PERIODICALLY FORCED CHAOTIC SYSTEM WITH SIGNUM NONLINEARITY

Transcription

1 International Journal of Bifurcation and Chaos, Vol., No. 5 () c World Scientific Publishing Compan DOI:.4/S PERIODICALLY FORCED CHAOTIC SYSTEM WITH SIGNUM NONLINEARITY KEHUI SUN,, and J. C. SPROTT, School of Phsics Science and Technolog, Central South Universit, Changsha 483, China Department of Phsics, Universit of Wisconsin Madison, WI 5376, USA kehui@csu.edu.cn sprott@phsics.wisc.edu Received Jul, 9; Revised August 3, 9 A sinusoidall-driven sstem with a simple signum nonlinearit term is investigated through an analtical analsis as well as dnamic simulation. To obtain the correct Lapunov eponents, the signum function is replaced b a sharpl varing continuous hperbolic tangent function. B phase portraits, Poincaré sections and bifurcation diagrams, the rich dnamic behaviors of this sstem are demonstrated, such as an onion-like strange attractor, pitchfork and attractor merging bifurcations, period-doubling routes to chaos, and chaotic transients in the case of small damping. Moreover, the chaos persists as the damping is reduced to zero. Kewords: Chaos; nonautonomous sstems; differential equations; bifurcations; signum function.. Introduction It has been known that man nonlinear functions can generate chaos. Some of the earliest eamples of chaos occur in periodicall forced nonlinear oscillators, which have the form ẍ + f(ẋ, ) =F sin ωt, where the function f(ẋ, ), contains at least one nonlinearit in the damping (ẋ) termortherestoring force () term. For eample, the van der Pol equation has an in the damping term [van der Pol, 96], while the Raleigh differential equation [Birkhoff & Rota, 978] is similar to the van der Pol equation but has an ẋ nonlinearit in the damping. More common and more widel studied is the sstem in which the damping is linear, but the restoring force contains a cubic nonlinearit, such as the Duffing oscillator [Duffing, 98], the Ueda oscillator [Ueda, 979], Duffing s two-well oscillator [Moon & Holmes, 979], as well as a variant of this function [Bonatto et al., 8] without the linear term in. Several hbrid chaotic forced oscillators have been studied b combining two cases such as the Raleigh Duffing oscillator [Haashi et al., 97] and the Duffing van der Pol oscillator [Ueda, 99]. Additionall, quadratic functions in the restoring force term can also suffice to give chaos in a forced oscillator [Tang et al., ]. Other sstems in which f(ẋ, ) is a more complicated nonlinear function have been studied [Scheffczk et al., 99]. A simpler nonlinearit is a piecewise linear function involving a single absolute value. Piecewise linear oscillators with more than two linear regions that produce chaos when periodicall forced have been etensivel studied, for eample, using Chua s diode [Murali et al., 994; Srinivasan, 8], and using damping control for a chaotic impact oscillator [Souza et al., 7]. Whereas the absolute value is a continuous piecewise-linear function, discontinuous functions sometimes arise in models of real applications. An eample is a harmonic oscillator 499

2 5 K. Sun & J. C. Sprott with Coulomb damping [Feen, 99]. For a forced oscillator, the simplest such case would be f(ẋ, ) = ẋ +sgn(), which is arguabl simpler than the so-called simplest sinusoidall-forced chaotic sstem studied in [Gottlieb & Sprott, ] which involves an 3 nonlinearit. Here, we are interested in the case where the onl nonlinearit is the signum function in the restoring force term. It is well known that the signum function describes a class of simple discontinuous switching structures, or one tpe of the most important nonsmooth structures. On one hand, a simple nonsmooth structure ma easil create comple phenomena, and therefore, nonsmooth technique has graduall become a powerful strateg for generating comple dnamics in simple autonomous sstems [Chen et al., 6]. On the other hand, switching schemes usuall can be simpl structured and easil designed with costeffective electronic circuit realizations. The plan of the paper is as follows. In Sec., we present the forced oscillator and its attractor. In Sec. 3, the sstem equations are briefl investigated through analtical solutions in the respective linear regions. In Sec. 4, we describe the dnamics and bifurcations of the forced sstem. Finall, we summarize the results and indicate future directions.. The Periodicall Forced Oscillator Consider the following nonautonomous differential equation ẍ + aẋ + b sgn() =F sin(ωt), () where, a, b, F, ω are positive constants, and sgn() is the signum function which is either + or depending on whether its argument is positive or negative, respectivel. This sstem resembles the Duffing oscillator ecept that the nonlinearit is in the low order term rather than in the high order term. For simplicit, we shall set the parameter ω = as described in [Thompson, 997], and regard a, b, and F as the control parameters. The onion-like strange attractor is observed b choosing appropriate parameters as shown in Fig. with the largest Lapunov eponent of.4, and the Kaplan Yorke dimension D KY =.34. It is instructive to eamine the solution of Eq. () in phase space. For this purpose, rewrite Eq. () withω =intermsof and as {ẋ =, () ẏ = a + b sgn()+f sin(t) Fig.. The onion-like attractor of Eq. () with a =.5, b =,F =., and ω =. If we let z = t, then the nonautonomous sstem is converted into an autonomous one, and the twodimensional sstem becomes three-dimensional ẋ = ẏ = a + b sgn()+f sin(z). (3) ż = 3. Analtical Stud 3.. Eplicit analtical solution Let us briefl investigate this sstem equation b an analtical solution in each of the two linear regions. Let D be the subspace where is positive and D be the subspace where is negative. D and D are represented as D = {(t,, ẋ ) >}, (4) D = {(t,, ẋ ) <}. In each of the regions D and D,Eq.() is linear when F =. The stabilit depends on the eigenvalues of the equation λ + aλ =,whichare found to be ( a ± a +4)/. Consequentl the equilibrium points in both the D and D regions are saddle points, either a > ora <. Then Eq. () can be represented as a single second-order inhomogeneous linear differential equation in each of the regions D and D as and ẍ + aẋ = F sin(ωt) b >, (5a) ẍ + aẋ = F sin(ωt)+b <. (5b) Equation () can be eplicitl integrated in terms of elementar functions in each of the regions D and

3 Periodicall Forced Chaotic Sstem with Signum Nonlinearit 5 D and matched across the boundaries to obtain the full solution as shown below. The general solution to Eq. (5a) can be easil written as (t) =Ae λt + Be λt + E sin ωt + E cos ωt b, (6) where A and B are constants of integration and E = F ( + ω ) [a ω ( + ω ) ] (7) Faω E = [a ω ( + ω ) ]. (8) Then ẋ(t) is obtained from Eq. (6) as ẋ(t) =Aλ e λ t + Bλ e λ t + E ω cos ωt E ω sin ωt. (9) From Eq. (), it follows that =ẋ(t). () The constants A and B in the above equations can be evaluated b solving both Eqs. (6) and(9) at a suitable initial state with and as initial conditions at time t = t. The constants A and B can be obtained from Eqs. (6) and(9) as A = e λ t λ λ [ λ +(E λ E ω)cosωt +(E ω + E λ )sinωt bλ ], () B = e λ t λ λ [ λ +(E λ E ω)cosωt +(E ω + E λ )sinωt bλ ], () where represent D and D regions respectivel. 3.. The associated Hamiltonian function Using the kinetic energ function / and the potential energ function /+b for sstem (), we can define the sstem Hamiltonian function as H(, ) = + b. (3) This function is piecewise defined, and is continuous but not differentiable at (and onl at) =. On the level set H(, ) =with =,ithas three zeros: =and, = ±b as shown in Fig.. It shows that sstem () is similar to the Duffing sstem for the case with the restoring force of the form.4 3 [Sprott, 3]. H(,) o Duffing Sstem () o - Fig.. The Hamiltonian function of sstem () andduffing oscillator with one potential well. 4. Dnamical Behaviors 4.. Calculating the largest Lapunov eponent Lapunov eponents are the best indicators to categorize the different classes of nonlinear phenomena. A positive Lapunov eponent confirms chaos. If the sstem equations are known, algorithms [Parker & Chua, 989] can readil be applied to compute the largest or all of the eponents. Alternativel, the method of embedded dimensions ma be applied to a time series resulting from simulations and eperiments to estimate their eponents [Wolf et al., 985]. But these approaches do not work satisfactoril for nonsmooth sstems, i.e. those where the vector field is not continuousl differentiable. For eample, if parameters a =.4, b =,F =., ω =, and initial condition [.765,.8, 5] in Eq. (), then the sstem is a limit ccle as shown in Fig. 3(a). But the calculated largest Lapunov eponent is.54, which is evidentl incorrect, and is an eample of the chaotic limit ccle parado as described in [Grantham & Lee, 993]. To calculate the largest Lapunov eponent of sstem () correctl, the signum function can be replaced b continuous equivalents [Gans, 995], sgn() tanh(n), where n is a large constant. The question is how to determine a suitable value for n. Wecalculate the largest Lapunov eponent versus n for states of limit ccle [Fig. 3(a)] and chaos [Fig. 3(c)] in sstem () respectivel, and the curves are presented in Figs. 3(b) and 3(d). As shown in Figs. 3(b) and 3(d), if n is a ver small number, then the hperbolic o

4 5 K. Sun & J. C. Sprott.8 Maimum LE log (n) (a) (b) - Maimum LE (c) 3 log (n) Fig. 3. Phase portraits and their maimum Lapunov eponent versus n. (a) Limit ccle. (b) Maimum Lapunov eponent for limit ccle. (c) Chaos. (d) Maimum Lapunov eponent for chaos. (d) tangent is far from the signum function. If n is a much larger number, the hperbolic tangent is nearl equal to the signum function. The results are unreliable in these limits. So, for sstem (), n can be chosen from 3 to, in which range the calculated Lapunov eponent is independent of n. After the signum function is replaced b tanh with n =, the largest Lapunov eponent of the limit ccle shown in Fig. 3(a) becomes Bifurcations and route to chaos To stud the dnamics of sstem (), three cases were considered as follows. () Fi b =andf =., and var a. The sstem is calculated numericall with a [,.45] for an increment of a =.5. The bifurcation diagram is shown in Fig. 4(a). A pitchfork bifurcation occurs at a =.365, and two attractors, denoted in blue and red, respectivel, coeist when the parameter is less than.365. An attractor-merging crisis bifurcation occurs in this case. The attractors grow larger as a decreases, until the collide at a =.7. Thus two separate attractors merge into one and remain bounded until a =.4. If the orbit is initiall confined to one of the attractors, the region occupied b it suddenl doubles in size at the crisis point. With an increase in the value of the parameter a, a reverse period-doubling phenomenon is observed. The largest Lapunov eponent of the sstem () was computed numericall for a [.5,.4] in increments of a =. and is shown in Fig. 4(d). It is evident that chaos eists for.5 a..

5 Periodicall Forced Chaotic Sstem with Signum Nonlinearit ma.8.6 ma a b (a) (b).4.. b ma Maimum LE. F a (c) F (d) Parameters Fig. 4. Bifurcation and largest LE diagrams with different control parameters for Eq. (). (a) b =, F =.. (b) a =.5, F =.. (c) a =,b =. (d) Diagram of largest LE..4 () Fi a =.5 and F =., and var b. The sstem is calculated numericall with b [.95,.35] for an increment of b =.5. The bifurcation diagram is shown in Fig. 4(b). A pitchfork bifurcation occurs at b =.4, and two attractors, denoted in blue and red, respectivel, coeist when the parameter is less than.4. An attractormerging crisis bifurcation occurs in this case. The attractors grow larger as b decreases, until the collide at b =.3. Thus two separate attractors merge into one and remain bounded until b =.99. If the orbit is initiall confined to one of the attractors, the region occupied b it suddenl doubles in size at the crisis point. For an increase in the value of the parameter b, a reverse period-doubling phenomenon is also observed. The largest Lapunov eponent of the sstem () was computed numericall for b [,.4] in increments of b =. and is shown in Fig. 4(d). It is evident that chaos eists for b.5. (3) Fi a =andb =,andvarf. The sstem is calculated numericall with F [.8,.] for an increment of F =.5. The bifurcation diagram is shown in Fig. 4(c). A pitchfork bifurcation occurs at about F =.855, and two attractors, denoted in blue and red, respectivel, coeist when the parameter is larger than.855. An attractor-merging crisis bifurcation occurs in this case. The attractors grow larger as F increases, until the collide at F =.4. Thus two separate attractors merge into one and remain bounded until F =.6. If the orbit is

6 54 K. Sun & J. C. Sprott initiall confined to one of the attractors, the region occupied b it suddenl doubles in size at the crisis point. With an increase in the value of parameter F, a period-doubling phenomenon is observed. The largest Lapunov eponent of the sstem () was computed numericall for F [.8,.6] in increments of F =. and is shown in Fig. 4(d). It is evident that chaos eists for.5 F (a ) (a ) (b ) (b ) (c ) (c ) Fig. 5. Phase portraits (left) and Poincaré sections (right) of Eq. (). (a) F =.65, one T -periodic solution. (b) F =.95, two T -periodic solutions. (c) F =,twot -periodic solutions. (d) F =., two 4T -periodic solutions. (e) F =.45, chaotic solution.

7 Periodicall Forced Chaotic Sstem with Signum Nonlinearit (d ) (d ) (e ) (e ) Fig. 5. (Continued) From the three cases above, it can be concluded that selection of appropriate values for the sstem control parameters can be used to suppress or generate chaos, and that the route to chaos is the same for each of the three parameters. To observe the route to chaos, the phase portraits and Poincaré sections are presented in Figs. 5(a) 5(e) b fiing a = b = but changing the control parameter F.ThePoincaré section is in the plane of z mod π =. Increasing the parameter F, we first observe one limit ccle in the phase portrait as seen in Figs. 5(a ), 5(b ) and one point in the Poincaré sections in Figs. 5(a ), 5(b ), then twocclesasshowninfig.5(c )andtwopointsin Fig. 5(c ), four ccles in Fig. 5(d ), four points in Fig. 5(d ), and finall an infinite number of ccles as shown in Figs. 5(e )and5(e ). The route to chaos is b period-doubling bifurcations. At the same time, when parameter F increases from.65 to.45, a pitchfork bifurcation and a pair of attractors are born as shown in Fig. 5(b), and then the two coeisting attractors epand as showninfigs.5(c)and5(d)andfinallmergeb an attractor-merging crisis bifurcation at the onset of chaos as shown in Fig. 5(e) Chaotic transients with small damping A phenomenon ver common in dnamical sstems is that the seem chaotic during some transient period, but finall fall onto a periodic attractor. This phenomenon is said to be a chaotic transient, which has been reported for the standard Lorenz sstem [Yorke & Yorke, 979]. Superlong chaotic transients (referred to as supertransients ) have been observed in nonlinear circuit eperiments [Zhu et al., ]. In such a case, trajectories starting from random initial conditions wander chaoticall for an arbitraril long time before settling into a final attractor that is usuall nonchaotic, and the average transient lifetime, which is a function of

8 56 K. Sun & J. C. Sprott ma log (t) (a) (b) Fig. 6. Bifurcation diagram and final attractor for Eq. () witha =.5, b = andf =.. (a) Chaotic transient. (b) The final attractor. the damping parameter and eternal driving force, was studied in [Shen et al., 997]. Such a situation also occurs in sstem (). Choosing initial conditions [,,z ] = [.4589,.46, ] and a small damping parameter a =.5, the chaotic transient is shown in Fig. 6(a). It is evident that the trajector is chaotic with man periodic windows until t>6 6, whereupon it finall becomes a limit ccle as shown in Fig. 6(b), and thereafter remains so. It is observed that the transient lifetime is a function of parameters a and F,andwhen a, the chaotic transient has an infinite lifetime The undamped oscillator If we allow the damping to be zero (a = ) in Eq. (), then two conservative sstems are given b ẍ + b sgn() =F sin ωt, (4) ẍ + b sgn() =F sin ωt. (5) Equation (5) with f() = sgn() is especiall interesting because it is arguabl simpler than the so-called simplest sinusoidall-forced chaotic sstem [Gottlieb & Sprott, ] which has a cubic nonlinearit. In fact, it is the special case of the simplest one with p = in [Gottlieb & Sprott, ]. A trajector of the conservative sstem (5) is shown in Fig. 7(a) for initial conditions [,,z ]= [,., ] and parameters b = F = ω =. The largest Lapunov eponent is.465 in this case. A Poincaré section in the -plane for t mod π =.5π for various initial values of in the range (a) Fig. 7. Trajector (a) and Poincaré section (b) of the conservative sstem in Eq. (5). (b)

9 Periodicall Forced Chaotic Sstem with Signum Nonlinearit 57 < < 9and =,z =.5π is shown in Fig. 7(b). The chaotic sea is small, as is tpical for such periodicall forced sstems [Sprott, 3]. 5. Conclusions This paper has reported and analzed a simple periodicall-forced sstem with a nonlinear function sgn(), which can generate chaos for appropriate parameters. The dnamical properties of this chaotic sstem have been analzed, including the Lapunov eponents, bifurcations, routes to chaos and chaotic transients. The results show it has abundant dnamic behaviors. Therefore, research on dnamics of nonlinear sstems with a signum nonlinearit has great theoretical significance and practical applications (such as liquid miing and secure communications). It is worthwhile to eplore further simple sstems with the signum function to create chaos for different demands. Acknowledgments This work was supported b the China Scholarship Council (No. 6A39), the National Nature Science Foundation of People s Republic of China (Grant No. 6674), and the National Science Foundation for Post-doctoral Scientists of People s Republic of China (Grant No ). We are grateful for discussions with Prof. George Rowlands. References Birkhoff, G. & Rota, G. [978] Ordinar Differential Equations, 3rd edition (John Wile and Sons: NY). Bonatto, C., Gallas, J. A. C. & Ueda, Y. [8] Chaotic phase similarities and recurrences in a dampeddriven Duffing oscillator, Phs. Rev. E 77, Chen, Q. F., Hong, Y. G. & Chen, G. R. [6] Chaotic behaviors and toroidal/spherical attractors generated b discontinuous dnamics, Phsica A 37, Duffing, G. [98] Erzwungene Schwingungen Bei Veranderlicher Eigenfrequenz (Vieweg: Braunschweig). Feen, H. [99] A nonsmooth coulomb friction oscillator, Phsica D 59, Gans, R. F. [995] When is cutting chaotic? J. Sound Vibr. 88, Gottlieb, H. P. W. & Sprott, J. C. [] Simplest driven conservative chaotic oscillator, Phs. Lett. A 9, Grantham, W. J. & Lee, B. [993] A chaotic limit ccle parado, Dn. Contr. 3, Haashi, C., Ueda, Y., Akamatsu, N. & Itakura, H. [97] On the behavior of self-oscillator sstems with eternal force, Trans. Instit. Electron. Commun. Engin. Japan 53-A, Moon, F. C. & Holmes, P. J. [979] A magetoelastic strange attractor, J. Sound Vibr. 65, Murali, K., Lakshmanan, M. & Chua, L. O. [994] Bifurcation and chaos in the simplest dissipative nonautonomous circuit, Int. J. Bifurcation and Chaos 4, Parker, T. S. & Chua, L. O. [989] Numerical Algorithms for Chaotic Sstems (Springer-Verlag). Scheffczk, C., Parlitz, U., Kurz, T., Knop, W. & Lauterborn, W. [99] Comparison of bifurcation structures of driven dissipative nonlinear oscillators, Phs. Rev. A 43, Shen, J., Yin, H. & Dai, J. [997] Dnamical behavior, transient chaos, and riddled basins of two charged particles in a Paul trap, Phs. Rev. A 55, Souza, S. L. T., Caldas, I. L. & Viana, R. L. [7] Damping control law for a chaotic impact oscillator, Chaos Solit. Fract. 3, Sprott, J. C. [3] Chaos and Time-Series Analsis (Oford Universit Press, Oford). Srinivasan, K. [8] Doubling bifurcation route to chaos in periodicall pulsed Murali Lakshmanan Chua (MLC) circuit, Int. J. Bifurcation and Chaos 8, Tang, K. S., Man, K. F., Zhong, G. Q. & Chen, G. R. [] Generating chaos via mid mid, IEEE Trans. CAS I 48, Thompson, J. M. T. [997] Nonlinear Mathematics and Its Applications, ed. Aston, P. J. (Cambridge Universit Press, Cambridge). Ueda, Y. [979] Randoml transitional phenomena in the sstem governed b Duffing s equation, J. Stat. Phs., Ueda, Y. [99] The Road to Chaos (Aerial Press, Santa Cruz). van der Pol, B. [96] On relaation-oscillations, The London, Edinburgh, and Dublin Philos. Mag. J. Sci. Ser. 7, Wolf, A., Swift, J., Swinne, H. & Vastano, J. [985] Determining Lapunov eponents from a time series, Phsica D 6, Yorke, J. A. & Yorke, E. D. [979] Metastable chaos: The transition to sustained chaotic oscillations in a model of Lorenz, J. Stat. Phs., Zhu, L., Raghu, A. & Lai, Y.-C. [] Eperimental observation of superpersistent chaotic transients, Phs. Rev. Lett. 86, 47 4.