. To get the existence of the global limit cycle for (4) on E, we construct an invariant region near the separating curve L0 on E. The method is to set two bourn lines L1 ; L2 on the cylinder E (See Fig. 2). L1 = P[ 1 P2 ∪ P[ 2 P3 ∪ P[ 3 P4 ∪ P[ 4 P5 ∪ P[ 5 P6 ∪ P[ 6 P7 ∪ P[ 7 P1 and L2 is the symmetric line of L1 about the origin point, L2 = Q[ 1 Q2 ∪ Q[ 2 Q3 ∪ Q[ 3 Q4 ∪ Q[ 4 Q5 ∪ Q[ 5 Q6 ∪ Q[ 6 Q7 ∪ Q[ 7 Q1 :
X.-J. Zhang, G.-X. Wang / Physica A 331 (2004) 467 – 476
x z
P1
473
P2
y
Q4
Q5 Q6
P3
dx/dt<0
Q7
P7 Q3 dx/dt>0
P6 P5
P4
Q2 Q1
Fig. 2. Invariant region on the cylinder.
Every part of L1 ; is described as follows: (1) P[ 1 P2 = {(x; y; z) ∈ E: x = xup ; A0 6 y 6 A} ; 3 P[ 2 P3 = {(x; y; z) ∈ E: x0 ¡ x ¡ xup ; y = x − x − 1 (x)} ; √ where x0 = 33 , 1 (x) = 2!A(a + ! − x), xup is the unique root of the equation A and a is the positive root of the equation x3 − x = A. P1 is the x3 − x − (x) = point xup ; A0 ; − A2 − A20 , P2 is the point (xup ; A; 0) and P3 is the point x0 ; −(A0 + A1 ); A2 − (A0 + A1 )2 with A1 = 2!A(a + ! − x0 ). It is easy to check that xup → a as ! → 0. P[ 3 P4 = {(x; y; z) ∈ E: − 2x0 6 x ¡ x0 ; y = 2 (x)}, where 2 (x) = −(A0 + A1 ) + √ (2) A!= 2(a + ! − x0 )(x−x0 ) and x = 2x0 is the bigger one of the two roots of equation x3 − x = A0 . P4 is the point − 2x0 ; −(A0 + A1 + A2 ); A2 − (A0 + A1 + A2 )2 with √ A2 = 3x0 A!= 2(a + ! − x0 ). 3 [ − 3 (x)}, where 3 (x) = (3) P4 P5 = {(x; y; z) ∈ E: − x1 6 x ¡ − 2x0 ; y = x − x 2!A(a + ! − x) + c with c = A0 + A + A − A and A = 2!A(a + ! + 2x0 ), and 2 3 3 1 3 −x1 is the unique root of x − x − 2!A(a + ! − x) − c = −A. Note that −x1 ¿ − a since c → 0 as ! → 0. 3 (4) P[ 5 P6 = {(x; y; z) ∈ E: x= −x1 ; −A 6 y 6 y0 }, where y0 = −x1 + x1 and P6 is the point − x1 ; y0 ; − A2 − y02 . P = {(x; y; z) ∈ E:−x1 6 x 6−x0 ; y = x3 −x}, where P7 is the point −x0 ; A0 ; (5) P[ 6 7 − A2 − A20 . (6) P[ 7 P1 is the vertical line connecting point P7 and point P1 . Thus, L1 is a closed curve on E and L1 , L2 form a closed region G on the cylinder.
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To prove the region G is invariant, we just need check that all orbits starting from points on the bourn lines L1 , L2 will enter G as time increases. Also because of the symmetry of L1 to L2 , we need only prove the statement valid on L1 . And the validness is obvious on the segments P[ 6 P7 and P[ 7 P1 . 1 P2 , P[ 5 P6 , P[ The orbits of system (4) intersecting P[ P point to the inside of G since 2 3
dy
−!z
! A2 − y2
= 3 =
d x P[ x − x + y P[ 1 (x) 2 P3 2 P3 P[ 2 P3
√
√ !A !A 2 = 3x2 − 1 − ¡ ¡ 3x − 1 + 2(a + ! − x) 2(a + ! − x)
1 (x)
:
The orbits of system (4) intersecting P[ 3 P4 points to the inside of G since
dy
−!z
= 3
d x P[ x − x + y P[ 3 P4 3 P4
! A2 − y 2
=
x − x3 + (A0 + A1 ) + a!= (a + ! − x0 )(x0 − x) √
¡
P[ 3 P4
A! A! ; = A1 2(a + ! − x0 )
where we use the property x − x3 + A ¿ 0 for −x0 6 x 6 x0 . On the segment P[ 4 P5 , the situation is the same as on P[ 2 P3 since the two segments are parallel to each other. Therefore, we :nish the proof of the invariance of region G. By PoincarPe–Bendixsion theorem on cylinder, there exists a limit cycle in region G. To get the uniqueness of the limit cycle, we again project system (4) onto the plane − with translation (5), but keep the time invariable t. The changed system on plane − now is d = x˙ − ! , F(; ) ; dt d = x˙ + ! , G(; ) : dt
(8)
Since transformation (5) is a homeomorphism, the idea here is to prove that any limit cycle of system (8) is stable. So there can not be more than one limit cycle on − plane as well as on the cylinder E. Let L1 , L2 and G be the image of curves L1 , L2 and region G, respectively, under transformation (5). Suppose L! be a limit cycle of system (4) on E and its image on − plane denoted by L! , which is a limit cycle of system (8). The stability of L! on E is equivalent to that of L! on − plane.
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Let V (; ) be the divergence of system (8) along L! on − plane, i.e., V (; ) = 9F(; )=9 + 9G(; )=9 = 2˙x + · 9˙x=9 + · 9˙x=9. So, by the basic theory of planar dynamical systems, we only need to prove V (; ) dt ¡ 0 : (9) L!
Note that 9˙x 9˙x 9 9˙x 9 9˙x 9˙x = · + · =· +· ; 9x 9 9x 9 9x 9 9 we have
9˙x 9˙x dt 2˙x(; ) + · +· 9 9 L! 9˙x 9˙x =2 dt = dt = − dx + F (x) dt ; 9x 9x L! L! L! L!
L!
V (; ) dt =
where F(x) = x3 − x; F (x) = 3x2 − 1. Since that F (x) ¡ 0 for |x| ¡ x0 , F (x) ¿ 0 for |x| ¿ x0 and that the limit cycle L! is symmetry respect to origin point, we get F (x) dt = F (x) dt + F (x) dt L!
L! ;|x|6x0
=2 −
L! ;|x|6x0 ; z¿0
L! ; x0 6|x|6xup 2
(1 − 3x ) dt +
2
L! ; x0 6|x|6xup ; z¿0
(3x − 1) dt
:
(10)
Firstly, L! ;|x|6x0 (1−3x2 ) dt ¿ t, where t is the time spent by the orbit L! moving from x = x0 to x = −x0 between lines P3 P4 and Q7 Q1 on the cylinder E. It is already known that the maximum diLerence of y between lines P3 P4 and Q7 Q1 on E is y = A1 + A2 ∼ o(!1=2 ), so t ∼ o(!−1=2 ) since y = A cos !t. Therefore, for small values of !, there exists a constant M such that (1 − 3x2 ) dt ¿ M!−1=2 : (11) L! ;|x|6x0 ; z¿0
On the other hand, 0¡
L! ; x0 6x6xup ; z¿0
=!−1
A
A0
(3x2 − 1) dt ¡
L! ;2x0 6x6xup ; z¿0
3x2 − 1 dy ! A2 − y 2
3x2 − 1 A0 dy 6 !−1 (12x02 − 2x0 ) arcsin : 2 2 A ! A −y
(12)
Therefore, (12), (13) and (10) implies inequality (8). Thus the uniqueness of limit cycle on E is proved.
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Acknowledgements This work was supported by Natural Sciences Founds of China (NSF10271009) and we thank Hong Gu and Zhu-Yu Wang for showing us some numerical results from their Master theses. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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