S. SabarathinamResearch Scholar

Under the guidance of

Prof. K. Thamilmaran Centre for Nonlinear Dynamics School of Physics Bharathidasan universityTiruchirappalli-620 024

Transient chaos in two coupled, dissipatively perturbed Hamiltonian Duffing oscillators

CollaboratorsProf. T. Kapitaniak , Dr P. Perlikowski A. Stefanski, L. Borkowski, P. Brzeski.Division of DynamicsLodz University of TechnologyLodz, Poland

Plan of talk

IntroductionTransient ChaosModelStability AnalysisNumerical AnalysisExperimental SetupExperimental ResultsConclusion

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 …

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.

A question is, then, can chaos be transient?

Transient Chaos

t*t

• 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.

Transient Chaos Arises…..

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

3 sin x bx x x f t

0.5, 1.0, 1.0, 0.42, 1.0b f

0, 0 - Double well potential

x

y

-----------------(1)

2

2

( ) 0d x dV xdt dx

Autonomous Duffing Equation

---------------------------------------------------- (2)

Here the potential

2 4

( )2 4x xV x

0, 0 ------------------------------- (3)

Kinetic Energy then

21( )2

dxT xdt

So the total Energy of our model can be written as..

T V

----------------------------- (4)

-------------------------------- (5)

2 4 2 421 1 2 2

1 2 1 2( , ) - - ( )2 4 2 4 2x x x x kV x x x x

Potential for two coupled autonomous Duffing Oscillators

Kinetic Energy then

2 21 2

1 21( , )2

dx dxT x xdt dt

---- (8)

---------------------------------- (9)

Two mutually coupled autonomous Duffing Equation

231

1 1 2 12 ( ) 0d x x x k x xdt

232

2 2 1 22 ( ) 0d x x x k x xdt

---------------- (6)

--------------- (7)

(i) For H = 0.105156

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.

Source: https://en.wikipedia.org/wiki/Kolmogorov-Arnold-Moser_theorem

(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.

(ii) For H = 0.167042

Non-Conservative System

2 21 2

1 2 1 21 1( , )2 2

dx dxD x x b bdt dt

231

1 1 1 1 2 12 ( ) 0d x b x x x k x xdt

232

2 2 2 2 1 22 ( ) 0d x b x x x k x xdt

------------------ (9)

----------------(10)

b – damping co-efficient

---------------- (11)

1 2, 0.0001b b b

Stability Analysis

Model Equation

1 2

32 2 1 1 1 3

3 4

34 4 3 3 3 1

( )

( )

x x

x bx x x k x xx x

x bx x x k x x

21

23

- 1 0 0

-3 - - - 0 0 0 - 1

0 - 3 - - -

x k b kJ

k x k b

Locally linearized the above equation

0.3, 0.85, 0.08k

--------------------------------------------------------------- (12)

-------------- (13)

--------------------------------------------------------------- (14)

------------- (15)

---------------- (16)

Eigen Values at

(1) b=0

1,2

3,4

0.87178

0.77459

i

i

(2) b=0.0001 (Positive Damping)

1,2

3,4

0.0005 0.87178

0.0005 0.77459

i

i

(3) b=-0.0001 (Negative Damping)

1,2

3,4

0.0005 0.87178

0.0005 0.77459

i

i

1 2

3 4

( , ) (0,0),( , ) (0,0)x xx x

Numerical observations

Numerical Analysis(1) For b=0.0

(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

(a)

(b)

Experimental Analysis

Mutually coupled Duffing oscillator Equations

1 2

32 2 1 1 1 3

3 4

34 4 3 3 3 1

( )

( )

x x

x bx x x k x xx x

x bx x x k x x

, , , and b k are the Control parameters.

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

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.

a

b

2

32a 2a 1a 1a1a 2a 1a 4a 4a 2a 2a 2a2

7a

1a2a 1b

8a 9a

0.01d v dv R RC C R R = R C + v vdt dt R R

R v vR

2

32b 2b 1b 1b1b 2b 1b 4b 4b 2b 2b 2b2

7b

1b1a 2a

8b 9b

0.01d v dv R RC C R R = R C + v vdt dt R R

R v vR

Circuit Equation

Parameters

4 7 1 2 4 8 1 2

1 1 4 9 1 2

1 0.01, ,

1 1- ,

R R C C R R C C

b kR C R R C C

(1) Zero Damping

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

Fig.: Experimentally observed Time series : The data acquisition is made using Agilent U2531A -4 GS/s.

1( 2.5 )R M (ii) For positive damping

Fig.: Experimentally observed Time series : The data acquisition is made using Agilent U2531A -4 GS/s.

1( 2.5 )R M (ii) For negative damping

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.