The SD oscillator and its attractors

The SD oscillator and its attractors Qingjie Cao, Marian Wiercigroch, Ekaterina Pavlovskaia Celso Grebogi and J Michael T Thompson Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, King s College, Aberdeen AB24 3UE, Scotland, UK We propose an archetypal system named SD oscillator and present the corresponding attractors. This systems behaves both smooth and discontinuous depending on the value of the smooth parameter. New dynamics are presented for both regimes, the attractors and the transition. Key words: SD oscillator, Lypunov exponent, chaos sea, resonance 1 Formulation of the oscillator This study is motivated by a growing recent interest in the non-smooth dynamics where various physical systems have been studied. Examples include problems from mechanical and civil engineering [1, 2], electronics [3, 4], control [5, 6], computer graphics [7, 8, 9], biology [10, 11], and others. Although some theoretical foundations have been laid in the work by Filippov [12], Feigin [13], Kunze [14], Peterka [15], Shaw and Holmes [16] and Nordmark [17], there is a large disparity between development and understanding of smooth and discontinuous (non-smooth) systems. We present an archetypal oscillator whose nonlinearity can be smooth or discontinuous depending on the value of the smoothness parameter α is proposed and studied, see Cao and et. al. [18, 19]. As the considered oscillator has properties of both a smooth and discontinuous system (at the limit), potentially a wealth of knowledge can be drawn from the well developed theory of continuous dynamics. Physically (as shown in Fig. 1a) this oscillator is similar to a snap-through truss system. It comprises a mass, m, linked by a pair of inclined elastic springs which are capable of resisting both tension and compression; each spring of stiffness k is pinned to a rigid support. This model is inspired on the elastic arch described by Thompson and Hunt in [20] (see Fig. 1b). Although the springs themselves provide linear restoring resistance, the resulting vertical force on the mass is strongly nonlinear because of changes to the geometric configuration. The equation of motion can be written as, mẍ + 2kX(1 L ) = 0, (1) X2 + l2 where L is the equilibrium length, X is the mass displacement and l is the half distance between the rigid supports. Now suppose system (1) is perturbed by a viscous damping and an external harmonic excitation of amplitude F 0 and frequency Ω. This leads to the following system, L mẍ + δẋ + 2kX(1 X2 + l ) = F 2 0 cosωt. (2) System (1) can be made dimensionless by letting ω 2 0 = 2k m, x = X L and α = l L 0, ẍ + ω 2 0x(1 1 ) = 0. (3) x2 + α2 1

(a) (b) F cos( t) 0 L m F cos( t) 0 X k k l l 2l Figure 1: (a) The dynamical model in a the form of a nonlinear oscillator, where a mass is supported by a pair of springs pinned to rigid supports and (b) a simple elastic arch. The smoothness parameter α defines not only the geometry of the oscillator (Fig.1a) but also has physical meaning. For α > 1 the system represents a pretentioned discrete elastic string, while if α = 0, the model corresponds to an oscillating mass supported by two parallel vertical springs. δ 2mω 0 Again system (2) can be written in a dimensionless form by letting τ = ω 0 t, f 0 = F0 2kL, ξ = and ω = Ω ω 0, x + 2ξx 1 + x(1 x2 + α ) = f 2 0 cosωτ, (4) where denotes differentiation with respect to τ. The nonlinear restoring force F(x) = ω 2 0x(1 1 x2 + α 2 ) is plotted for ω 0 = 1 in Fig. 2a for different values of parameter α. The solid line represents the discontinuous case α = 0, the dotted, the dashed dotted and the dashed lines mark the smooth cases, for α = 0.01, 0.5, and 0.75 respectively. 2 Unperturbed Oscillator When α = 0 system (3) can be written in the following form, ẍ + ω 2 0(x Sign(x)) = 0. (5) It is worth reiterating here that the discontinuous dynamics is obtained by decreasing of the smoothness parameter α to 0. To examine the influence of parameter α on the dynamics of (3) we construct the bifurcation diagram, depicted in Fig. 2b. The system undergoes a supercritical pitchfork bifurcation at α = 1 where the stable branch x = 0 bifurcates into two stable branches withat x = ± 1 α 2. The stationary x = 0 state is now unstable, exhibiting the standard hyperbolic structure. The Hamiltonian for system (3) can be written as H(x, y) = 1 2 y2 + 1 2 ω2 0x 2 ω 2 0 x2 + α 2 + ω 2 0α, (6) where ẋ = y. With the help of the Hamiltonian function (6), the trajectories can be classified and analysed. For both continuous and discontinuous cases, the phase portraits of systems (3) and (5) are plotted for different values of the Hamiltonian H(x, y) = E. For instance, for the smooth nonlinearity, α = 0.5, the dynamic behaviour of double-well is similar to that of the Duffing oscillator [21], shown in Fig. 3a. For α = 0, the behaviour is singular, as shown in Fig. 3b: the orbits for E > 0 are 2

Figure 2: (a) Nonlinear restoring force, F(x); solid line marks the discontinuous case for α = 0, dotted, dash-dotted and dashed curves are the smooth cases for α = 0.1, 0.5 and α = 0.75 respectively; (b) bifurcation diagram of system (3). Figure 3: Phase portraits; (a) smooth case for α = 0.5 and (b) discontinuous case for α = 0. comprised of two large segments of circles with their centres located at ( 1, 0) and (1, 0) connected at x = 0. The case of E < 0 is represented by two families of circles. It is the most interesting that E = 0 for the discontinuous system (5) is made up of two circles centred at (±1, 0) connecting at the singular point (0, 0), which form special singular homocliniclike orbits. The structure around the point indicates a saddle-like behavior. The hyperbolicity at origin (0, 0) is lost due to the the tangency of the stable and unstable eigen directions. This isolated singularity has neither eigenvalue nor eigenvector. The pair of circles excluding the point (0, 0) are not the manifolds of the singularity, but the flow along these circles approaches the point as x 0, and it will be trapped by the singularity. The solution of the special homoclinic like orbits can be formulated as where (x ± (t), y ± (t)) = (±1 ± cosω 0 t, sinω 0 t). Γ = {(x ± (t), y ± (t)), t ( π ω 0, π ω 0 )} {(0, 0)}, (7) 3 Perturbed attractors: from smooth to discontinuous Numerical simulations have been carried out for system (4), assuming f 0 = 0.8, ξ = 0.01 2 and ω = 0.75 2. Figure 4 shows bifurcation diagrams constructed for x sampled stroboscopically at 3

phase zero versus control parameter α as α decreases from 1 to 0 (Fig. 4a) and α increases from 0 to 0.3 and from 0.5 to 1 and decreases from 0.5 to 0.3 (Fig. 4b). The system has co-existing periodic attractors and a strange chaotic attractor for α > 0. However, for α = 0 the system exhibits chaotic and periodic solutions with unusual high periods. These behaviours can be controlled by the strength of the damping. The main feature of these attractors is their topological similarity since they are associated with the hyperbolic stationary state. This similarity also holds for α > 0. The chaotic attractor can become a chaotic sea with islands representing quasi periodic behaviour for α = 0 and ξ = 0. Figure 4: (Color online) Bifurcation diagrams for x versus α constructed for: (a) Decreasing from α = 1 and following the attractors starting with period 1 (black), period 3 (green), 5 (blue) and 7 (red) respectively, (b) first, increasing from α = 0 to 0.3 following the attractor starting with the initial condition (1,0). Second, decreasing α from 0.5 to 0.3 and increasing from α from 0.5 to 1, following the attractors starting with two period 4 solutions respectively. Figure 4a also shows the co-existing periodic solution of period one, three, five and seven at α = 1 (orbits are shown in Fig. 5a) and their bifurcations under decreasing α. Other co-existing periodic solutions are found for α (0, 0.1), as shown in Fig. 4a and Fig. 4b. These two co-existing period 2 solutions are symmetrical and the orbits for α = 0.05 are presented in Fig. 5b. For α (0.41, 0.52) a chaotic attractor co-exists with two period 4 solutions and the corresponding trajectories and the chaotic attractor are shown in Fig. 5c and 5d, respectively. In addition to the co-existence of different attractors, the system exhibits chaotic transients throughout. This behaviour can be characterized by a chaotic saddles, see [22, 23, 24]. The transient and the final periodic attractor are shown in Figs. 6b and 6c for α = 0.01 and α = 0.001 respectively. The set of Poincaré maps shown in Fig. 6 for α = 0.1, 0.01, 0.001 and 0 shows the topological similarity now associated with the discontinuity at the origin. The co-existing period 2 solutions persist for α = 0 and the chaotic saddle becomes a chaotic attractor for α = 0 as shown in Fig. 6d. The behaviour of this chaotic attractor can be controlled by the damping ratio ξ. Other attractors are presented in Fig. 7a for ξ = 0.005, Fig. 7b for ξ = 0.0125 and Fig. 7c for ξ = 0.025. A high period periodic attractor is shown in Fig. 7d having period 23 for ξ = 0.028. For both α = 0 and ξ = 0, a chaotic sea and quasi-periodic behaviour can also be inferred via a semi-analytical method, see [25]. Figure 8 shows the chaotic sea together with a pair of quasi-period 2 solutions, a pair of quasi-period 5 solutions, the quasi-period 9 and the quasi-period 11 solutions, and also the corresponding series of fixed points. The largest Lyapunov exponents for all the chaotic attractors presented in this paper have been calculated using the chaos synchronization method, see [26] for instance, as shown in the captions for the corresponding figures. 4

Figure 5: Co-existence of periodic motions: (a) Period 1 (thick line), period 3 (dash line), 5 (dotted line) and 7 (thin line) respectively for α = 1.0, (b) a pair of period two solutions for α = 0.05, (c) and (d) represent the pair of period four and the chaotic attractor for α = 0.5 with the largest Lyapunov exponent 0.1065. 4 Conclusion and discussions A smooth or discontinuous nonlinear oscillator has been proposed. The nonlinearity can be smooth or discontinuous depending on the value of the smoothness parameter α. A mathematical model of the oscillator has been developed and investigated. It has been found that the unperturbed smooth system has the double-well characteristic which is similar to the Duffing oscillator. The discontinuous system exhibits a saddle-like singularity connecting two circles forming the homoclinic-like orbits. Under perturbation, for large α, the system exhibits complex co-existence of periodic attractors and also periodic solutions with a strange chaotic attractor which disappears at α 0.1. Decreasing α further creates chaotic transients around a chaotic saddle, which leads to periodic orbits as time increases. This chaotic transient becomes a chaotic attractor for α = 0. The attractor can deform in shape or bifurcate to a high period periodic attractor depending on the strength of damping, or even can become a chaotic sea with islands of quasi-periodic trajectories for ξ = 0. The presented oscillator is being actively studied by the authors in two main directions. Firstly, the peculiar properties at the limit of α = 0 are being analysed in more details to better understand the bifurcation structures under varying damping and external forcing. This research relates to the studies of preloaded oscillators, e.g. [2, 27]. The second pursued direction is to develop analytical measures (e.g. construction of Melnikovian) to predict the border of chaos. This research bears a significant analogy to the predictions made for the Duffing oscillator, see e.g [28, 29]. 5

Figure 6: (a) Chaotic attractor for α = 0.1 with the largest Lyapunov exponent 0.0812, (b) chaotic saddle leading to period 2 solution for α = 0.01 and (c) chaotic saddle leading to period 2 solution for α = 0.001 (d) chaos for α = 0 with the largest Lyapunov exponent 0.0480. 5 Acknowledgement The first author acknowledges the financial support from EPSRC under Grant No.GR/R85556, and from the Scottish Enterprize. References [1] Pavlovskaia, E.E., Wiercigroch, M., Woo, K.-C. and Rodger, A.A. Meccanica 38, 85-97 (2003). [2] Pavlovskaia, E., Karpenko, E.V. and Wiercigroch, M. J. Sound Vib. 276, 361-379 (2004). [3] Banerjee, S. and Chakrabarty, K. IEEE Trans. on Power Electronics, 13(2), 252-260 (1998). [4] Banerjee, S., Ott, E, Yorke, J.A. and Yuan, G.H. Power Electronics Specialists Conference, 1337-1344 (1997). [5] Slotine, J.J., Sastry, S.S. International Journal of Control 38(2), 465-492 (1983). [6] Richard, P.Y., Cormerais, H., Buisson, J. Nonlinear Analysis, Theory, Methods and Applications 65(9), 1751-1772 (2006). [7] Luo, A., Burkhardt, H. International Journal of Computer Vision 15(3), 171-188 (1995). 6

Figure 7: Poincaré sections for f 0 = 0.8, ω = 0.75 2, α = 0; (a) chaotic attractor for ξ = 0.005 with the largest Lyapunov exponent 0.0525, (b) chaotic motion for ξ = 0.0125 with the largest Lyapunov exponent 0.0505, (c) chaotic motion for ξ = 0.025 with the largest Lyapunov exponent 0.0470 and (d) high periodic motion of period 23 for ξ = 0.028. [8] Nishita, Tomoyuki, Sederberg, Thomas W., Kakimoto, Masanori Computer Graphics (ACM) 24(4), 337-345 (1990) [9] Gu, X., He, Y., Qin, H. Graphical Models 68(3), 237-254 (2006). [10] Kribs-Zaleta, C.M. Mathematical Biosciences 192(2), 137-152 (2004). [11] Drulhe, S., Ferrari-Trecate, G., De Jong, H., Viari, A. Lecture Notes in Computer Science 3927, 184-199 (2006). [12] Filippov, A. F. Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers, (1988). [13] Feigin, M.I. Forced vibrations of nonlinear systems with discontinuities, Nauka: Moscow (1994). In Russian. [14] Kunze, M. Non-Smooth Dynamical Systems, Springer-Verlag: Berlin Heidelberg New York (2000). [15] Peterka, F. Introduction to vibration of mechanical systems with internal impacts, Academia Publishing House: Prague (1981). In Czech. [16] Shaw, S.W. and Holmes, P.J. J. Sound Vib. 90(1), 129-155 (1983). 7

4 y 2 0-2 -4-6 -4-2 0 2 x Figure 8: Poincare section for f0 = 0.8, ω = 0.75 2, α = 0: chaotic sea, the series of resonant fixed points (periodic) and the quasi-periodic (islands) trajectories for ξ = 0. [17] Nordmark, A.B. J. Sound Vib. 145, 279-297 (1991). [18] Q. Cao, M. Wiercigroch, E E Pavlovskaia, C Grebogi and JMT Thompson, Phys. Rev. E 74, 046218 (2006). [19] Q. Cao, M. Wiercigroch, E E Pavlovskaia, C Grebogi and JMT Thompson, Phil. Roy. Ser. A (2007 in press). [20] Thompson, J.M.T., and Hunt, G.W., A general theory of elastic stability, John Wiley & Sons, London, (1973). [21] Guckenheimer, J. and Holmes, P., Nonlinear Oscillation, Dynamical System and Bifurcation of Vector Fields, Spinger-Verlag, New York, (1983). [22] Lai, Y.C. and Winslow R.L., Phys. Rev. Lett. 74, 5208 (1995). [23] Tel T., Phys. Lett. A 119, 65 (1986). [24] Grebogi C., Ott E. and Yorke J.A., Physica D 7,181 (1983). [25] Pavlovskaia, E. and Wiercigroch, M., Chaos Solitons & Fractals, 19, 151 (2004). [26] Stefanski, A. and Kapitaniak, T., Chaos Solitons & Fractals, 15, 233 (2003). [27] Peterka, F. Chaos, Solitons & Fractals, 18(1), 79-88 (2003). [28] Ueda,Y., Yoshida, S., Stewart, H.B. & Thompson, J.M.T. Phil. Trans. R. Soc. Lond., A 332, 169-186 (1990). [29] Lansbury, A.N., Thompson, J.M.T. & Stewart, H.B. Int. J. Bifn & Chaos 2, 505-532 (1992). 8