72 research outputs found
Negative-coupling resonances in pump-coupled lasers
We consider coupled lasers, where the intensity deviations from the steady
state, modulate the pump of the other lasers. Most of our results are for two
lasers where the coupling constants are of opposite sign. This leads to a Hopf
bifurcation to periodic output for weak coupling. As the magnitude of the
coupling constants is increased (negatively) we observe novel amplitude effects
such as a weak coupling resonance peak and, strong coupling subharmonic
resonances and chaos. In the weak coupling regime the output is predicted by a
set of slow evolution amplitude equations. Pulsating solutions in the strong
coupling limit are described by discrete map derived from the original model.Comment: 29 pages with 8 figures Physica D, in pres
Cascades of subharmonic stationary states in strongly non-linear driven planar systems
The dynamics of a one-degree of freedom oscillator with arbitrary polynomial
non-linearity subjected to an external periodic excitation is studied. The
sequences (cascades) of harmonic and subharmonic stationary solutions to the
equation of motion are obtained by using the harmonic balance approximation
adapted for arbitrary truncation numbers, powers of non-linearity, and orders
of subharmonics. A scheme for investigating the stability of the harmonic
balance stationary solutions of such a general form is developed on the basis
of the Floquet theorem. Besides establishing the stable/unstable nature of a
stationary solution, its stability analysis allows obtaining the regions of
parameters, where symmetry-breaking and period-doubling bifurcations occur.
Thus, for period-doubling cascades, each unstable stationary solution is used
as a base solution for finding a subsequent stationary state in a cascade. The
procedure is repeated until this stationary state becomes stable provided that
a stable solution can finally be achieved. The proposed technique is applied to
calculate the sequences of subharmonic stationary states in driven hardening
Duffing's oscillator. The existence of stable subharmonic motions found is
confirmed by solving the differential equation of motion numerically by means
of a time-difference method, with initial conditions being supplied by the
harmonic balance approximation.Comment: 37 pages, 11 figures, revised material on chaotic motio
Predicting incipent instabilities and bifurcations of nonlinear dynamical systems modelling compliant off-shore structures
For engineers, the two most important aspects of dynamical analysis are high amplitude resonance vibrations and structural stability, i.e. whether a steady state solution is stable under small perturbations. For the former case, a novel and simple method based on Poincare mapping technique has been devised to predict an imminent flip bifurcation. This bifurcation represents the beginning of the second order subharmonic response. For the latter case, we discovered that while classical quantitative analytical techniques work well in establishing the 'local' structural stability of a steady state solution, the global geometric structure of the catchment region can alter dramatically such that even an initial condition close to the steady state can diverge from it rather than being attracted. This phenomenon known as fractal basin boundary occurs when the invariant manifolds of the saddle separating the steady state solution from any remote attractor cross. The critical point in which the invariant manifolds just touch can be accurately predict by the Melinkov's method. Because of the complicated interwoven nature of the invariant manifolds, it is called a tangle. If the invariant manifolds are originated from the same saddle, the crossing is known as a homoclinic tangle, if originated from different saddle, a heteroclinic tangle. The critical point is then known as homoclinic or heteroclinic tangency. Tangles are also intimately related to chaotic behaviour. The creation and destruction of chaotic attractors have been observed through a series of homoclinic and heteroclinic tangency. In fact, after the invariant manifolds of an inverting saddle cross, the unstable manifold becomes the chaotic attractor. This leads us to believe that all chaotic attractors are topologically the same
Pattern formation in 2-frequency forced parametric waves
We present an experimental investigation of superlattice patterns generated
on the surface of a fluid via parametric forcing with 2 commensurate
frequencies. The spatio-temporal behavior of 4 qualitatively different types of
superlattice patterns is described in detail. These states are generated via a
number of different 3--wave resonant interactions. They occur either as
symmetry--breaking bifurcations of hexagonal patterns composed of a single
unstable mode or via nonlinear interactions between the two primary unstable
modes generated by the two forcing frequencies. A coherent picture of these
states together with the phase space in which they appear is presented. In
addition, we describe a number of new superlattice states generated by 4--wave
interactions that arise when symmetry constraints rule out 3--wave resonances.Comment: The paper contains 34 pages and 53 figures and provides an extensive
review of both the theoretical and experimental work peformed in this syste
A study of poststenotic shear layer instabilities
Imperial Users onl
Nonlinear modification of the laser noise power spectrum induced by a frequency-shifted optical feedback
In this article, we study the non-linear coupling between the stationary
(i.e. the beating modulation signal) and transient (i.e. the laser quantum
noise) dynamics of a laser subjected to frequency shifted optical feedback. We
show how the noise power spectrum and more specifically the relaxation
oscillation frequency of the laser are modified under different optical
feedback condition. Specifically we study the influence of (i) the amount of
light returning to the laser cavity and (ii) the initial detuning between the
frequency shift and intrinsic relaxation frequency. The present work shows how
the relaxation frequency is related to the strength of the beating signal and
the shape of the noise power spectrum gives an image of the Transfer Modulation
Function (i.e. of the amplification gain) of the nonlinear-laser dynamics.The
theoretical predictions, confirmed by numerical resolutions, are in good
agreements with the experimental data.Comment: in Physical Review, American Physical Society (APS), 201
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