1. 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
Collaborators
Prof. T. Kapitaniak , Dr P. Perlikowski A. Stefanski, L. Borkowski, P. Brzeski.
Division of Dynamics
Lodz University of Technology
Lodz, Poland

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

3. 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 …

4. 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

5. 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.

6. 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.

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

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

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

10. 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 =
Source:

11. (ii) For H =
(iii) For H =
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.

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

13. Stability Analysis

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

15. Eigen Values at
(1) b=0
(2) b= (Positive Damping)
(3) b= (Negative Damping)

16. Numerical observations

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

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

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

20. Experimental Analysis

21. Mutually coupled Duffing oscillator Equations
are the Control parameters.

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

23. 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.

24. Circuit Equation
Parameters

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

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

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

28. 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.

29. 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.