Pizza Seminar 07-11-2014

by Amartya Sarkar

AN OSCILLATOR’S SLOW MARCH TO DEATH

Background 2

Types of periodic solutions – Limit Cycles and Centres

2 1 4 3(2 3) 0n nx n x x x Limit Cycles - Isolated closed, initial-condition-independent trajectory in phase space.

Strictly nonlinear phenomenon.

Self-sustaining oscillations

e.g. Van der Pol oscillator, glycolytic oscillator

Centres - Family of initial-condition-dependent periodic orbits.

Non-hyperbolic fixed points.

e.g. Lotka –Volterra, Duffing

Special type – Isochronous oscillators

History 3

Generalized Riccati Equation a.k.a Emden Equation a.k.a. Liénard type equation

33 0x xx x

2

2 1 0.ndx x

dt

n=0 :

Otherwise :

Scalar field equation in 1-D φ4 field theory

Model for pellet fusion process. Ervin et al.

An equation governing spherically symmetric expansion or collapse of relativistically gravitating mass. McVittie et al.

1-D analogue of the Yang-Mill’s boson ‘gauge theory’

2 1 4 3(2 3) 0n nx n x x x

Facts of the Case

4

Recast as an classical mechanical problem of a particle moving in a 1-D potential :

(1)

Interested in the behavior of the solution of (1) with varying values of α.

n=0, α = 3.0. Equation is linearizable, possesses 8 symmetries and is completely integrable. And the solution is aperiodic.

n=0, α = 0.1, the solution turns out to be periodic. Numerical observation. Similar behavior for higher values of n.

2 1 4 3 0n nx x x x

The Big Question?

5

Why and how does the periodic solution

vanish?

Investigation

Jacobi’s last multiplier.

Renormalization Group

technique.

Scaling behavior study.

Hidden symmetry

reduction.

Runge-Kutta 4th order to analyze phase trajectories for varying n and α.

Finding the critical αc where periodic solution vanishes.

Data collapse.

6

Analytic techniques Numerical studies

Investigation

Jacobi’s last multiplier.

Renormalization Group

technique.

Scaling behavior study.

Hidden symmetry

reduction.

Runge-Kutta 4th order to analyze phase trajectories for varying n and α.

Finding the critical αc where periodic solution vanishes.

Data collapse.

7

Analytic techniques Numerical studies

Renormalization group approach

8

The initial condition can be

placed anywhere on the phase

path.

Split time interval t – t0 as t –

τ + τ – t0 .

Absorb τ – t0 containing terms

into renormalized A and θ.

x(t) has to be independent of τ

gives rise to flow equations.

Flow equations can be used to

get perturbative solutions to

oscillators.

2 ( , )x x F x x

*Chen, Goldenfeld and Oono, PRL 73,

1311 (1994)

Observations

9

Bilagrangian structure and hence associated

Hamiltonian one, when α = 2n +3 .

For every possible n there is a critical αc beyond

which periodic behavior ceases to exist.

As α approaches the critical value, the time period,

‘T’ diverges. RG suggests :

Bottleneck at origin. Periodic orbit has 2 widely

separated time scales.

0.5

.cT

CLUEs

10

The periodic orbits are dependent on initial

conditions. Yet the critical value αc is independent of

initial conditions.

The trajectories for different “initial conditions” can

be collapsed onto a single universal orbit.

Omnipresence of the number ‘2n+2’ in various

analytical expressions for the system.

The numerically determined critical value ‘αc’ for

different ‘n’ suspiciously close to 2(2n+2)1/2 .

Clinching Evidence 11

(a) Periodic orbits corresponding to various initial conditions xi = C and yi = 0. (b) All the data points have collapsed onto a single orbit as x

and y are respectively scaled as x/C and y/C4 .

A Different Angle

12

Use “Hidden Symmetry Reduction” and recast Eqn (1) into a different

form via two consecutive transformations.

1st one - a generalization of Riccati transformation increasing the

order of the differential equation.

2nd one - decreasing back the order of the differential equation.

In the new co-ordinates ask the question: what is the critical value for

the parameter ?

The answer : αc =2(2n+2)1/2 .

* A. Sarkar, P. Guha, A. Ghose-Choudhury, J. K. Bhattacharjee, A. K. Mallik and P. G. L. Leach, “On

the properties of a variant of the Riccati system of equations”, Jour. Phys. A: Math. Theor. 45 (41),

415101. (2012)

Reconstructing Events

31-01-2012 BOSEFEST 2012

13

As a dynamical system:

Equation for the trajectory:

Scaling form:

x → x/γ and y → y/γ4

Keeps the trajectory invariant.

2 1 4 3n n

x v

v vx x

4 32 1

nndv x

xdx v

Figure showing a typical trajectory and the isoclines : x = 0 and αv = -x2n+2 .

EPILOGUE 14

RG flow equations reveal a curious fact.

Add a linear term to Eqn (1) → Both amplitude and

phase flows become null, suggesting we have an

‘Isochronous Oscillator’.

Numerical tests confirm that the above system executes

‘amplitude’- independent oscillations at frequency ω, for

even very large values of n.

2 1 4 3 2(2 3) 0n nx n x x x x

Linear term

J. K. Bhattacharjee

Partha Guha

Anindya Ghose-Chowdhury

A. K. Mallik

Acknowledgements