SD oscillator to study transitions from smooth to discontinuous dynamics M. Wiercigroch, Q. Cao, E.E Pavlovskaia, C. Grebogi, and J.M.T. Thompson Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland, UK 1 Introduction An archetypal oscillator whose nonlinearity can be smooth or discontinuous depending on the value of the smoothness parameter α is proposed and studied. As the considered oscillator has properties of both a smooth and discontinuous system (at the limit), poten- tially 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 [1] (see Fig. 1b). Although the springs themselves provide linear restoring resistance, the re- sulting vertical force on the mass is strongly nonlinear because of changes to the geometric configuration. Without loss of generality, the dimensionless equa- tion of motion can be written as, x ′′ +2ξx ′ + x(1 - 1 √ x 2 + α 2 )= f 0 cos ωt, (1) where α = l L , L and l is the equilibrium length and the half distance between the rigid supports, x is the unit mass displacement, f 0 and frequency ω is the forcing amplitude and ξ is the damping coefficient. It is worth noticing here that the transition occurs on (1) from the smooth system to the discontinuous dynamics system when the smoothness parameter α is decreased to 0. 2 Unperturbed system The unperturbed 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 unsta- ble, exhibiting the standard hyperbolic structure and l l m k k F t cos( ) W 0 X L 2l F t cos( ) W 0 (a) (b) 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 Hamiltonian can be written as by letting x ′ = y. H (x, y)= 1 2 y 2 + 1 2 ω 2 0 x 2 - ω 2 0  x 2 + α 2 + ω 2 0 α, (2) Figure 2: Phase portraits; (a) smooth case for α =0.5 and (b) discontinuous case for α = 0. The trajectories can be classified and analysed for both smooth and discontinuous cases, the phase por- traits are plotted for different values of the Hamiltonian H (x, y)= E for α =0.5 and α =0.5 respectively. The dynamic behaviour of double-well is similar to that of the Duffing oscillator [2]. For α = 0, the structure around the point (0, 0) indicates a saddle-like behav- ior. The trajectory mading up of two circles, E = 0, centred at (±1, 0) together with (0, 0) forms special singular homoclinic-like orbits. 1