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Pizza Seminar 07-11-2014
by Amartya Sarkar
AN OSCILLATOR’S SLOW MARCH TO DEATH
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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
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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
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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
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The Big Question?
5
Why and how does the periodic solution
vanish?
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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
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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
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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)
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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
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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 .
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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 .
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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)
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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 .
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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
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J. K. Bhattacharjee
Partha Guha
Anindya Ghose-Chowdhury
A. K. Mallik
Acknowledgements