Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing oscillators S. Sabarathinam Research Scholar Under the guidance of Prof. K. Thamilmaran Centre for Nonlinear Dynamics School of Physics Bharathidasan university Tiruchirappalli-620 024 Collaborators Prof. T. Kapitaniak , Dr P. Perlikowski A. Stefanski, L. Borkowski, P. Brzeski. Division of Dynamics Lodz University of Technology Lodz, Poland

Plan of talk Introduction Transient Chaos Model Stability Analysis Numerical Analysis Experimental Setup Experimental Results Conclusion

Chaos Chaos is characterized by Lyapunov Exponent Spectrum 0 - 1 Test Chaos is a phenomena of appearance of bounded, non periodic evolution, in a completely deterministic nonlinear dynamical system with high sensitivity dependence on initial condition. Chaos is characterized by Lyapunov Exponent Spectrum 0 - 1 Test FFT spectrum …

A question is, then, can chaos be transient? Transient Chaos The system trajectory evolves on a strange chaotic repeller (chaotic saddle) for significantly long period of time, t* say and afterward, for t > t*, converges to the regular attractor. The value of t* will of course vary from trajectory to trajectory and may be very sensitive to the initial conditions, but representative average of t* can be used to describe the phenomena of transient chaos. t* t

Transient Chaos Arises….. • Chemical reactions in closed containers can lead to thermal equilibrium only. However, the transients can be chaotic if one begins sufficiently far from equilibrium states. • Certain epidemiological data, e.g., on the spread of chickenpox, can be consistently and meaningfully interpreted only in terms of transient chaos. • The so-called shimmy (an irregular dancing motion) of the front wheels of motorcycles and airplanes, which can lead to disastrous incidents, turns out to be a manifestation of transient chaos. • Satellite encounters and the escapes from major planets are chaotic transients. • The trapping of advected material or pollutant around obstacles, often seen in the wake of pillars or piers, is a consequence of transient chaos. • In nanostructures, today a cutting-edge field of science and engineering, the classical dynamics of electrons bear the signature of transient chaos.

Applications The main application is control and maintenance of transient chaos for desirable system performance. The collection and analysis of transient chaotic time series for probing the system also applicable in many areas of science and Engineering.

The Non-Autonomous Duffing Oscillator -----------------(1) y x

Autonomous Duffing Equation ---------------------------------------------------- (2) Here the potential ------------------------------- (3) Kinetic Energy then ----------------------------- (4) So the total Energy of our model can be written as.. -------------------------------- (5)

Two mutually coupled autonomous Duffing Equation ---------------- (6) --------------- (7) Potential for two coupled autonomous Duffing Oscillators ---- (8) Kinetic Energy then ---------------------------------- (9)

Kolmogorov Arnold Moser (KAM) theorem The motion of an integrable system is confined to a doughnut shaped surface, an invariant torus. Different initial conditions of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting any of the coordinates of an integrable system would show that they are quasi-periodic. (i) For H = 0.105156 Source: https://en.wikipedia.org/wiki/Kolmogorov-Arnold-Moser_theorem

(ii) For H = 0.167042 (iii) For H = 4.87512 Fig.: Surface of section in different energy levels of four to six hundred sets of randomly generated initial conditions shows the KAM island exists in the quasiperiodic motions of k=0.08.

Non-Conservative System ------------------ (9) ----------------(10) b – damping co-efficient ---------------- (11)

Stability Analysis

Locally linearized the above equation Model Equation --------------------------------------------------------------- (12) -------------- (13) --------------------------------------------------------------- (14) ------------- (15) Locally linearized the above equation ---------------- (16)

Eigen Values at (1) b=0 (2) b=0.0001 (Positive Damping) (3) b=-0.0001 (Negative Damping)

Numerical observations

Numerical Analysis (1) For b=0.0 (a) (b) (a) Time series of (t-x) plane, (b) Blow up of the colored region (Red)

(a) (b) (1) For b=0.0001 (Positive Damping) (a) Time series of (t-x) plane, (b) Extended time series of the colored area

(a) (b) (1) For b= -0.0001 (Negative Damping) (a) Time series of (t-x) plane, (b) Extended time series of the colored area

Experimental Analysis

Mutually coupled Duffing oscillator Equations are the Control parameters.

Fig. : Schematic diagram of two mutually coupled autonomous Duffing oscillator

a b Fig. : Real time Hardware experimental construction of mutually coupled Duffing oscillator. Blue wire indicating the coupling between the Duffing oscillators, (a,b) are the negative damping resistors.

Circuit Equation Parameters

(1) Zero Damping Fig.: Numerical and Experimental comparison of Phase Portraits and Time series for Zero damping case.

(ii) For positive damping Fig.: Experimentally observed Time series : The data acquisition is made using Agilent U2531A -4 GS/s.

(ii) For negative damping Fig.: Experimentally observed Time series : The data acquisition is made using Agilent U2531A -4 GS/s.

We have investigated the Surface of Section (SOS) of the Hamiltonian system (Duffing oscillator) with different energy levels. Small perturbation (-,+) in the Hamiltonian system makes the system’s phase space into non-conservative. So the phase space may be stable or unstable depends on its perturbations. The transient dynamics of two coupled nearly Hamiltonian Duffing oscillators is studied by numerical and hardware electronic circuit. The coupled Duffing oscillator exhibits transient chaos in both positive and negative damping.

References S. Sabarathinam, K. Thamilmaran L. Borkowski, P. Perlikowski, A. Stefanski, T. Kapitaniak, "Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing oscillators." Communications in Nonlinear Science and Numerical Simulation 18, (2013) 3098. M. A. Lieberman and K. Y. Tsang,“Transient Chaos in Dissipatively Perturbed, Near-Integrable Hamiltonian Systems” , Phys. Rev. Lett. 55, 1985. P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, ’’Routed to complex dynamics in a ring of unidirectionally coupled systems ’’ Chaos, 20 (1), 2010. P. Perlikowski, B. Jagiello, A. Stefanski, T. Kapitaniak, ‘Experimental observation of ragged synchronizability’’ Phys. Rev E –Statistical Nonlinear,and soft matter Physics, 78, 2008. A. S. Pikovsky, ”Escape exponent for transient chaos and chaotic scattering in nonhyperbolic Hamiltonian systems”, J. Phys -A Math Gen, 25 , 1992. U. E. Vincent, A. Kenfack, “Synchronization and bifurcation structures in coupled periodically forced non-identical Duffing oscillators” Phys Scr, 77, 2008.