Prediction of Horseshoe Chaos in Duffing-Van Der Pol Oscillator Driven By Different Periodic Forces

Transcription

1 RESEARCH INVENTY: International Journal of Engineering and Science ISSN: , Vol. 1, Issue 5 (October 1), PP Prediction of Horsesoe Caos in Duffing-Van Der Pol Oscillator Driven By Different Periodic Forces L. Ravisankar 1, V. Ravicandran, V. Cinnatabi 3 1,,3 (Departent of Pysics, Sri K.G.S. Arts College, Srivaikunta-68619, Tailnadu, India) Abstract: An analytical tresold condition for te prediction of onset of orsesoe caos is obtained in te Duffing van der Pol oscillator driven by different periodic forces using te Melnikov etod. Te external periodic forces considered are sine wave, square wave, syetric saw-toot wave, rectified sine wave and odulus of sine wave. Melnikov tresold curve is drawn in a paraeter space. Analytical predictions are deonstrated troug direct nuerical siulation. Nuerical investigations including coputation of stable and unstable anifolds of saddle and easuring te tie elapsed between two successive transverse intersections are used to detect onset of orsesoe caos. Keywords Duffing- van der Pol oscillator, Horsesoe caos, Melnikov function, periodic forces, caos I. Introduction Most of te previous works on te nonlinear oscillator systes eploy excitations of te sinusoidal type only. It is of considerable interest to study te syste under influence of nonsinusoidal excitation suc as square wave, syetric and asyetric saw-toot waves, rectified and odulus of sine waves. Recently any studies ave sown tat te effect of different kinds of periodic forcing on tese systes is considerable and tey can cange te dynaical beaviours drastically. For exaple, onset of ooclinic caos by a periodic string of pulses odulated by Jacobian elliptic function and periodic δ-function [1], generation of caotic beavior by a distorted force [], anti-control of caos by certain periodic forces [3], suppression of caos by δ- pulse [4], stocastic resonance, nonescape dynaics and occurrence of orsesoe caos wit different periodic forces [5-7] ave been reported. Horsesoe is te occurrence of transverse intersection of stable and unstable anifolds of a saddle fixed point in te Poincaré ap and is a global penoenon. Its appearance can be predicted analytically by eploying te Melnikov tecnique [8]. Tis tecnique essentially gives a criterion for a transverse intersection of te stable and unstable anifolds of ooclinic / eteroclinic orbits wic iply orsesoe caos. Even toug te orbits created by te orsesoe ecanis are unstable, global inforation can be obtained analytically using te Melnokov tecnique. Te essence of tis etod is to calculate te so-called Melnikov integral, wic can be used to predict te regions in te paraeter space were Sale-orsesoe caos occurs [8-1]. It is well known tat te existence of orsesoe does not iply tat trajectories will be asyptotically caotic. Te asyptotically caotic otion is caracterized by positive Lyapunov exponent. However, te orbits created by orsesoe ecanis display an extree sensitive dependence on initial conditions and possibly exibit eiter a caotic transient before settling to stable orbits or a strange attractor [8]. In any dynaical systes [9,1] onset of caos as been found to occur near te Melnokov-tresold curve. Consequently, te Melnikov-tresold curve is considered as a lower tresold for te onset of caos. Te Melnikov tecnique was firstly applied by Holes [11] to study te caotic attractor of a periodically driven Duffing oscillator wit negative linear stiffness. Recently, tis etod as been applied to certain nonlinear systes. Cao and Cen [1,13] studied control of ooclinic and eteroclinic bifurcations in double well and tree- well oscillators by a second sinusoidal forces. Nbendjo et al [14] analysed te control of escape and orsesoe caos in a aronically excited particle fro a catastropic single-well four-well potential. Dana et al [15] reported experiental evidence of sil nikov-type ooclinic caos in asyetry induced Cua s oscillator. Tang et al [16] studied te relationsip between te order of caos of Melnikov function and te aronic coponents of a syste. Tresold values of caotic otion under te periodic and quasiperiodic perturbation in Duffing and Duffing-van der Pol oscillators are obtained by Melnikov etod [17, 18]. Rajasekar et al [19, ] investigated te possibility of controlling orsesoe and asyptotic caos in te Duffing-van der Pol oscillator using Melnikov etod. Effect of bounded noises as also been studied in certain oscillators [1-5]. 17

2 Prediction of Horsesoe Caos in Duffing-Van In tis paper we study bot analytically and nuerically te effect of te sape of periodic forces on orsesoe caos in Duffing-van der Pol oscillator equation x px ( 1 x ) x x 3 F( t) Te otivation for our interest in tis syste is tat it as wide range of applications in pysics and biology. Eq. (1) is an alternative for of Bonoeffer-van der Pol oscillator [6, 7] driven agnetic oscillator [8] and also describe te dynaics of carged density in te plasa of a rf gas discarge. Recently, Ravicandran et al [9] studied te effect of te sape of periodic forces and second periodic forces on orsesoe caos in Duffing oscillator. Our objective ere is to explore te possibility of occurrence of orsesoe caos using bot analytical and nuerical tecniques. In our present analysis we use Melnikov analytical etod to study te influence of sape of periodic forces on ooclinic orbits. Te paper is organized as follows. In section we obtain te Melnikov tresold condition for te transverse intersection of ooclinic orbits for te syste (1) separately for eac of te periodic forces. In section 3 te effect of sape of periodic forces on orsesoe dynaics is analyzed using te Melnikov etod. Te analytical predictions is deonstrated troug direct nuerical siulations. Controlling of asyptotic caos is also studied. Finally, section 4 contains te concluding rearks. II. Preliinaries And Calculation Of Melnikov Function We consider te linearly perturbed double-well Duffing-van der Pol (DVP) oscillator driven by an external periodic force F(t) in te for (1) x y,. y x x 3 [ p x(1 x ) F( t)], (a) (b) were F(t) is an external periodic force, ε is a sall paraeter, > and β >. Figure 1 depicts various periodic forces considered in our present work. Te unperturbed part of te syste () (ε=) as one saddle point ( ) = (,) and two centre type fixed points ( ) = (,). Te two ooclinic orbits connecting te saddle to itself are given by W ( x ( t), y ( t)) ( sec t, sec t tant), t t (3) Stable anifolds ( ) and unstable anifolds ( ) of ooclinic orbits are indicated in Fig.. Periodic orbits are nested witin and outside te ooclinic orbits. For, te stable and unstable brances of ooclinic orbits join sootly. Wen te dissipative perturbation is included te stable anifold unstable anifold Wu transverse intersections of Ws and do not join. However, above certain critical aplitude of te external periodic force Ws and Wu occurs. Te presence of suc intersections iplies tat te Poincaré ap 18

3 Prediction of Horsesoe Caos in Duffing-Van Figure 1: Wave for of various periodic forces (a) sine wave (b) square wave (c) rectified sine wave (d) syetric saw-toot wave (e) asyetric saw-toot wave and (f) odulus of sine wave. For all te forces period is π/ω, ω = 1 and aplitude f is 1. as te so-called orsesoe caos [8, 3]. Eventoug te orbits created by te orsesoe ecanis are unstable tey can exert a draatic influence on te beavior of orbits wic pass close to te point of intersection. Te appearance of transverse intersections of ooclinic orbits can be predicted analytically by te Melnikov tecnique. In order to apply te Melnikov tecnique, in general, we rewrite te given equation of otion into te following standard for x f ( x, y) g ( x, y, t) 1 1 y f ( x, y) g ( x, y, t) (4a) (4b) were g 1 and g are periodic in t wit period T, te Melnikov function is given by [3] T M ( t o ) f ( X g X t trace D f X S ds d ( )) ( ( ), ) exp[ [ X ( ( )) ] (5) were X ( x, y ) represents ooclinic orbits of te unperturbed systes, f g f 1 g f g 1 and D denotes te partial derivatives wit respect to X ( x, y). X Figure : Pase trajectories of te unperturbed syste. Te stable ( W s ± ) and unstable anifold (W u ± ) parts of ooclinic orbits connecting saddle to itself are indicated. Te analytical expression for te ooclinic orbits is given byeq. (3). For te DVP Eq. () te Melnikov function is M( t ) y [ py (1 x ) F( t )] d. (6) In te following we calculate te Melnikov function for te syste () wit different periodic forces. Sine wave For te syste () driven by te force F( t) fsin sin t, te Melnikov integral is worked out as M sin ( t ) A f sin Bcost (7a) were 4 p 3 4 A 1, 3 5 B sec (7b) 19

4 Prediction of Horsesoe Caos in Duffing-Van Square wave For te square wave F( t) F sq ( t ) f sq sgn(sin t) were sgn( y ) is sign of y. Te Fourier series is F sq 4 f ( t) sin(n 1) t n1 (n 1) Using its Fourier series we obtain M ( t ) A f sq B n cos(n 1) t sq n1 (8a) (8b) were 4 (n 1) B n sec Syetric saw-toot wave Te nuerical ipleentation of te force is given by (8c) F sst ( t ) 4 f sst t, T 4 f sst t, T 4 f sst t 4 f sst, T t, t 3, 3 t, (9a) were T is te period of te force and t is taken as od. Its Fourier series ( 1) 8 ( 1) sin( 1) ( ) n f n t Fsst t. n1 ( n 1) Te Melnikov function is M sst ( t ) A f sst B n cos(n 1) t n1 were 8 ( 1) n1 (n 1) B n sec B (n 1) (9b) (9c) (9d) Asyetric saw-toot wave For te asyetric saw-toot wave, f t ast, T Fast ( t) F t ast f t ast f, ast T were t is taken as od π / ω. Te Fourier series of F ast (t) is given by t, t, (1a)

5 Prediction of Horsesoe Caos in Duffing-Van We obtain were Rectified sine wave For te rectified sine wave f ( 1) n1 sin nt F ast ( t) n1 n M ast ( t ) A f ast Bn cos nt n1 Bn ( 1) n 1 n sec (1b) (1c) (1d) F rec ( t) F ( t ) rec f rec, sin t, t t (11a) Te Fourier series of rectified sine wave is f F ( t) rec rec f rec 1 n1 n cos nt 11b) for t and for t Te Melnikov function is M rec ( t ) A frec BnCn sin nt n1 were and C n B n (n1) sec (n) C n can be evaluated nuerically. 1 ( n ) tan sin nd (11c) 11d) (11e) Modulus of sine wave For te odulus of sine wave Its Fourier series is Melnikov function is were Fsw ( t) F sw ( t ) f sw sin( t f 4 f F sw ( t) M sw n cos nt n1 (4n 1) t ( t ) A fsw Bcos B sec 4 ) (1a) (1b) (1c) (1d) 1

6 Prediction of Horsesoe Caos in Duffing-Van III. Effect of Sape Of Periodic Forces On Horsesoe Caos In tis section we copute te Melnikov tresold values for te onset of orsesoe caos and copare it wit nuerical predictions. For te sine wave, te condition for transverse intersection of stable and and unstable anifolds is f sin f 4 p cos 3 Eq. (13) is te necessary condition for te occurrence of orsesoe caos. Te sufficient condition requires te existence of siple zeros of M ( t ). Equality sign in Eq. (13) corresponds to tangential intersection. For oter forces except sine and odulus of sine waves we can not write te sufficient condition for te existence of siple zeros of M ( t ) as like Eq. (13) since M ( t ) is a convergent series. However, te occurrence of ooclinic bifurcation can be studied by easuring te tie τ M elapsed between two successive zeros of M t ). We consider sufficiently large nuber of ters say, 5 ters in te suation of te equation ( for M ( t ). For a fixed value f, t is varied fro to T were If te sign of M t ) reains sae in tis tie interval ten tere is no zero of ) ( (13) T is period of te external force. M t and τ M is assued as ( Figure 3: f M versus n, te nuber of ters in te suation in Eq. (8b ) for α = 1, β =5., p =.4 and ω = 1 wen te syste (1) is driven by te square wave force. Variation in f M converges to a constant value wit increase in n. infinity. τ M is calculated for a range of aplitude of te external force. Te value of f, at wic first tie M ( t ) canges sign and tereby giving finite τ M is te Melnikov tresold value for ooclinic bifurcation.figure 3 sows te plot of f versus n, te nuber of ters in te suation in Eq. (8b) for te square wave force. f converses to a constant value wit increase in n. For n > 1, te variation in f is negligible. Siilar result is found for te oter forces also. In our nuerical calculation of f we fix n =5. Figure 4 sows te variation of 1/τ M versus f for te syste () diven by different periodic forces. Solid curve represents te inverse of first intersection tie, 1/τ M of stable and unstable brances of ooclinic orbits Dased curve corresponds to te orbit W. Horsesoe caos does not occur wen 1/τ M is zero and it occurs in te region were 1/τ M >. Figure 5 sows te plot of te tresold curve for orsesoe caos in te ( f ) paraeter plane. Below te tresold curve no transverse intersection of stable and unstable anifolds of te saddle occurs and above te tresold curve te transverse intersection of te orbits of te saddle occurs. Tis figure clearly illustrates te effect of te various forces. Soot variation of f is found for all te forces. Te tresold curves are non-intersecting except rectified and odules of sine wave forces. Te variation of wit ω is siilar for all te periodic forces. Aong te six forces f is axiu for te rectified sine wave and iniu for te square wave. Tus onset of orsesoe caos can be eiter delayed or advanced by an appropriate coice of periodic force. We verify te analytical predictions by nuerically coputing te stable and unstable anifolds of te saddle. Te unstable anifolds are obtained by nuerically integrating te Eq. () by te fourt-order Runge W. f

7 Prediction of Horsesoe Caos in Duffing-Van Kutta etod in te forward tie for a set of 9 initial conditions cosen around te perturbed saddle point. Te stable anifolds are obtained by integrating te equation of otion in reverse tie. In Fig. 6 we plotted te orbits of te saddle for two values of f for te forces-one for f < f and anoter for f > f. For clarity only part of te anifolds are sown. In te left-side sub plots, te stable and unstable orbits are well separated. In te rigt-side subplots for f > f, we can clearly notice transverse intersection of orbits at certain places. Figure 4: Variation of 1/τ M versus f for te syste (1) driven by te forces (a) sine wave (b) square wave (c) syetric saw-toot wave (d) asyetric saw-toot wave (e) rectified sine wave and (f) odulus of sine wave. Te values of te oter paraeters in Eq. (1) are α = 1, β =5., p =.4 and ω = 1. Continuous curve is for positive sign and dased curve is for negative sign of M(t ). Figure 5: Melnikov tresold curves for orsesoe caos in te (f - ω) plane for te syste (1) driven by te forces (a) sine wave (b) square wave (c) syetric saw-toot wave (d) asyetric saw-toot wave (e) rectified sine wave and (f) odulus of sine wave. Te values of te oter paraeters in Eq. (1) are α = 1, β =5., p =.4 and ω = 1. 3

8 Prediction of Horsesoe Caos in Duffing-Van Figure 6: Nuerically coputed stable and unstable anifolds of te saddle fixed point of te syste (1). Te syste is driven by (a-b) sine wave (c-d) square wave (e-f) syetric saw-toot wave (g-) asyetric sawtoot wave (i-j) rectified sine wave and (k-l) odulus of sine wave. Te values of te oter paraeters in Eq.(1) are α = 1, β =5., p =.4 and ω = 1. Left side subplots are for f < f M wile te rigt side subplots are for f > f M. Figure 6 : continued IV. CONCLUSION In te present paper we considered te Duffing-van der Pol oscillator driven by different periodic forces. We studied te effect of te sape of periodic forces on orsesoe caos. Applying Melnikov analytical etod, we obtained te tresold condition for onset of orsesoe caos, tat is, transverse intersections of stable and unstable brances of ooclinic orbits. Melnikov tresold curve is drawn in a paraeter space. Aong te six forces, f is axiu for te rectified sine wave force and is iniu for te square wave. Analytical prediction of orsesoe caos is found to be in good agreeent wit nuerical siulation for all te forces. Wit te good agreeent obtained between teoretical and nuerical predictions we epasize tat te Melnikov analysis can be successfully used to predict te onset of caos in te presence of weak periodic perturbation. 4

9 Prediction of Horsesoe Caos in Duffing-Van References [1] R. Cacon, Caos and geoetrical resonance in te daped pendulu subjected to periodic pulses, J. Mat. Pys. 38, 1997, [] K. Konisi, Generating caotic beaviours in an oscillator driven by periodic forces, Pys. Lett.A,, 3, 3, -6 [3] Z.M. Ge and W.Y. Leu, Anti-control of caos of two degrees of freedo loud speaker and syncronization of different order systes, Caos, Solitons and Fractals,, (4), [4] A.Y.T. Leung and L. Zengrong, Suppressing caos for soe nonlinear oscillators, Int. J. Bifur. and Caos 14, 4, [5] V.M. Gandiati, K. Murali and S. Rajasekar, Stocastic resonance wit different periodic forces in overdaped two coupled anaronic oscillators, Caos, Solitons and Fractals, 3, 6, [6] V. Ravicandran, V. Cinnatabi and S. Rajasekar, Study of nonescape dynaics in Duffing oscillator wit four different periodic orces, Indian Journal of Pysics, 86(1), 1, [7] V. Ravicandran, V. Cinnatabi and S. Rajasekar, Effect of rectified and odulated sine forces on caos in Duffing oscillator, Indian J. Pys., 83(11), (9) [8] J. Guckeneier and P. Holes, Nonlinear oscillations, Dynaical syste and bifurcations of vector fields (Springer-Verlag, 1983,New York) [9] M. Bartuccelli, P.L. Cristiansen, N.F. Pedersen and M.P. Soersen, Prediction of Caos in Josepson junction by te Melnikov function, Pys. Rev. B, 33, 1986, [1] M. Bartuccelli, P.L. Cristiansen, N.F.Pedersen and M.Salerno, Horsesoe caos in te space-independent double sine-gordon, Wave Motion, 8, 1986, [11] P.J. Holes, A nonlinear oscillator wit a strange attractor, Pilos trans R Soc, A9, 1979, [1] H Cao, Priary resonant optial control for ooclinic bifurcations in single-degree of-freedo nonlinear oscillators, Caos, Solitons and Fractals, 4, 5, [13] H. Cao and G. Cen, Global and local control of ooclinic and eteroclinic bifurcations, Int. J.Bifur. and Caos, 15, 5, [14] B.R.N. Nbendjo, Y. Salissou, P. Woafo, Active control wit delay of catastropic otion and orsesoe caos in a single-well Duffing oscillator, Caos, Solitons and fractals, 3, 5, [15] S.K. Dana, S. Cakraborty and G. Anantakrisna, Hooclinic bifurcation in Cua s circuit, Praana J. of Pys., 64, 5, [16] Y. Tang, F. Yang, G. Cen and T. Zou, Classification of ooclinic tangencies for periodically perturbed systes, Caos, Solitons and Fractals, 8, 6, [17] Z. Zing and R. Wang, Coplex dynaics in Duffing syste wit two external forcings, Caos, Solitons and Fractals, 3, 5, [18] Z. Zing, Z. Yang and T. Jiang, Coplex dynaics in Duffing-Van der Pol equation, Caos, Solitons and Fractal,s 7, 6, [19] S. Rajasekar, Controlling of caos by weak periodic perturbations in Duffing oscillator, Praana J. of Pys., 41, 1993, [] S. Rajasekar, S. Partasaraty, and M. Laksanan, Prediction of orsesoe caos in BVP and DVP oscillators, Caos, Solitons and ractals,, 199, 71-8 [1] W.Y. Liu, W.Q. Zu and Z.L. Huang, Effect of bounded noise on caotic Duffing oscillator under paraetric excitation, Caos, Solitons and Fractals, 1,1, [] Z.H.Liu, W.Q.Zu, Hooclinic bifurcation and caos in siple pendulu under bounded noise excitations, Caos, Solitons and Fractas,, 4, [3] X. Yang, W. Xu and Z. Sun, Effect of bounded noise on te caotic otion of a Duffing-van der Pol oscillator in a Ф 6 potential, 7 6, [4] X. Yang, W. Xu, Z. Sun and T. Fang, Effect of bounded noise on caotic otion of a triple-well potential syste, Caos, Solitons and Fractals, 5, 5, [5] W.Q. Zu and Z.H. Liu, Hooclinic bifurcation and caos in coupled siple pendulu and aronic oscillator under bounded noise excitation, Int.J. Bifur.Caos, 15, 5, 33=43 [6] S. Rajasekar and M. Laksanan, Controlling of caos in BVP oscillator, Int.J. Bifur. Caos,, 199, 1-4 [7] S. Rajasekar and M. Laksanan, Algorits for controlling caotic otion : application for te BVP oscillator, Pysica D 67, 1993, 8-3 [8] K.Wiesenfeld and B.McNaara, Sall-signal aplification in bifurcating dynaical systes, Py. Rev. A, 33, 1986, [9] V. Ravicandran, V. Cinnatabi and S. Rajasekar and C.H. Lai, arxiv:88.46vi [nlin.cd] Aug.8 [3] S.Wiggins Global bifurcations and Caos (Springer, Berlin, 1988) 5