Synchronization dynamics of two different dynamical systems

Chaos, Solitons & Fractals 44 (2011) 362–380

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Synchronization dynamics of two different dynamical systems Albert C.J. Luo ⇑, Fuhong Min Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

a r t i c l e

i n f o

Article history: Received 14 September 2010 Accepted 14 December 2010 Available online 21 April 2011

a b s t r a c t In this paper, synchronization dynamics of two different dynamical systems is investigated through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) are presented. Using such a synchronization theory, the synchronization of a controlled pendulum with the Duffing oscillator is systematically discussed as a sampled problem, and the corresponding analytical conditions for the synchronization are presented. The synchronization parameter study is carried out for a better understanding of synchronization characteristics of the controlled pendulum and the Duffing oscillator. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are presented to illustrate the analytical conditions. The synchronization of the Duffing oscillator and pendulum are investigated in order to show the usefulness and efficiency of the methodology in this paper. The synchronization invariant domain is obtained. The technique presented in this paper should have a wide spectrum of applications in engineering. For example, this technique can be applied to the maneuvering target tracking, and the others. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction One has been interested in the synchronization in dynamical systems since the 17th century. In 1673, Huygens [1] described the synchronization of two pendulum clocks with weak synchronization, which is about two modal shapes of vibration. If the coupled two pendulums have small oscillations with the same initial conditions or the zero initial phase difference, the two pendulums will be synchronized. If the initial phase difference is 180°, the anti-synchronization of two pendulums can be observed. For a general case, the motion of the two pendulums will be combined by the synchronization and anti-synchronization modes of vibration. The recent progress on the Huygens’ synchronization was presented in Leonov et al. [2] in 2010. In such an edited book, mathematical methods and synchronization of dynamical systems relative to Huygens coupling were discussed. Based on discontinuous

⇑ Corresponding author. Tel.: +1 618 650 5389; fax: +1 618 650 2555. E-mail address: [email protected] (A.C.J. Luo). 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2010.12.011

dynamical systems theory, Kuznetsov and Leonov [3] presented Lyapunov quantities to determine the limit cycles of two-dimensional dynamical systems. Four basic synchronizations of two or more dynamical systems are: (i) identical or complete synchronization, (ii) generalized synchronization, (iii) phase synchronization, and (iv) anticipated and lag synchronization and amplitude envelope synchronization. For any synchronization of two or more system, at least one constraint exists for synchronicity, and such synchronization may experience the asymptotic stability characteristics. For instance, in 2010, Kuznetsov et al. [4] used Lyapunov method to investigate the phase synchronization of two metronomes. Once the two or more systems form a synchronization state at one or more specific constraints, such a state should be stable (e.g. [2,5,6]). In recent decades, one extended the synchronization concept from the traditional points of view. In 1989, the synchronization of two dynamical systems with common signals was studied from the sub-Lyapunov exponents in Pecora and Carroll [7]. The common signals were adopted as constraints for such two dynamical systems. Carroll and Pecora [8] used the synchronized circuits to simulate

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

such synchronization of chaotic responses. Such efforts caused a lot of attention on control methods and schemes of the synchronization of two dynamical systems with constraints. In 1992, Pyragas [9] presented two methods for chaos control with a small time continuous perturbation to obtain a synchronization of two chaotic dynamical systems. In 1995, a generalized synchronization of chaos in directionally coupled chaotic systems was discussed in Rulkov et al. [10]. To construct chaotically synchronized systems, Kocarev and Parlitz [11] decomposed the given systems into the active and passive dynamical systems. In 1996, Pyragas [12] studied the weak and strong synchronizations of chaos by the coupling strength of two dynamical systems. In 1997, an adaptive synchronization of chaos was used for secure communication in Boccaletti et al. [13], and Abarbanel et al. [14] used a small force to control a dynamical system to the given orbits. In 1998, Pyragas [15] discussed the genealized synchronization of chaos. In 1999, Yang and Chua [16] employed linear transformations to discuss the generalized synchronization of two dynamical systems. In 2002, Boccaletti et al. [17] gave a review on the synchronization of chaotic systems and clarified definitions and concepts of dynamical system synchronization. In 2004, Campos and Urias [18] mathematically described multi-modal synchronization with chaos, and introduced a multi-valued, synchronized function. In 2005, Chen [19] investigated the synchronization of two different chaotic systems. Such synchronization is based on the error dynamics of the slave and master systems. The active control functions were used to remove nonlinear terms, and the Lyapunov function was used to determine the stability of the synchronization. Lu and Cao [20] used the similar technique of Chen [19] to discuss the adaptive complete synchronization of two identical or different chaotic systems with unknown parameters. The Lyapunov function is given by a nonlinear function of errors of the two dynamical systems. Fujiwara and Kurths [21] discussed the spectral universality of phase synchronization in non-identical oscillator networks through the eigenvalue analysis of the linearized systems. From the aforementioned literature survey, one like to construct the error dynamical systems and the Lyapunov method is used to determine the corresponding asymptotic stability. In 2009, Luo [22] presented an alternative way for the synchronization of two different dynamical systems, and a theory for synchronization of dynamical systems with specific constraints was developed through the theory of discontinuous dynamical systems (e.g. [22–24]). In such a theory, the G-functions were introduced to determine the switchability of a flow from one domain to another in discontinuous dynamical systems. When two different dynamical systems cannot produce the error dynamics or the slave and master systems are dependent on the time, the synchronization of the slave and master systems is not asymptotic, and the Lyapunov functions cannot be used. In recent years, one considered a sliding mode control to investigate the synchronization of two different dynamical systems. In 2004, Bowsong et al. [25] presented controlled synchronization of chaotic systems with uncertainties through a sliding mode control, and a sliding mode law was introduced by an error function. In 2007, Chen et al.

363

[26] used the sliding mode control to investigate the synchronization of Rossler systems, and the simple switching surface with errors was introduced. Based on the switching surface, the Lyapunov method was used to determine the asymptotic stability. In 2009, Roopaei and Jahromi [27] used the adaptive sliding mode approach to investigate the synchronization of two similar nonlinear oscillators. Since the sliding mode approach did not give necessary and sufficient conditions, and the stability should be determined by the Lyapunov method. Thus, only the asymptotic synchronization can be obtained. In addition, the theory for discontinuous dynamic systems based on Filippov’s theory can be referred to Filippov [28,29] and Yakubovich et al. [30]. Such a theory is based on differential inclusion, which has difficulty to be applied to the synchronization of two dynamical systems. Once such a theory of discontinuous dynamical systems is used to the synchronization, the approximate Lyapunov quantities may be adopted. However, the theory of discontinuous dynamical systems in Luo [22–24] provides the necessary and sufficient conditions for synchronization, de-synchronization and penetration (instantaneous synchronization) of the two dynamical systems. Therefore, non-synchronization, partial and full synchronizations of two dynamical systems can be easily obtained. In 2010, Luo and Min [31] used the theory of discontinuous dynamical systems in Luo [23,24] to discuss the synchronization of the periodically forced, damped Duffing oscillator with the periodically excited pendulum. The periodically excited pendulum possesses librational motions, rotational motions, and separatrix chaotic motions. The rotational and separatrix motions may have infinite displacements. However, the motion of the Duffing oscillator is finite. Thus, the controlled Duffing oscillator can only synchronize with the librational chaotic motions in the periodically excited pendulum. If the Duffing oscillator is considered as a master system, the corresponding synchronization mechanism was investigated in Luo and Min [32,33]. From such an investigation, a general theory for dynamical systems synchronization will be presented herein, and the synchronization of the Duffing oscillator and pendulum will be investigated in order to show the usefulness and efficiency of the methodology. In this paper, the synchronization theory of two different dynamical systems will be presented through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) will be presented. Furthermore, the synchronization of the controlled pendulum with the Duffing oscillator will be discussed as an example, and the analytical conditions of synchronizations will be presented, and the parameter study for such synchronizations will be carried out. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions will be illustrated.

2. Synchronization theory In this section, the synchronization theory of two different dynamical systems will be presented, and the corre-

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sponding necessary and sufficient conditions for synchronization, de-synchronization, and instantaneous synchronization will be obtained. 2.1. Dynamical system synchronizations As in Luo [22], consider two dynamic systems as

y_ ¼ Fðy; t; pÞ 2 Rn

ð1Þ

and

x_ ¼ F ðx; t; qÞ 2 Rn

ð2Þ

where F ¼ ðF 1 ; F 2 ; . . . ; F n ÞT ; y ¼ ðy1 ; y2 ; . . . ; yn ÞT and p ¼ ðp1 ; p2 ; . . . ; pk ÞT ; F ¼ ðF 1 ; F 2 ; . . . ; F n ÞT ; x ¼ ðx1 ; x2 ; . . . ; xn ÞT and q ¼ ðq1 ; q2 ; . . . ; qk ÞT . The vector functions F and F can be time-dependent or time-independent. Consider a time interval I12  ðt1 ; t2 Þ 2 R and domains U y # Rn and U x # Rn . For initial conditions (t0, x0) 2 I12  Ux and (t0, y0) 2 I12  Uy, the corresponding flows of the two systems are x(t) = U(t, x0, t0, q) and y(t) = U(t, y0, t0, p) for (t, x) 2 I12  Ux and (t, y) 2 I12  Uy with p 2 U p # Rk and q 2 U q # Rk . The semi-group properties of two flows hold (i.e., U(t + s, x0, t0, q) = U(t,U(s, x0, t0, q), s, q) with x(t0) = U(t0, x0, t0, q), and U(t + s, y0, t0, p) = U(t, U(s, y0, t0, p), s, p) with y(t0) = U(t0, y0, t0, p)). To investigate the synchronization of the two systems in Eqs. (1) and (2), the slave and master systems are defined as follows: Definition 1. A system in Eq. (2) is called a master system if its flow y(t) is independent. A system in Eq. (1) is called a slave system of the master system if its flow x(t) is ~ ðtÞ of the master system. constrained by the flow x From the foregoing definition, a slave system is constrained by a master system via specific conditions. Such a phenomenon is called the synchronization of the slave and master systems under such specific conditions. To make this concept clear, a formal definition is given. Definition 2. If a flow y(t) of the slave system in Eq. (1) is constrained with a flow x(t) of a master system in Eq. (2) through the following function:

uðxðtÞ; yðtÞ; t; kÞ ¼ 0; k 2 Rn0 ;

ð3Þ

for time t 2 ½tm1 ; tm2 , then the slave system is said to be synchronized with the master system in the sense of Eq. (3) for time t 2 ½tm1 ; tm2 , denoted by the (n:n)-dimensional synchronization of the slave and master systems in the sense of Eq. (3). If t m2 ! 1, the slave system is said to be absolutely synchronized with the master system in the sense of Eq. (3) for time t 2 ½tm1 ; 1Þ. Two special cases are given as follows: (i) For n = n, such a synchronization is called an equidimensional system synchronization in the sense of Eq. (3) for t 2 ½t m1 ; t m2 . (ii) For n = n, such a synchronization is called an absolute, equi-dimensional system synchronization in the sense of Eq. (3) for t 2 ½t m1 ; 1Þ.

If n – n, the (n: n)-synchronization is called a nonequi-dimensional system synchronization. Under a certain rule in Eq. (3), it is interesting that a slave system can follow another completely different master system to synchronize. From the proceeding definition, it can be seen that the slave system is synchronized with the master system under a constraint condition. In fact, constraint conditions for such a synchronization phenomenon can be more than one. In other words, the slave system can be synchronized with the master system under multiple constraints. Thus, the definition for the synchronization of a slave system with a master system under multiple constraints is given as follows: Definition 3. An n-dimensional slave system in Eq. (1) is called to be synchronized with an n-dimensional master system in Eq. (2) of the (n: n; l)-type (or an (n: n; l)-synchronization) if there are l-linearly independent functions u(j)(x(t), y(t), t,kj) (j 2 L and L ¼ f1; 2; . . . ; lgÞ to make two flows x(t) and y(t) of the master and slave systems satisfy





uðjÞ xðtÞ; yðtÞ; t; kj ¼ 0 for kj 2 Rnj and j 2 L

ð4Þ

for time t 2 ½tm1 ; tm2 . If t m2 ! 1, the synchronization of the slave and master systems is called an absolute (n: n; l)-synchronization in the sense of Eq. (4) for time t 2 ½t m1 ; 1Þ. The six special cases are given as follows: (i) For l = n, the synchronization of the slave and master systems is called to a complete, (n: n; n)-synchronization in the sense of Eq. (4) for t 2 ½tm1 ; tm2 . (ii) For l = n and tm2 ! 1, the synchronization of the slave and master systems is called an absolute, complete (n: n; n)-synchronization in the sense of Eq. (4) for t 2 ½tm1 ; 1Þ. (iii) If n = n > l, the synchronization of the slave and master systems is called an equi-dimensional (n: n; l)synchronization in the sense of Eq. (4) for t 2 ½tm1 ; tm2 . (iv) If n = n > l and t m1 ! 1, the synchronization of the slave and master systems is called an absolute, equi-dimensional, (n:n;l)-synchronization in the sense of Eq. (4) for t 2 ½t m1 ; 1Þ. (v) If n = n = l, the synchronization of the slave and master systems is called a complete, equi-dimensional (n: n; n)-synchronization (simply called a synchronization) in the sense of Eq. (4) for t 2 ½t m1 ; tm2 . (vi) If n = n = l and t m1 ! 1, the synchronization of the slave and master systems is called an absolute, complete, equi-dimensional (n: n; n)-synchronization (simply called an absolute synchronization) in the sense of Eq. (4) for t 2 ½t m1 ; 1Þ. 2.2. Synchronization conditions The synchronization between two dynamical systems can be investigated in the vicinity of synchronization boundary. Since the master system is independent of the synchronization constraint, only the slave system should be controlled to satisfy the synchronization constraints. Thus, the controlled slave system is discontinuous under

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the jth- synchronization constraint. The corresponding domains and boundary for the controlled slave system can be defined by

 )  u ðxðtÞ; yð1;jÞ ðtÞ; t; kj Þ > 0;  j ; ¼ yð1;jÞ   uj is C rj  continuous ðr j P 1Þ  ( )  u ðxðtÞ; yð2;jÞ ðtÞ; t; kj Þ < 0;  j ; ¼ yð2;jÞ   uj is C rj  continuous ðr j P 1Þ (

Xð1;jÞ Xð2;jÞ

€zðaj ;jÞ ¼

ð15Þ The combination of Eqs. (14) and (15) gives a dynamical system in phase space of ðz; z_ Þ, i.e., for j 2 L

ð5Þ

 ðaj ;jÞ

 y_ ðaj ;jÞ ¼ F yðaj ;jÞ ; t; pðaj ;jÞ on Xðaj ;jÞ ;   y_ ð0;jÞ ¼ Fð0;jÞ yð0;jÞ ; t; kj on @ Xð12;jÞ :

zðaj ;jÞ ¼ uj xðtÞ; yðaj ;jÞ ðtÞ; t; kj

ð7Þ ð8Þ

for j 2 L:

ð9Þ

On the boundary @ Xðaj bj ;jÞ ,

z

ð0;jÞ



¼ uj xðtÞ; y

ð0;jÞ



ðtÞ; t; kj ¼ 0 for j 2 L:

ð10Þ

If the two systems do not synchronize each other, the new variables (zj – 0, j = 1, 2, . . ., l) will change with time t. The corresponding time-change rate is given by

  z_ ðaj ;jÞ ¼ Duj x; yðaj ;jÞ ; t; kj @ uj @ uj @ uj x_ þ ða ;jÞ y_ ðaj ;jÞ þ @x @t @y j

¼

n n X X @ uj @ uj ðaj ;jÞ @ uj þ x_ p þ y_ : ðaj ;jÞ q @x @t p @y p¼1 q¼1 q

z_ ðaj ;jÞ ¼

p¼1



@ uj F p ðx; t; qÞ þ @xp

 ða ;jÞ  F q j yðaj ;jÞ t; pðaj ;jÞ

nuj ¼

@yðaj ;jÞ

¼

9   z_ ðaj ;jÞ ¼ gðaj ;jÞ zðaj ;jÞ ; t for j 2 L; > = : x_ ¼ F ðx; t; qÞ 2 Rn ; >   ; y_ ðaj ;jÞ ¼ F ðaj ;jÞ yðaj ;jÞ ; t; pðaj ;jÞ 2 Rn :

ð17Þ

For a better understanding of such a discontinuous dynamical system, the boundary and domains in phase space are defined as

@ Nð12;jÞ ¼ Nð1;jÞ \ Nð2;jÞ     ¼ zð0;jÞ ; z_ ð0;jÞ wj zð0;jÞ ; z_ ð0;jÞ ¼ zð0;jÞ ¼ 0  R;

ð18Þ

and

Nð1;jÞ ¼

 

 zð1;jÞ ; z_ ð1;jÞ zð1;jÞ > 0  R2 ;   zð1;jÞ ; z_ ð1;jÞ zð2;jÞ < 0  R2 :

ð19Þ

ðaj ;jÞ q¼1 @yq

@ uj þ : @t

@ uj

@ uj @ uj ; ða ;jÞ ; . . . ; ða ;jÞ ðaj ;jÞ j @y1 @y2 @yn j

ð12Þ

!T

ð13Þ :

Using Eqs. (13) and (12) becomes

z_ ðaj ;jÞ ¼ nuj  F ðx; t; qÞ þ nuj  Fðaj ;jÞ ðyðaj ;jÞ ; t; pðaj ;jÞ Þ þ

s

  d zð0;jÞ ¼ Ds uj xðtÞ; yð0;jÞ ðtÞ; t; kj ¼ 0 for s ¼ 1; 2; . . . s dt

ð20Þ

Thus, the synchronization boundary is determined by

@ uj

  @ uj @ uj @ uj @ uj T ; ;...; ; ¼ @x @x1 @x2 @xn @ uj

ða ;jÞ ða ;jÞ T T ; z_ ðaj ;jÞ . Letting gðaj ;jÞ ¼ g 1 j ; g 2 j ,

gives

ð11Þ

Two new normal vectors are defined as

nuj ¼

ðaj ;jÞ

uj(x(t), y(0,j)(t), t,kj) = 0 on the synchronization boundary

Substitution of Eqs. (2) and (7) into Eq. (11) yields n X



where z ¼ z one obtains

Nð2;jÞ ¼

¼

n X

ð16Þ

ðaj ;jÞ

For simplicity, a new variable is introduced in domain Xðaj ;jÞ



  @u  nuj  F ðx; t; pÞ þ nuj  Fðaj ;jÞ yðaj ;jÞ ; t; qðaj ;jÞ þ @t j ;   ðaj ;jÞ  ða ;jÞ  ð a ;jÞ D > ¼ g2 z j ; t  Dt g 1 j zðaj ;jÞ ; t > > h i > > @ uj > ðaj ;jÞ ðaj ;jÞ ðaj ;jÞ D ¼ Dt nuj  F ðx; t; pÞ þ nuj  F ðy ; t; q Þ þ @t : ;

ð6Þ

From the domains and boundary, the corresponding equations for the controlled slaved system become for j 2 L

9 > > > > > > =

ða ;jÞ z_ ðaj ;jÞ ¼ g 1 j ðzðaj ;jÞ ; tÞ

€zðaj ;jÞ

@ Xð12;jÞ ¼ Xð1;jÞ \ Xð2;jÞ  ( )  u ðxðtÞ; yð0;jÞ ðtÞ; t; kj Þ ¼ 0  j ¼ yð0;jÞ  :  uj is C rj  continuous ðr j P 1Þ



    @ uj D : nuj  F ðx; t; qÞ þ nuj  Fðaj ;jÞ yðaj ;jÞ ; t; pðaj ;jÞ þ Dt @t

@ uj : @t ð14Þ

If the vector fields in different domains Xðaj ;jÞ ðaj ¼ 1; 2Þ are distinguishing, z_ ðaj ;jÞ is discontinuous. Similarly, for each domain Xðaj ;jÞ , one obtains

zð0;jÞ ¼ 0; z_ ð0;jÞ ¼ 0 for j 2 L; x_ ¼ F ðx; t; qÞ 2 Rn ;   y_ ð0;jÞ ¼ Fð0;jÞ yð0;jÞ ; t; kj 2 Rn :

ð22Þ

  The domains and boundary in phase space of zðjÞ ; z_ ðjÞ are sketched in Fig. 1(a) and the location for switching may not be continuous (i.e., zðaj ;jÞ – zðbj ;jÞ – zð0;jÞ ¼ 0Þ because the vector fields of the resultant system are discontinuous (or z_ ðaj ;jÞ – z_ ðbj ;jÞ – z_ ð0;jÞ ¼ 0Þ, but the boundary in such a phase space is independent of time. However, the boundaries and domains in phase space of the controlled slave system in Eqs. (8) and (9) are shown in Fig. 1. The boundary varying with time is presented, but switching points for a flow are continuous. However, such flows will be controlled by the vector fields g(1,j)(z(1,j), t) and g(2,j)(z(2,j), t). The dynamical systems in phase space ðz; z_ Þ are:

  9 z_ ðKj ;jÞ ¼ gðKj ;jÞ zðKj ;jÞ ; t for j 2 L; Kj ¼ 0; aj > = x_ ¼ F ðx; t; qÞ 2 Rn ; ; >   ; _yðKj ;jÞ ¼ FðKj ;jÞ yðKj ;jÞ ; t; pðKj ;jÞ 2 Rn

ð23Þ

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Fig. 1. A partition of phase space: (a) ðz; z_ Þ for the jth-synchronization boundary, and (b) the controlled slave system. Two dashed lines (curves) are infinitesimally close to the boundary with the dotted line (curve).

where

  ða ;jÞ   ða ;jÞ   T 9 > > gðaj ;jÞ zðaj ;jÞ ; t ¼ g 1 j zðaj ;jÞ ; t ; g 2 j zðaj ;jÞ ; t > > > > > > in Naj ðaj 2 f1; 2gÞ; =  

     : ðaj ;jÞ ðaj ;jÞ ðaj ;jÞ ðbj ;jÞ ðbj ;jÞ ð0;jÞ z ;t 2 g z ;t ; g z ;t g > > > > > on @ Nð12;jÞ for non-stick; > > >   ; gð0;jÞ zðaj ;jÞ ; t ¼ ð0; 0ÞT on @ Nð12;jÞ for stick:

ð24Þ

ð12;jÞ

ð27Þ

where

The normal vector of oN(12,j) is computed from Eq. (18), i.e., T

n@ Nð12;jÞ ¼ ð1; 0Þ

T

and Dn@ Nð12;jÞ ¼ ð0; 0Þ ;

ð25Þ

where D() = D()/Dt. From Luo [22,23], the corresponding two G-functions are computed by

   ð0;aj Þ  ða ;jÞ  ða ;jÞ  G@ Nð12;jÞ z j ; t ¼ n@Nð12;jÞ  gðaj ;jÞ zðaj ;jÞ ; t ¼ g 1 j zðaj ;jÞ ; t ;    ð1;aj Þ  ða ;jÞ  ða ;jÞ  G@ Nð12;jÞ z j ; t ¼ n@Nð12;jÞ  Dgðaj ;jÞ zðaj ;jÞ ; t ¼ g 2 j zðaj ;jÞ ; t : ð26Þ With G-functions, the sufficient and necessary conditions

ða ;jÞ ð0;jÞ ð0;jÞ for a passable flow at zm ; t m with zm j ¼ zm ¼ zm for the boundary oN(12,j) are given by Luo [34], i.e.,

9



ð0;1Þ ð1;jÞ ð1;jÞ ð1;jÞ = G@ Nð12;jÞ zm ; tm ¼ g 1 zm ; tm < 0; >



for Nð1;jÞ ! Nð2;jÞ ð0;2Þ ð2;jÞ ð2;jÞ ð2;jÞ ; zm ; tmþ < 0 > G@ Nð12;jÞ zm ; tmþ ¼ g 1 9



ð0;1Þ ð1;jÞ ð1;jÞ ð1;jÞ = G@ Nð12;jÞ zm ; tmþ ¼ g 1 zm ; tmþ > 0; >



for Nð2;jÞ ! Nð1;jÞ ; ð0;2Þ ð2;jÞ ð2;jÞ ð2;jÞ ; zm ; tm ¼ g 1 zm ; tm > 0 > G@ N

ða ;jÞ

g1 j

ða ;jÞ

zm j ; t m ¼ nuj  F ðxm ; tm ; qÞ þ nuj ða ;jÞ

@u j  Fðaj ;jÞ ym j ; tm ; pðaj ;jÞ þ : @t

ð28Þ

The foregoing condition gives the sufficient and necessary conditions for the controlled slave system synchronizing with the master system under the j th-synchronization condition, and the current states of the controlled slave system will be switched from one domain to the other through such a synchronization condition. Such a flow to the synchronization boundary is called an instantaneous penetration synchronization between two dynamical systems. The sufficient and necessary conditions for a stick flow (or sink flow, or sliding flow) on the synchronization boundary oN(12,j) are obtained from Luo [22,23], i.e.,

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380





9 ð0;1Þ ð1;jÞ ð1;jÞ ð1;jÞ G@Nð12;jÞ zm ; t m ¼ g 1 zm ; tm < 0; > =

on @ Nð12;jÞ



> ð0;2Þ ð2;jÞ ð2;jÞ ð2;jÞ zm ; tm > 0 ; G@Nð12;jÞ zm ; t m ¼ g 1 ð29Þ

From the foregoing condition, the controlled slave systems will stick with the master system under the jthsynchronization condition. This phenomenon is called the synchronization of the controlled slave system with the master system under the jth-synchronization condition. Similarly, the sufficient and necessary conditions for a source flow on the boundary oN(12,j) are given in Luo [22,23], i.e.,

9



ð1;jÞ ð1;jÞ ð1;jÞ zm ; t mþ ¼ g 1 zm ; tmþ > 0; > = on @ Nð12;jÞ :



> ð0;2Þ ð2;jÞ ð2;jÞ ð2;jÞ zm ; tmþ < 0 ; G@Nð12;jÞ zm ; t mþ ¼ g 1

9 ða ;jÞ

ða ;jÞ

ð0;a Þ ða ;jÞ ð1Þaj G@ Nðaj b ;jÞ zm j ; tmþ ¼ ð1Þaj g 1 j zm j ; t mþ < 0; > > > > i j > ðb ;jÞ

ðb ;jÞ

= ð0;bj Þ ðbj ;jÞ j j zm ; tm ¼ 0; G@ Nða b ;jÞ zm ; tm ¼ g 1 i j > > ðb ;jÞ

ðb ;jÞ

> > ð1;b Þ ðb ;jÞ ; ð1Þbj G@ Nðaj b ;jÞ zm j ; tm ¼ ð1Þbj g 2 j zm j ; t m < 0 > i j

ð33Þ

for de-synchronization appearance pertaining to the instantaneous synchronization and, 9 ða ;jÞ

ða ;jÞ

ð0;a Þ ða ;jÞ ð1Þaj G@ Nðaj b ;jÞ zm j ; tmþ ¼ ð1Þaj g 1 j zm j ; t mþ < 0; > > > > i j > ðb ;jÞ

ðb ;jÞ

= ð0;bj Þ ðbj ;jÞ j j zm ; tm ¼ 0; G@ Nða b ;jÞ zm ; tm ¼ g 1 i j > > ðb ;jÞ

ðb ;jÞ

> > ð1;b Þ ðb ;jÞ ; ð1Þbj G@ Nðaj b ;jÞ zm j ; tm ¼ ð1Þbj g 2 j zm j ; t m < 0: > i j

ð34Þ

ð0;1Þ G@Nð12;jÞ

ð30Þ For this case, the controlled slave system

will not synchroð0;jÞ nize with the master system at zm ; t m for the synchronization boundary oN(12,j) relative to the jth-synchronization condition. This phenomenon is called the desynchronization of the controlled slave system with the master system under the jth-synchronization condition. The appearance and disappearance of three synchronization states of the two dynamical systems to the jthsynchronization condition in Eq. (4) can be determined from Luo [19,20], which are for the switching bifurcation of three states of synchronizations between the two dynamical systems. (i) The sufficient and necessary conditions of synchronization appearance from the instantaneous penetration synchronization are ða ;jÞ

ða ;jÞ

9 ð0;aj Þ ða ;jÞ ð1Þaj G@ Nð12;jÞ zm j ; tm ¼ ð1Þaj g 1 j zm j ; t m > 0; > > > > > =



ð0;bj Þ ðbj ;jÞ ðbj ;jÞ ðbj ;jÞ G@ Nð12;jÞ zm ; t m ¼ g 1 zm ; tm ¼ 0; > > > ðb ;jÞ

ðb ;jÞ

> > bj ð1;bj Þ bj ðbj ;jÞ j j ð1Þ G@Nð12;jÞ zm ; t m ¼ ð1Þ g 2 zm ; tm < 0: ; ð31Þ

The sufficient and necessary conditions for synchronization vanishing from the jth-synchronization boundary are 9



ð0;a Þ ða ;jÞ ða ;jÞ ða ;jÞ ð1Þaj G@ Nðaj b ;jÞ zm j ; tm ¼ ð1Þaj g 1 j zm j ; tm > 0; > > > > i j > >



= ð0;bj Þ ðbj ;jÞ ðbj ;jÞ ðbj ;jÞ G@ Nða b ;jÞ zm ; t m ¼ g 1 zm ; tm ¼ 0; i j > > >



> > ðbj ;jÞ ðbj ;jÞ bj ð1;bj Þ bj ðbj ;jÞ ; ð1Þ G@N zm ; t m ¼ ð1Þ g 2 zm ; tm < 0: > ðai bj ;jÞ

ð32Þ

The appearance and vanishing conditions for the synchronization relative to the instantaneous synchronization in Eq. (31) are the vanishing and appearance conditions for the instantaneous penetration synchronization relative to the synchronization, respectively. (ii) From Luo [22–24], the sufficient and necessary conditions are

367

for the vanishing of the de-synchronization pertaining to the instantaneous synchronization. (iii) From Luo [22–24], the sufficient and necessary switching conditions between the synchronization and de-synchronization of the controlled slave and master systems on the jth-synchronization boundary are 9 ða ;jÞ

ða ;jÞ

ð0;a Þ ða ;jÞ > G@ Nðaj b ;jÞ zm j ; t m ¼ g 1 j zm j ; tm ¼ 0 > > > i j >



> > ðaj ;jÞ > aj ð1;aj Þ aj ðaj ;jÞ ðaj ;jÞ zm ; t m < 0; > ð1Þ G@ Nða b ;jÞ zm ; tm ¼ ð1Þ g 2 = i j



ð0;bj Þ ðbj ;jÞ ðbj ;jÞ ðbj ;jÞ > > zm ; tm ¼ 0; G@ Nða b ;jÞ zm ; tm ¼ g 1 > > i j > >



> > ðbj ;jÞ ðbj ;jÞ bj ð1;bj Þ bj ðbj ;jÞ ; zm ; tm ¼ ð1Þ g 2 zm ; t m < 0: > ð1Þ G@ N ðai bj ;jÞ

ð35Þ

Similarly, the sufficient and necessary switching conditions between two instantaneous penetration synchronizations at the jth-synchronization boundary for aj – bj ða ;jÞ

ða ;jÞ

9 ð0;aj Þ ða ;jÞ G@Nð12;jÞ zm j ; tm ¼ g 1 j zm j ; t m ¼ 0 for aj 2 f1; 2g; > > > > > ða ;jÞ



> > aj ð1;aj Þ aj ðaj ;jÞ ðaj ;jÞ j > = zm ; t m < 0; ð1Þ G@ Nð12;jÞ zm ; tm ¼ ð1Þ g 2



ð0;bj Þ ðbj ;jÞ ðbj ;jÞ ðbj ;jÞ > zm ; tm ¼ 0 for bj 2 f1; 2g; > G@Nð12;jÞ zm ; tm ¼ g 1 > > > >

ðb ;jÞ

> ðbj ;jÞ > bj ð1;bj Þ bj ðbj ;jÞ j ; ð1Þ G@ Nð12;jÞ zm ; t m ¼ ð1Þ g 2 zm ; tm < 0: ð36Þ

A flow of the controlled slave system, tangential to the synchronization boundary oN(12,j) is another instantaneous synchronization (or tangential synchronization), and the corresponding sufficient and necessary conditions are 9 ða ;jÞ

ða ;jÞ

ð0;aj Þ ða ;jÞ = G@Nð12;jÞ zm j ; tm ¼ g 1 j zm j ; t m ¼ 0 for aj 2 f1; 2g; > ða ;jÞ



: aj ð1;aj Þ aj ðaj ;jÞ ðaj ;jÞ j > ; zm ; t m < 0: ð1Þ G@ Nð12;jÞ zm ; tm ¼ ð1Þ g 2 ð37Þ

3. An application This section will apply the synchronization theory to investigate the synchronization of two totally different dynamical systems.

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3.1. Master and slave systems Consider a periodically forced, damped Duffing oscillator with a twin well potential as a master system, i.e.,

€x þ d1 x_  a1 x þ a2 x3 ¼ A0 cos xt:

ð38Þ

A periodically driven pendulum is considered as a slave system, i.e.,

€ þ a0 sin y ¼ Q 0 cos Xt: y

ð39Þ

Since the Duffing oscillator has periodic and chaotic motions, the control laws should be exerted to enforce the controlled pendulum to synchronize with motions in the Duffing oscillator. Before the control laws are exerted, the state variables are introduced for the Duffing oscillator and pendulum as

x ¼ ðx1 ; x2 ÞT

and y ¼ ðy1 ; y2 ÞT

ð39Þ

and the vector fields are defined as

F ðx; tÞ ¼ ðx2 ; F ðx; tÞÞT

and Fðy; tÞ ¼ ðy2 ; Fðy; tÞÞT :

ð40Þ

So the Duffing oscillator is described by

x_ ¼ F ðx; tÞ;

ð41Þ

and F ðx; tÞ

¼ d1 x2 þ a1 x1  a2 x31 þ A0 cos xt:

ð42Þ

The equation of motion for the pendulum without control is

y_ ¼ Fðy; tÞ;

ð43Þ

where

y2  y_ 1

for y1 > x1

ð47Þ

and y2 > x2 ;

F 1 ðy; tÞ ¼ y2  k1 F 2 ðy; tÞ ¼ a0 sin y1 þ Q 0 cos Xt þ k2 for y1 > x1



 ð48Þ

and y2 < x2 :

F 1 ðy; tÞ ¼ y2 þ k1 F 2 ðy; tÞ ¼ a0 sin y1 þ Q 0 cos Xt þ k2

 ð49Þ

for y1 < x1 and y2 < x2 : F 1 ðy; tÞ ¼ y2 þ k1 F 2 ðy; tÞ ¼ a0 sin y1 þ Q 0 cos Xt  k2

 ð50Þ

for y1 < x1 and y2 > x2 : From the above regions, there are two constraint surfaces, and four domains Xa (a = 1, 2, 3, 4) in phase space of the controlled pendulum are defined as

X1 X2 X3 X4

¼ fðy1 ; y2 Þjy1  x1 ðtÞ > 0; y2  x2 ðtÞ > 0g; ¼ fðy1 ; y2 Þjy1  x1 ðtÞ > 0; y2  x2 ðtÞ < 0g; ¼ fðy1 ; y2 Þjy1  x1 ðtÞ < 0; y2  x2 ðtÞ < 0g;

ð51Þ

¼ fðy1 ; y2 Þjy1  x1 ðtÞ < 0; y2  x2 ðtÞ > 0g:

The boundaries oXab (a, b = 1, 2, 3, 4; a – b) of the four domains are

where

x2  x_ 1

F 1 ðy; tÞ ¼ y2  k1 F 2 ðy; tÞ ¼ a0 sin y1 þ Q 0 cos Xt  k2

and Fðy; tÞ ¼ a0 sin y1 þ Q 0 cos Xt:

ð44Þ

With a control law, equations of motion for the controlled pendulum become

y_ ¼ Fðy; tÞ  uðx; y; tÞ

ð45Þ

where

uðx; y; tÞ ¼ ðu1 ; u2 ÞT ; u1 ¼ k1 sgnðy1  x1 Þ and u2 ¼ k2 sgnðy2  x2 Þ;

ð46Þ

@ X12 ¼ X1 \ X2 ¼ fðy1 ; y2 Þjy2  x2 ðtÞ ¼ 0; y1  x1 ðtÞ > 0g; @ X23 ¼ X2 \ X3 ¼ fðy1 ; y2 Þjy1  x1 ðtÞ ¼ 0; y2  x2 ðtÞ < 0g; @ X34 ¼ X3 \ X4 ¼ fðy1 ; y2 Þjy2  x2 ðtÞ ¼ 0; y1  x1 ðtÞ < 0g; @ X14 ¼ X1 \ X4 ¼ fðy1 ; y2 Þjy1  x1 ðtÞ ¼ 0; y2  x2 ðtÞ > 0g: ð52Þ Notice that Xa is the closure of the domain Xa (a = 1, 2, 3, 4). The sub-domains and boundaries for the controlled pendulum are sketched, and for simplicity, the two boundaries in phase space can be separated, as shown in Fig. 2. The corresponding domains and boundaries are labeled, and the dashed curves give the boundaries. The intersection point of the two boundaries is labeled by a filled circular symbol. From the afore-defined domains, the equation of motion for the controlled pendulum in the a-domain is

  y_ ðaÞ ¼ FðaÞ yðaÞ ; t ; 3.2. Discontinuous description In despite of periodic or chaotic motions in the Duffing oscillator, the controlled pendulum under a control will follow the master system (i.e., the Duffing Oscillator) to be synchronized. Consider the master system to be independent, and only the slave system is controlled. Under the control, the slave system will become discontinuous. The two control laws make the controlled pendulum in Eq. (45) possess four regions, and the corresponding vector fields are defined by F(y, t) = (F1(y, t), F2(y, t))T. The components in such vector fields for the controlled pendulum are given as follows:

ð53Þ

where ðaÞ

ðaÞ

FðaÞ ðyðaÞ ; tÞ ¼ ðF 1 ; F 2 ÞT ; ða Þ F 1 ðyðaÞ ; tÞ ða Þ F 1 ðyðaÞ ; tÞ ða Þ F 2 ðyðaÞ ; tÞ ða Þ F 2 ðyðaÞ ; tÞ

¼ ¼ ¼ ¼

ðaÞ y2 ðaÞ y2

 k1

for a ¼ 1; 2;

þ k1

for a ¼ 3; 4;

ðaÞ a0 sin y1 ðaÞ a0 sin y1

þ Q 0 cos Xt  k2

for a ¼ 1; 4;

þ Q 0 cos Xt þ k2

for a ¼ 2; 3: ð54Þ

From Luo [34], the dynamical systems on the boundaries @ Xab ¼ Xa \ Xb are

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

369

Fig. 2. Two boundaries in the absolute coordinates: (a) velocity, and (b) displacement.

  y_ ðabÞ ¼ FðabÞ yðabÞ ; xðtÞ; t ; x_ ¼ F ðx; tÞ

ð56Þ

X1 ðtÞ ¼ fðz1 ; z2 Þjz1 X2 ðtÞ ¼ fðz1 ; z2 Þjz1 X3 ðtÞ ¼ fðz1 ; z2 Þjz1 X4 ðtÞ ¼ fðz1 ; z2 Þjz1

where



T ðabÞ ðabÞ F ¼ F1 ; F2 ;  ðabÞ  ðabÞ  ðabÞ  F 1 y ; t ¼ y2 ðtÞ ¼ x2 ðtÞ and F 2 yðabÞ ; t ¼ x_ 2 ðtÞ ðabÞ

ð57Þ with ðabÞ y1

¼ x1 ðtÞ and

on @ Xab ðabÞ

y1

> 0; z2 > 0g; > 0; z2 < 0g; < 0; z2 < 0g; < 0; z2 > 0g:

@ X12 ðtÞ ¼ fðz1 ; z2 Þjz2 ¼ 0; z1 > 0g; @ X23 ðtÞ ¼ fðz1 ; z2 Þjz1 ¼ 0; z2 < 0;g; @ X34 ðtÞ ¼ fðz1 ; z2 Þjz2 ¼ 0; z1 < 0g;

¼ x2 ðtÞ

ð60Þ

ð61Þ

@ X14 ðtÞ ¼ fðz1 ; z2 Þjz1 ¼ 0; z2 > 0;g:

for ða; bÞ ¼ ð2; 3Þ; ð1; 4Þ;

¼ x1 ðtÞ þ C

on @ Xab

ðabÞ y2

The domains and boundaries in the relative coordinates become

ðabÞ

and y2

¼ x2 ðtÞ

for ða; bÞ ¼ ð1; 2Þ; ð3; 4Þ:

ð58Þ

The boundary flow of x(t) is generated by the Duffing system, which implies that the boundaries change with time. Without loss of generality, the relative coordinates are introduced by

z 1 ¼ y 1  x1

and z_ 1  z2 ¼ y2  x2 :

ð59Þ

From the above definition, the boundaries in the relative coordinates are constant. Thus, the domains and boundaries are sketched in Fig. 3. Based on such boundaries and domains, the analytical conditions for the aforementioned synchronization will be developed. The controlled pendulum system in the relative coordinates is expressed by

  z_ ðaÞ ¼ gðaÞ zðaÞ ; x; t with x_ ¼ F ðx; tÞ;

ð62Þ

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A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

Fig. 3. Two boundaries in the relative coordinates: (a) velocity, and (b) displacement.

ðabÞ

where



T ðaÞ ðaÞ g zðaÞ ; x; t ¼ g 1 ; g 2 ;  ðaÞ  ðaÞ ðaÞ g 1 z ; x; t ¼ z2  k1 for a ¼ 1; 2;  ða Þ  ðaÞ g 1 zðaÞ ; x; t ¼ z2 þ k1 for a ¼ 3; 4;     ða Þ g 2 zðaÞ ; x; t ¼ G zðaÞ ; x; t  k2 for a ¼ 1; 4;    ða Þ  g 2 zðaÞ ; x; t ¼ G zðaÞ ; x; t þ k2 for a ¼ 2; 3:  ðaÞ

z1

ðabÞ z1



ð63Þ



  ðaÞ G zðaÞ ; x; t ¼ a0 sin z1 þ x1 þ Q 0 cos Xt þ d1 x2 ð64Þ

The equations of motion on the boundary in the relative coordinates become

  z_ ðabÞ ¼ gðabÞ zðabÞ ; x; t ; with x_ ¼ F ðx; tÞ;

ð65Þ

where

  ðabÞ ðabÞ T gðabÞ zðabÞ ; x; t ¼ g 1 ; g 2 ;   ðabÞ  ðabÞ ðabÞ  ðabÞ g 1 z ; x; t ¼ z2 ¼ 0 and g 2 z ; x; t ¼ 0 with

¼ 0 on @ Xab

for ða; bÞ ¼ ð2; 3Þ; ð1; 4Þ;

ðabÞ z2

¼C

and

¼ 0 on @ Xab

for ða; bÞ ¼ ð1; 2Þ; ð3; 4Þ: ð67Þ

with

 a1 x1 þ a2 x31  A0 cos xt:

ðabÞ

¼ 0 and z2

ð66Þ

3.3. Analytical conditions for synchronization As in Luo [34], before the analytical conditions are developed, the G-functions will be given in the relative coordinates for a = i, j and (i, j) 2 {(1, 2), (2, 3), (3, 4), (1, 4)}, i.e., for zm 2 oXij at t = tm,

 ða Þ G@ Xij ðzm ; x; t m Þ ¼ nT@ Xij  gðaÞ ðzm ; x; t m Þ  gðijÞ ðzm ; x; t m Þ ;

ð68Þ

 ð1;aÞ G@ Xij ðzm ; x; t m Þ ¼ nT@ Xij  DgðaÞ ðzm ; x; t m Þ  DgðijÞ ðzm ; x; t m Þ : ð69Þ ðaÞ G@ Xij ðzm ; x; t m Þ

ð1;aÞ G@ Xij ðzm ; x; tm Þ

where and are the zero-order and first-order G-functions of the flow in the domain Xa (a 2 {i, j}) at the boundary oXij (i, j = 1–4). From Eq. (61), the normal vectors of the relative boundaries are

n@ X12 ¼ n@X34 ¼ ð0; 1ÞT

and n@ X23 ¼ n@ X14 ¼ ð1; 0ÞT :

ð70Þ

371

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

From Eqs. (62)–(67), the corresponding G-functions for the boundary are ðaÞ

ða Þ

ða Þ

ðaÞ

ða Þ

ða Þ

G@ X12 ðzm ; x; tm Þ ¼ G@ X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; t m Þ; G@ X23 ðzm ; x; tm Þ ¼ G@ X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ; ð71Þ ð1;aÞ

ð1;aÞ

ðaÞ

ð1;aÞ

ð1;aÞ

ðaÞ

9 ð2Þ ð2Þ G@ X23 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ < 0; = for zm 2 @ X23 ; ð3Þ ð3Þ G@ X23 ðzm ; x; t mþ Þ ¼ g 1 ðzm ; x; t mþ Þ < 0 ; 9 ð1Þ ð1Þ G@ X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ > 0; = for zm 2 @ X14 : ð4Þ ð4Þ G@ X14 ðzm ; x; t mþ Þ ¼ g 1 ðzm ; x; t mþ Þ > 0 ; ð78Þ

G@ X12 ðzm ; x; tm Þ ¼ G@ X34 ðzm ; x; t m Þ ¼ Dg 2 ðzm ; x; tm Þ;

The conditions of a flow grazing to the boundaries of oX12, oX34, oX23 and oX14 for the controlled pendulum are

G@ X23 ðzm ; x; tm Þ ¼ G@ X14 ðzm ; x; t m Þ ¼ Dg 1 ðzm ; x; tm Þ; ð72Þ where   ðaÞ  ð aÞ  Dg 1 zðaÞ ; x; t ¼ g 2 zðaÞ ; x; t for a ¼ 1; 2; 3; 4;     ðaÞ Dg 2 zðaÞ ; x; t ¼ DG zðaÞ ; x; t



ðaÞ ð aÞ ¼ a0 z2 þ x2 cos z1 þ x1  Q 0 X sin Xt þ d1 F 2 ðx; tÞ  a1 x2 þ 3a2 x21 x2 þ xA0 sin xt; for a ¼ 1; 2; 3; 4:

ð0;aÞ

ðaÞ

G@ X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; t m Þ ¼ 0; ð1;aÞ

ðaÞ

ð1Þa G@ X12 ðzm ; x; tm Þ ¼ ð1Þa Dg 2 ðzm ; x; tm Þ < 0 for zm 2 @ X12 in Xa ða 2 f1; 2gÞ; ð0;aÞ

ðaÞ

G@ X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; t m Þ ¼ 0; ð1;aÞ

   ðaÞ  ðaÞ  ðaÞ  G@ X12 zðaÞ ; x; t ¼ G@ X34 zðaÞ ; x; t ¼ g 2 zðaÞ ; x; t ;    ðaÞ  ðaÞ  ðaÞ  G@ X23 zðaÞ ; x; t ¼ G@ X14 zðaÞ ; x; t ¼ g 1 zðaÞ ; x; t ;    ð1;aÞ  ð1;aÞ  ða Þ  G@ X12 zðaÞ ; x; t ¼ G@ X34 zðaÞ ; x; t ¼ Dg 2 zðaÞ ; x; t ;    ð1;aÞ  ð1;aÞ  ða Þ  G@ X23 zðaÞ ; x; t ¼ G@ X14 zðaÞ ; x; t ¼ Dg 1 zðaÞ ; x; t :

ðaÞ

for zm 2 @ X34 in Xa ða 2 f3; 4gÞ; ð79Þ

ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

G@X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ > 0 G@X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ > 0; ð4Þ

for zm 2 @ X12 ; ) for zm 2 @ X34 :

ð4Þ

ð75Þ ð2Þ

ð2Þ

ð3Þ

ð3Þ

ð1Þ

ð1Þ

ð4Þ

ð4Þ

G@X23 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ < 0; G@X12 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ > 0 G@X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ < 0;

) for zm 2 @ X23 ; ) for zm 2 @ X14 :

G@X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ > 0

The conditions of a flow passing through the boundaries of @ X12, oX34, oX23 and oX14 for the controlled pendulum are ð1Þ

ð1Þ

ð2Þ

ð2Þ

) for zm 2 @ X12 ;

G@X12 ðzm ; x; t mþ Þ ¼ g 2 ðzm ; x; tmþ Þ < 0

ð77aÞ ð3Þ

ð3Þ

G@X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ > 0; ð4Þ G@X34 ðzm ; x; t mþ Þ

¼

ð4Þ g 2 ðzm ; x; tmþ Þ

>0

ð1;aÞ

ðaÞ

ð0;aÞ

ðaÞ

G@ X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ ¼ 0

9 =

ðaÞ

for zm 2 @ X14 in Xa ða 2 f1; 4gÞ: ð80Þ The conditions for onset of a sliding flow on the boundaries of @ X12, @ X34, @ X23 and @ X14 for the controlled pendulum are

9 ð0;1Þ ð1Þ > G@ X12 ðzm ;x;t m Þ ¼ g 2 ðzm ;x;t m Þ < 0; > > = ð0;2Þ ð2Þ from X1 ! @ X12 ; G@ X12 ðzm ;x;t m Þ ¼ g 2 ðzm ;x;t m Þ ¼ 0; > > > ; ð1;2Þ ð2Þ G@ X12 ðzm ;x;t m Þ ¼ Dg 2 ðzm ;x;tm Þ < 0 9 ð0;3Þ ð3Þ G@ X34 ðzm ;x;t m Þ ¼ g 2 ðzm ;x;t m Þ > 0; > > > = ð0;4Þ ð4Þ from X3 ! @ X34 : G@ X34 ðzm ;x;t m Þ ¼ g 2 ðzm ;x;t m Þ ¼ 0; > > > ; ð1;4Þ ð4Þ G@ X34 ðzm ;x;t m Þ ¼ Dg 2 ðzm ;x;tm Þ > 0 ð81Þ

ð76Þ

G@X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ < 0;

ð1Þa G@ X23 ðzm ; x; tm Þ ¼ ð1Þa Dg 1 ðzm ; x; tm Þ > 0 ;

ð1;aÞ

)

G@X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ < 0

ðaÞ

9 =

ð1Þa G@ X14 ðzm ; x; tm Þ ¼ ð1Þa Dg 1 ðzm ; x; tm Þ < 0 ;

The conditions of a flow sliding on the boundaries of oX12, oX34, oX23 and oX14 for the controlled pendulum are

G@X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ < 0;

ð0;aÞ

G@ X23 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ ¼ 0 for zm 2 @ X23 in Xa ða 2 f2; 3gÞ;

ð74Þ

)

ð1Þa G@ X34 ðzm ; x; tm Þ ¼ ð1Þa Dg 2 ðzm ; x; tm Þ > 0

ð73Þ

The G-functions in domains with respect to the boundary are defined as

)

) for zm 2 @ X34 : ð77bÞ

9 ð0;2Þ ð2Þ G@ X23 ðzm ;x;t m Þ ¼ g 1 ðzm ;x;t m Þ < 0; > > > = ð0;3Þ ð3Þ from X2 ! @ X23 ; G@ X23 ðzm ;x;t m Þ ¼ g 1 ðzm ;x;t m Þ ¼ 0; > > > ð1;3Þ ð3Þ ; G@ X23 ðzm ;x;t m Þ ¼ Dg 1 ðzm ;x;tm Þ < 0 9 ð0;4Þ ð4Þ G@ X14 ðzm ;x;t m Þ ¼ g 1 ðzm ;x;t m Þ > 0; > > > = ð0;1Þ ð1Þ from X4 ! @ X14 : G@ X14 ðzm ;x;t m Þ ¼ g 1 ðzm ;x;t m Þ ¼ 0; > > > ð1;1Þ ð1Þ ; G@ X14 ðzm ;x;t m Þ ¼ Dg 1 ðzm ;x;tm Þ > 0 ð82Þ The conditions for vanishing of a sliding flow from the boundaries of oX12, oX34, oX23 and oX14 to a domain for the controlled pendulum are

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ðbÞ

ð1Þb G@ X12 ðzm ; x; t m Þ ¼ ð1Þb g 2 ðzm ; x; t m Þ > 0 ð0;aÞ

ðaÞ

G@ X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; t m Þ ¼ 0

9 > > > > > =

from @ X12 ! Xa ; ð1;aÞ ðaÞ > ð1Þa G@ X12 ðzm ; x; tm Þ ¼ ð1Þa Dg 2 ðzm ; x; t m Þ < 0; > > > > ; for zm 2 @ X12 ; a; b 2 f1; 2g and b – a 9 b ð0;bÞ b ðbÞ ð1Þ G@ X34 ðzm ; x; t m Þ ¼ ð1Þ g 2 ðzm ; x; t m Þ < 0 > > > > > ð0;aÞ ðaÞ = G@ X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; t m Þ ¼ 0 from @ X34 ! Xa ; ðaÞ a ð1;aÞ a > ð1Þ G@ X34 ðzm ; x; tm Þ ¼ ð1Þ Dg 2 ðzm ; x; t m Þ > 0; > > > > ; for zm 2 @ X34 ; a; b 2 f3; 4g and b – a ð83Þ 9 ð0;bÞ ðbÞ ð1Þb G@ X23 ðzm ; x; t m Þ ¼ ð1Þb g 1 ðzm ; x; t m Þ < 0; > > > > > ð0;aÞ ðaÞ = G ðz ; x; t Þ ¼ g ðz ; x; t Þ ¼ 0 @ X23

m

m

1

m

from @ X23 ! Xa ; ð1;aÞ ðaÞ > ð1Þa G@ X23 ðzm ; x; tm Þ ¼ ð1Þa Dg 1 ðzm ; x; t m Þ > 0 > > > > ; for zm 2 @ X23 ; a; b 2 f2; 3g and b – a 9 ð0;bÞ ðbÞ ð1Þb G@ X14 ðzm ; x; t m Þ ¼ ð1Þb g 1 ðzm ; x; t m Þ > 0; > > > > > ð0;aÞ ðaÞ = G@ X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; t m Þ ¼ 0 from @ X14 ! Xa : ðaÞ a ð1;aÞ a > ð1Þ G@ X14 ðzm ; x; tm Þ ¼ ð1Þ Dg 1 ðzm ; x; t m Þ < 0 > > > > ; for zm 2 @ X14 ; a; b 2 f1; 4g and b – a ð84Þ

3.4. Synchronization invariant sets and mechanism From the sliding flow of the controlled pendulum on the separation boundary, the conditions for the complete synchronization of the controlled pendulum with the Duffing oscillator occurs at the intersection of the two separation boundaries (zm = 0), and the corresponding conditions are ð1Þ

ð1Þ

ð1Þ

ð2Þ G@ X12 ðzm ; x; t m Þ

¼

ð2Þ g 2 ðzm ; x; tm Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

ð3Þ

ð3Þ

ð4Þ

ð4Þ

> 0;

G@ X23 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ < 0 G@ X23 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ > 0; G@ X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ > 0 G@ X34 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ < 0; ð4Þ G@ X14 ðzm ; x; t m Þ

¼

ð4Þ g 1 ðzm ; x; tm Þ

>0

g 2 ðzm ; x; tm Þ ¼ þk1 > 0;

for zm 2 @ X12 \ @ X14 on X1 ; ) for zm 2 @ X12 \ @ X23 on X2 ; ) for zm 2 @ X23 \ @ X34 on X3 ; ) for zm 2 @ X34 \ @ X14 on X4 : ð85Þ

From Eq. (63), four basic functions are defined as    ðaÞ  ðaÞ g 1 zðaÞ ; x; t  g 1 zðaÞ ; x; t ¼ z2  k1 in Xa for a ¼ 1; 2;  ðaÞ    ðaÞ ðaÞ ðaÞ g 2 z ; x; t  g 1 z ; x; t ¼ z2 þ k1 in Xa for a ¼ 3; 4;  ðaÞ     ðaÞ  ðaÞ g 3 z ; x; t  g 2 z ; x; t ¼ G zðaÞ ; x; t  k2 in Xa for a ¼ 1; 4;      ðaÞ  g 4 zðaÞ ; x; t  g 2 zðaÞ ; x; t ¼ G zðaÞ ; x; t þ k2 in Xa for a ¼ 2; 3: ð86Þ

where

Gðx; tÞ ¼ a0 sin x1 þ Q 0 cos Xt þ d1 x2  a1 x1 þ a2 x31

  ðaÞ G zðaÞ ; x; t ¼ a0 sinðz1 þ x1 Þ þ Q 0 cos Xt þ d1 x2 ð87Þ

The synchronization conditions in Eq. (85) become

g 1 ðzm ; x; tm Þ ¼ z2m  k1 < 0; g 2 ðzm ; x; tm Þ ¼ z2m þ k1 > 0; g 3 ðzm ; x; tm Þ ¼ Gðzm ; x; t m Þ  k2 < 0; g 4 ðzm ; x; tm Þ ¼ Gðzm ; x; t m Þ þ k2 > 0:

ð90Þ

If k1 > 0 and k2 > 0, the first two equations of Eq. (88) can be automatically satisfied, and the third and fourth equations give the synchronization invariant set as

k2 < Gðx; t m Þ < k2 :

ð91Þ

In the small neighborhood of the synchronization of zm = 0, the attractivity conditions can be given for jz  zmj < e, i.e., 0 6 z2 < k 1 0 6 z2 < k 1

and Gðz; x; tÞ < k2 for z1 2 ½0; 1Þ in X1 ; and  k2 < Gðz; x; tÞ for z1 2 ½0; 1Þ in X2 ;

 k1 < z2 6 0 and  k2 < Gðz; x; tÞ for z1 2 ð1; 0 in X3 ;  k1 < z2 6 0 and Gðz; x; tÞ < k2 for z1 2 ð1; 0 in X4 : ð92Þ

z 1

z 2

from which and are obtained, the initial conditions for the controlled pendulum synchronization can be determined by

and y2 ¼ z 2 þ x2 :

ð93Þ

The conditions for vanishing of synchronization are for ðaÞ zðaÞ ðt m Þ ¼ zm ¼ zm



9 ðaÞ ðaÞ > g 1 zm ; x; tm ¼ z2m  k1 ¼ 0; > > > =

ðaÞ ðaÞ Dg 1 zm ; x; t m ¼ Gðzm ; x; tm Þ > 0; > >

> > ðbÞ ðbÞ ; g 2 zm ; x; t m ¼ z2m þ k1 > 0 for ða; bÞ ¼ fð1; 4Þ; ð2; 3Þg;

ð94Þ

from zm+e = y1  x1 > 0, and



9 ðaÞ ðaÞ > g 1 zm ; x; tm ¼ z2m  k1 < 0; > > > =

ðbÞ ðbÞ g 2 zm ; x; t m ¼ z2m þ k1 ¼ 0; > >



> > ðbÞ ðbÞ Dg 2 zm ; x; tm ¼ G zm ; x; t m < 0 ; for ða; bÞ ¼ fð1; 4Þ; ð2; 3Þg;

ð95Þ

from zm+e = y1  x1 < 0. The conditions for vanishing of synchronization are for ðaÞ zðaÞ ðt m Þ ¼ zm ¼ zm

where

 a1 x1 þ a2 x31  A0 cos xt:

ð89Þ

g 3 ðzm ; x; tm Þ ¼ Gðx; t m Þ  k2 < 0; g 4 ðzm ; x; tm Þ ¼ Gðx; t m Þ þ k2 > 0:

y1 ¼ z 1 þ x1

)

ð1Þ

G@ X12 ðzm ; x; t m Þ ¼ g 2 ðzm ; x; tm Þ < 0

g 1 ðzm ; x; tm Þ ¼ k1 < 0;

 A0 cos xt:

m

G@ X14 ðzm ; x; t m Þ ¼ g 1 ðzm ; x; tm Þ < 0;

Letting zm = 0, the synchronization conditions of the controlled pendulum with the Duffing oscillator are

ð88Þ





9 ðaÞ ðaÞ g 3 zm ; x; tm ¼ G zm ; x; tm  k2 ¼ 0; > > > > =

ðaÞ ðaÞ Dg 3 ðzm ; x; tm Þ ¼ DG zm ; x; tm > 0; > >



> > ðbÞ ðbÞ ¼ G z ; x; t þk >0 ; g z ; x; t 4

m

m

m

m

for ða; bÞ ¼ fð1; 2Þ; ð4; 3Þg; from z_ mþe ¼ y2  x2 > 0, and

2

ð96Þ

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9 ðaÞ ðaÞ g 3 zm ; x; t m ¼ G zm ; x; t m  k2 < 0; > > > > =



ðbÞ ðbÞ g 4 zm ; x; t m ¼ G zm ; x; tm þ k2 ¼ 0; > >



> > ðbÞ ðbÞ ; g 4 zm ; x; t m ¼ DG zm ; x; t m < 0 for ða; bÞ ¼ fð1; 2Þ; ð4; 3Þg;

from zm+e = y1  x1 < 0. The conditions for onset of synchronization are for ðaÞ zðaÞ ðtm Þ ¼ zm ¼ zm

ð97Þ

from z_ mþe ¼ y2  x2 < 0. The conditions for onset of synchronization are for ðaÞ zðaÞ ðtm Þ ¼ zm ¼ zm



9 ðaÞ ðaÞ > g 1 zm ; x; t m ¼ z2m  k1 ¼ 0; > > > =



ða Þ ða Þ Dg 1 zm ; x; tm ¼ G zm ; x; tm > 0; > >

> > ðbÞ ðbÞ ; g 2 zm ; x; t m ¼ z2m þ k1 > 0; for ða; bÞ ¼ fð1; 4Þ; ð2; 3Þg;

for ða; bÞ ¼ fð1; 2Þ; ð4; 3Þg;

ð100Þ

from z_ me ¼ y2  x2 > 0, and

9 ðaÞ ðaÞ g 3 ðzm ; x; t m Þ ¼ Gðzm ; x; t m Þ  k2 < 0; > = ðbÞ

ðbÞ

g 4 ðzm ; x; tm Þ ¼ Gðzm ; x; t m Þ þ k2 ¼ 0;

ð98Þ

ðbÞ

ðbÞ

g 4 ðzm ; x; tm Þ ¼ DGðzm ; x; tm Þ < 0 for ða; bÞ ¼ fð1; 2Þ; ð4; 3Þg;

from zme = y1  x1 > 0, and



9 ðaÞ ðaÞ > g 1 zm ; x; t m ¼ z2m  k1 < 0; > > > =

ðbÞ ðbÞ g 2 zm ; x; t m ¼ z2m þ k1 ¼ 0; > >



> > ðbÞ ðbÞ Dg 2 zm ; x; t m ¼ G zm ; x; t m < 0 ; for ða; bÞ ¼ fð1; 4Þ; ð2; 3Þg;





9 ðaÞ ðaÞ g 3 zm ; x; t m ¼ G zm ; x; t m  k2 ¼ 0; > > > > =



ða Þ ðaÞ Dg 3 zm ; x; t m ¼ DG zm ; x; t m > 0; > >



> > ðbÞ ðbÞ g 4 zm ; x; t m ¼ G zm ; x; tm þ k2 > 0 ;

> ; ð101Þ

from z_ me ¼ y2  x2 < 0. 3.5. Parameter studies

ð99Þ

From the previous analytical conditions, parameter studies will be carried out for a better understanding of the synchronization of two dynamical systems.

Fig. 4. Synchronization scenario of switching points versus control parameter k2 for periodic motion in right potential well: (a) switching displacement, (b) switching velocity, (c) switching phase, and (d) switching points on the synchronized periodic orbit. (Control parameter: k1 = 1. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.16, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519.) (Initial condition: x1 = y1  0.62208842 and x2 = y2  0.5307933.) (FS: Full synchronization, PS: Partial synchronization, NS: Non-synchronization.) (A: Synchronization appearance; V: Synchronization vanishing.)

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Consider a periodic motion of the Duffing oscillator with a set of parameters and initial condition, i.e.,

Duffing : a1 ¼ a2 ¼ 1:0; d1 ¼ 0:25; A0 ¼ 0:16; x ¼ 1:0; Pendulum : a0 ¼ 1:0; Q 0 ¼ 0:275; X ¼ 2:18517; Initial condition : x1 ¼ y1  0:62208842 and x2 ¼ y2  0:5307933; t 0 ¼ 0 ðrightÞ Initial condition : x1 ¼ y1  0:5954405 and x2 ¼ y2  0:4317926; t 0 ¼ 0 ðleftÞ: ð102Þ The synchronization switching scenario of the controlled slave system (i.e., controlled pendulum) varying with control parameter k2 is presented in Fig. 4 with k1 = 1 plus the other parameters and initial conditions in Eq. (102). Based on the parameters in Eq. (102), the Duffing oscillators have two periodic motions existing in two potential wells. Acronyms ‘‘FS’’, ‘‘PS’’ and ‘‘NS’’ represent full, partial and non synchronizations, respectively. A and V denote synchronization appearance and vanishing, respectively. The switching displacement, velocity and phases for the controlled pendulum to synchronize with a periodic motion of the Duffing oscillator are illustrated in Fig. 4(a)–(c) in the right potential well and Fig. 5(a)–(c) in the left potential well, respectively. The distribution of synchronization appear-

ance and vanishing on the trajectory of periodic motion in phase plane is presented in Fig. 4(d) in the right potential well and Fig. 5(d) in the left potential well. For the synchronization scenario (k1 = 1), the partial synchronization of the controlled pendulum with the periodic motion of the Duffing oscillator lies in k2 2 (0.602, 1.288). If k2 2 (0, 0.602), no synchronization can be observed. If k2 2 (1.288, 1), the controlled pendulum will fully synchronize with the periodic motion in the Duffing oscillator. The skew symmetry of the switching responses of the synchronization appearance and disappearance is observed. Consider the parameters for the Duffing oscillator and the pendulum

Duffing : a1 ¼ a2 ¼ 1:0; Pendulum : a0 ¼ 1:0;

d1 ¼ 0:25; Q 0 ¼ 0:275;

x ¼ 1:0; X ¼ 2:18517: ð103Þ

The control parameter maps (k1, k2) for the synchronization of the controlled pendulum with periodic and chaotic motions of the Duffing oscillator are presented in Fig. 6(a) and (b), respectively. The shaded part is a partial synchronization zone. To illustrate the control parameter map for the controlled pendulum synchronizing with the Duffing oscillator, the following excitation amplitude of the

Fig. 5. Synchronization scenario of switching points versus control parameter k2 for periodic motion in the left potential well: (a) switching displacement, (b) switching velocity, (c) switching phase, and (d) switching points on the synchronized periodic orbit. (Control parameter: k1 = 1. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.16, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519). (Initial condition: x1 = y1   0.3130951, x2 = y2  0.3196078.) (FS: Full synchronization, PS: Partial synchronization, NS: Non-synchronization.) (A: Synchronization appearance; V: Synchronization vanishing.)

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Fig. 6. Control parameter map of (k1, k2) for the synchronicity of the Duffing oscillator and controlled pendulum: (a) period-1 large motion (A0 = 0.454) with initial condition (x1 = y1  0.3646916 and x2 = y2  1.2598329), and (b) chaotic motion (A0 = 0.265) with initial condition (x1 = y1   0.4180597 and x2 = y2  0.2394332). (Duffing: a1 = a2 = 1.0, d1 = 0.25, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519.) (FS: Full synchronization, PS: Partial synchronization, NS: Non-synchronization.)

Duffing oscillator and the corresponding initial conditions will be used with parameters in Eq. (103)

A0 ¼ 0:454;

x1 ¼ y1  0:3646916;

 1:2598329;

ð104Þ

for a periodic motion, and

A0 ¼ 0:265;

t 0 ¼ 0:0

x2 ¼ y2  0:5307933;

t0 ¼ 0:0ðrightÞ



x1 ¼ y1  0:5954405; xx2 ¼ y2  0:4317926; t0 ¼ 0:0ðleftÞ A0 ¼ 0:16 for period-1 motion  A0 ¼ 0:325; x1 ¼ y1  0:4751176; for period-3 motion: x2 ¼ y2  0:6524778; t0 ¼ 0:0 ð106Þ

x1 ¼ y1  0:4180597 and x2 ¼ y2

 0:2394332;

(103), the corresponding excitation amplitude and initial conditions are: x1 ¼ y1  0:6220842;

and x2 ¼ y2

t0 ¼ 0:0

Fig. 7. Control parameter maps of (k1, k2) for the synchronicity of the Duffing oscillator and controlled pendulum: (a) period-1 small motion (A0 = 0.16) with initial conditions (x1 = y1  ±0.6220842 and x2 = y2  ±0.5307933) and (b) global period-3 motion (A0 = 0.325) with initial condition (x1 = y1  0.4751176 and x2 = y2  0.6524778). (Duffing: a1 = a2 = 1.0, d = 0.25, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519.) (FS: Full synchronization, PS: Partial synchronization, NS: Nonsynchronization.)

ð105Þ

for a chaotic motion. The lower boundary of the partial synchronization of chaotic motion is rough compared to the periodic motion. However, the upper boundaries for chaotic and periodic motion synchronization are independent of control parameter k1. As k1 is relatively small, the non-synchronization area is very large for such a periodic motion, and the non-synchronization area is very irregular for chaos. To further look into the control parameters for synchronization, the control parameter map of (k1, k2) for the period-1 and periodic-3 motions are presented in Fig. 7(a) and (b), respectively. With parameters in Eq.

The maps for partial synchronizations of two periodic motions are different as presented in Fig. 6(a). In Fig. 6(a), the periodic motion is outside of the separatrix of the Duffing oscillator. However, in Fig. 7(a), the period-1 motion is in the two potential wells of the Duffing oscillator. In other words, the period-1 motion is inside of the separatrix. In Fig. 7(b), the period-3 motion can cross the separatrix and exist inside and outside of separatrix. The non-synchronization area for period-1 motion is larger than the period-3 motion, and the lower boundary of the partial synchronization is more irregular than the period-1 motion. In addition to the control parameters, the system parameter effects to the synchronization are discussed.

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For the Duffing oscillator, the damping is a very important factor. The parameter maps of (d1, k2) for two excitation amplitudes A = 0.4 and 0.454 are presented in Fig. 8 for k1 = 1.0. The shaded area in the parameter map is also for partial synchronization. For A0 = 0.4, the synchronization of the controlled pendulum with periodic motions in the Duffing oscillator is in the range of d1 2 (0.0, 0.24) and (0.55, 1). The synchronization with chaos lies in the range of d1 2 (0.24, 0.55) and d1 = 0, which can be observed in Fig. 8(a). For A0 = 0.454, the synchronization of periodic motion is in the range of d1 2 (0.0, 0.27) and (0.63, 1), and the synchronization of chaotic motions is for d1 2 (0.27, 0.63) and d1 = 0, as shown in Fig. 8(b). With varying excitation amplitude, the synchronization of the controlled pendulum with the Duffing oscillator will be changed. In additions, excitation frequency is another important factor to change motion characteristics of the Duffing oscillator. Thus, the effects of excitation frequency and amplitude to the synchronization are discussed for k1 = 1.0 and d1 = 0.25. In Fig. 9(a) and (b), the parameter maps of (x,k2) and (A0, k2) for two system synchronization are presented. In Fig. 9(a), with increasing excitation fre-

Fig. 9. Control parameter maps for the synchronicity of the Duffing oscillator and controlled pendulum: (a) (k2, x) with (A0 = 0.454) and (b) (k2,A0) with (x = 1.0). (Duffing: a1 = a2 = 1.0 and d1 = 0.25. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519.) (FS: Full synchronization, PS: Partial synchronization, NS: Non-synchronization.)

Fig. 8. Control parameter maps of (k2, d1) for the synchronicity of the Duffing oscillator and controlled pendulum: (a) A0 = 0.40, and (b) A0 = 0.454. (Duffing: a1 = a2 = 1.0, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18519.) (FS: Full synchronization, PS: Partial synchronization, NS: Non-synchronization.)

quency, the partial synchronization area is bigger than with decreasing excitation frequency, which hints by use of arrow in the parameter map. The main extra area is hatched. This phenomenon is caused by the backbone characteristics of the Duffing oscillator. For excitation frequency near the primary resonance, the range of k2 are much bigger than other excitation frequency. That is, the two ranges of x 2 (0.613, 1.084) and (0.613, 0.995) are for increasing and deceasing excitation frequency, respectively. The small peaks near the superharmonic or subharmonic resonant frequencies also appear in the parameter map. The smooth boundary of the partial synchronization is for periodic motion, and the rough boundary is for chaotic motion. For large excitation frequency (x > 1.45), periodic motion synchronization can be observed. In Fig. 9(b), with increasing excitation amplitude, the partial synchronization area in the parameter map of (A0, k2) is smaller than with decreasing excitation amplitude. In other words, the main synchronization of periodic motions is in range of A0 2 (0, 0.263) and (0.453, 0.6) for increasing excitation amplitude. However, for deceasing excitation amplitude, the range for periodic motion synchronization is A0 2 (0, 0.263) and (0.418, 0.6). Other periodic motion

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

synchronization is embedded in the range of excitation amplitude for chaotic motion synchronization. The initial conditions are listed because the different parameters are different for parameter maps in Figs. 8 and 9. After discussing the synchronization parameter characteristics of the controlled pendulum and the Duffing oscillator, synchronization illustrations between the two different dynamical systems are very important. The partial synchronization of the controlled pendulum with a periodic motion in the Duffing oscillator is presented in Fig. 10 with the following parameters and initial conditions as

Control parameter : k1 ¼ 1; k2 ¼ 0:8; Duffing : a1 ¼ a2 ¼ 1:0; d1 ¼ 0:25; A0 ¼ 0:325; x ¼ 1:0; Pendulum : a0 ¼ 1:0; Q 0 ¼ 0:275; X ¼ 2:18517; Initial condition : x1 ¼ y1  0:41571176 and x2 ¼ y2  0:6524778; t0 ¼ 0: ð107Þ In Fig. 10(a), the time-history of velocity for the periodic motion of the Duffing oscillator is depicted by the solid curve, but the velocity response for the controlled pendulum is given by the dashed curve. The corresponding

377

G-functions for controlled pendulum are plotted in Fig. 10(b). The shaded portions are for synchronization. The non-shaded regions are for non-synchronization. If the G-function of g4 is dashed curve, the controlled pendulum is in domain Xa (a = 1, 4), which does not synchronize with the periodic motion in the Duffing oscillator, and the velocity of the controlled pendulum is greater than that of the Duffing oscillator. If the G-function of g3 is dashed curve, the controlled pendulum is in domain Xa (a = 2, 3), which does not synchronize with the periodic motion in the Duffing oscillator, and the velocity of the controlled pendulum is less than that of the Duffing oscillator. If the initial conditions are different for the Duffing oscillator and controlled pendulum, the non-synchronization for displacement can be observed. To check whether the synchronized portion of the trajectory is in the synchronization invariant domain or not, the invariant domain is superimposed on phase plane, as shown in Fig. 10(c). The synchronized portions of the trajectory are in the invariant domain, and the non-synchronized portion is in the synchronization invariant domain, the controlled pendulum should satisfy the corresponding conditions for the synchronization appearance and vanishing in the previous

Fig. 10. The partial synchronization of the Duffing oscillator and controlled pendulum for periodic motion: (a) velocity responses, and (b) G-function responses. (c) phase plane with the invariant domain, (d) switching points for synchronization appearance and vanishing. (Control parameters: k1 = 1 and k2 = 0.8. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.325, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18517.) (Initial condition: x1 = y1  0.4751176 and x2 = y2  0.6524778.) (S: Synchronization; N: Non-synchronization.) Hollow and filled circular symbols are synchronization appearance (A) and vanishing (V), respectively.

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Fig. 11. The partial synchronization of the Duffing oscillator and controlled pendulum for chaotic motion: (a) velocity responses, (b) G-function responses. (c) phase plane with the invariant domain, and (d) switching points for synchronization appearance and vanishing. (Control parameters: k1 = 1 and k2 = 0.9. Duffing: a1 = a2 = 1.0, d1 = 0.25, A0 = 0.265, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18517.) (Initial condition: x1 = y1  0.4180597 and x2 = y2  0.2394332.) (S: Synchronization; N: Non-synchronization.) Hollow and filled circular symbols are synchronization appearance (A) and vanishing (V), respectively.

section (i.e., Eqs. (94)–(101)). To observe the existence of the synchronization pattern for long time, the switching points for the synchronization and non-synchronization are presented for 10,000 periods of the Duffing oscillator, as shown in Fig. 10(d). In order to compare with the synchronizations of periodic and chaotic motions, the partial synchronization of the controlled pendulum with a chaotic motion in the Duffing oscillator is presented in Fig. 11 with the following parameters and initial conditions as

Control parameter : k1 ¼ 1; Duffing : a1 ¼ a2 ¼ 1:0;

k2 ¼ 0:9;

d1 ¼ 0:25;

A0 ¼ 0:265;

Table 1 Input data for illustrations (a1 = a2 = 1.0, d1 = 0.25, x = 1.0 and a0 = 1.0, Q0 = 0.275, X = 2.18517, t0 = 0).

x ¼ 1:0; Pendulum : a0 ¼ 1:0;

Q 0 ¼ 0:275;

11(a)–(c), the velocity and G-function responses for the periodic and chaotic motion synchronization are similar. The switching points for periodic motion are on periodic orbit. However, the switching points for the appearance and disappearance of chaotic motion synchronization are much chaotic, as shown in Fig. 11(d). For more illustrations of synchronization, the partial and full synchronizations of the controlled pendulum with period-1 and period-2 motions are presented through trajectories in phase plane in Figs. 12 and 13. Consider the parameters (a1 = a2 = 1.0, d1 = 0.25, x = 1.0) for the Duffing oscillator and (a0 = 1.0, Q0 = 0.275, X = 2.18517)

X ¼ 2:18517;

Initial condition : x1 ¼ y1  0:4180597 and x2 ¼ y2  0:2394332;

t 0 ¼ 0: ð108Þ

The time-history of velocity, G-function responses, trajectory in phase planes and switching points for appearance and vanishing of synchronization are presented in Fig. 11(a)–(d), respectively. From Figs. 10(a)–(c) and

(k1, k2, A0)

Initial condition (x1, x2) = (y1, y2)

Fig. 12(a)

(1.0, 0.8, 0.16)

Fig. 12(b)

(1.0, 0.8, 0.18)

Fig. 13(a)

(1.0, 1.5, 0.16)

Fig. 13(b)

(1.5, 1.5, 0.18)

(0.6220842, 0.5307933) (right) (0.5954405, 0.4317926) (left) (0.3838119, 0.3680388) (right) (0.8838948, 0.6275368) (left) (0.6220842, 0.5307933) (right) (0.5954405, 0.4317926) (left) (0.3838119, 0.3680388) (right) (0.8838948, 0.6275368) (left)

PS PS FS FS

A.C.J. Luo, F. Min / Chaos, Solitons & Fractals 44 (2011) 362–380

Fig. 12. The partial synchronization of the Duffing oscillator and controlled pendulum for periodic motion: (a) period-1 motion (k2, A0) = (0.8, 0.16) with x1 = y1  0.6220842, x2 = y2  0.5307933 (right), and x1 = y1  0.5954405, x2 = y2  0.4317926 (left); (b) period-2 motion (k2,A0) = (0.8, 0.183) with x1 = y1  0.3130951, x2 = y2  0.3196078 (right) and x1 = y1  0.5954405, x2 = y2  0.4317926 (left). (Control parameters: k1 = 1. Duffing: a1 = a2 = 1.0, d1 = 0.25, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18517.) Hollow and filled circular symbols are synchronization appearance (A) and vanishing (V), respectively. S: synchronization.

for pendulum with other parameters and initial conditions in Table 1. In Fig. 12(a) and (b), the partial synchronization of period-1 and period-2 motions are presented for k2 = 0.8. The partial synchronization patterns for such periodic motions are clearly observed. However, for k2 = 1.5, the controlled pendulum can fully synchronize with the same periodic motions in the Duffing oscillator, as shown in Fig. 13(a) and (b). The trajectories for the Duffing oscillator and the controlled pendulum are identical. The Duffing oscillator is represented by the solid curves, and the controlled pendulum is denoted by the circular symbols. 4. Conclusions In this paper, the synchronization theory of two different dynamical systems was presented. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) were obtained. Such a synchronization theory

379

Fig. 13. The full synchronization of the Duffing oscillator and controlled pendulum for periodic motions: (a) period-1 motion (k2, A0) = (1.5, 0.16) with x1 = y1  0.6220842, x2 = y2  0.5307933 (right), and x1 = y1  0.5954405, x2 = y2  0.4317926 (left); (b) period-2 motion (k2,A0) = (1.5, 0.183) with x1 = y1  0.3130951, x2 = y2  0.3196078 (right) and x1 = y1  0.5954405, x2 = y2  0.4317926 (left). (Control parameters: k1 = 1. Duffing: a1 = a2 = 1.0, d1 = 0.25, x = 1.0. Pendulum: a0 = 1.0, Q0 = 0.275, X = 2.18517).

was applied to the synchronization of the controlled pendulum with the Duffing oscillator, and the analytical conditions for the synchronization were achieved. Furthermore, the parameter study for the synchronization of the controlled pendulum and the Duffing oscillator was carried out. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are illustrated. The synchronization of the Duffing oscillator and pendulum are investigated herein to show the usefulness and efficiency of the methodology in this paper. The synchronization invariant domain is obtained for the first time. The technique presented in this paper should have a wide spectrum of applications in engineering. For instance, this technique can be applied to the maneuvering target tracking, and the others.

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