1,296 research outputs found
Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
In this article, we present a brief overview of some of the recent progress
made in identifying and generating finite dimensional integrable nonlinear
dynamical systems, exhibiting interesting oscillatory and other solution
properties, including quantum aspects. Particularly we concentrate on Lienard
type nonlinear oscillators and their generalizations and coupled versions.
Specific systems include Mathews-Lakshmanan oscillators, modified Emden
equations, isochronous oscillators and generalizations. Nonstandard Lagrangian
and Hamiltonian formulations of some of these systems are also briefly touched
upon. Nonlocal transformations and linearization aspects are also discussed.Comment: To appear in Eur. Phys. J - ST 222, 665 (2013
Conjugate coupling induced symmetry breaking and quenched oscillations
Spontaneous symmetry breaking (SSB) is essential and plays a vital role many
natural phenomena, including the formation of Turing pattern in organisms and
complex patterns in brain dynamics. In this work, we investigate whether a set
of coupled Stuart-Landau oscillators can exhibit spontaneous symmetry breaking
when the oscillators are interacting through dissimilar variables or conjugate
coupling. We find the emergence of SSB state with coexisting distinct dynamical
states in the parametric space and show how the system transits from symmetry
breaking state to out-of-phase synchronized (OPS) state while admitting
multistabilities among the dynamical states. Further, we also investigate the
effect of feedback factor on SSB as well as oscillation quenching states and we
point out that the decreasing feedback factor completely suppresses SSB and
oscillation death states. Interestingly, we also find the feedback factor
completely diminishes only symmetry breaking oscillation and oscillation death
(OD) states but it does not affect the nontrivial amplitude death (NAD) state.
Finally, we have deduced the analytical stability conditions for in-phase and
out-of-phase oscillations, as well as amplitude and oscillation death states.Comment: Accepted for publication in Europhysics Letter
On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator
Using the modified Prelle- Singer approach, we point out that explicit time
independent first integrals can be identified for the damped linear harmonic
oscillator in different parameter regimes. Using these constants of motion, an
appropriate Lagrangian and Hamiltonian formalism is developed and the resultant
canonical equations are shown to lead to the standard dynamical description.
Suitable canonical transformations to standard Hamiltonian forms are also
obtained. It is also shown that a possible quantum mechanical description can
be developed either in the coordinate or momentum representations using the
Hamiltonian forms.Comment: 19 page
Extended Prelle-Singer Method and Integrability/Solvability of a Class of Nonlinear th Order Ordinary Differential Equations
We discuss a method of solving order scalar ordinary differential
equations by extending the ideas based on the Prelle-Singer (PS) procedure for
second order ordinary differential equations. We also introduce a novel way of
generating additional integrals of motion from a single integral. We illustrate
the theory for both second and third order equations with suitable examples.
Further, we extend the method to two coupled second order equations and apply
the theory to two-dimensional Kepler problem and deduce the constants of motion
including Runge-Lenz integral.Comment: 18 pages, Article dedicated to Professor F. Calogero on his
70thbirthda
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