8,976 research outputs found
Adaptive Detection of Instabilities: An Experimental Feasibility Study
We present an example of the practical implementation of a protocol for
experimental bifurcation detection based on on-line identification and feedback
control ideas. The idea is to couple the experiment with an on-line
computer-assisted identification/feedback protocol so that the closed-loop
system will converge to the open-loop bifurcation points. We demonstrate the
applicability of this instability detection method by real-time,
computer-assisted detection of period doubling bifurcations of an electronic
circuit; the circuit implements an analog realization of the Roessler system.
The method succeeds in locating the bifurcation points even in the presence of
modest experimental uncertainties, noise and limited resolution. The results
presented here include bifurcation detection experiments that rely on
measurements of a single state variable and delay-based phase space
reconstruction, as well as an example of tracing entire segments of a
codimension-1 bifurcation boundary in two parameter space.Comment: 29 pages, Latex 2.09, 10 figures in encapsulated postscript format
(eps), need psfig macro to include them. Submitted to Physica
A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations
The worst situation in computing the minimal nonnegative solution of a
nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when
the corresponding linearizing matrix has two very small eigenvalues, one with
positive and one with negative real part. When both these eigenvalues are
exactly zero, the problem is called critical or null recurrent. While in this
case the problem is ill-conditioned and the convergence of the algorithms based
on matrix iterations is slow, there exist some techniques to remove the
singularity and transform the problem to a well-behaved one. Ill-conditioning
and slow convergence appear also in close-to-critical problems, but when none
of the eigenvalues is exactly zero the techniques used for the critical case
cannot be applied.
In this paper, we introduce a new method to accelerate the convergence
properties of the iterations also in close-to-critical cases, by working on the
invariant subspace associated with the problematic eigenvalues as a whole. We
present a theoretical analysis and several numerical experiments which confirm
the efficiency of the new method
Non-symmetric gravity waves on water of infinite depth
Two different numerical methods are used to demonstrate the existence of and calculate non-symmetric gravity waves on deep water. It is found that they appear via spontaneous symmetry-breaking bifurcations from symmetric waves. The structure of the bifurcation tree is the same as the one found by Zufiria (1987) for waves on water of finite depth using a weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water-wave problem
Convergence Time Towards Periodic Orbits in Discrete Dynamical Systems
We investigate the convergence towards periodic orbits in discrete dynamical
systems. We examine the probability that a randomly chosen point converges to a
particular neighborhood of a periodic orbit in a fixed number of iterations,
and we use linearized equations to examine the evolution near that
neighborhood. The underlying idea is that points of stable periodic orbit are
associated with intervals. We state and prove a theorem that details what
regions of phase space are mapped into these intervals (once they are known)
and how many iterations are required to get there. We also construct algorithms
that allow our theoretical results to be implemented successfully in practice.Comment: 17 pages; 7 figure
Deterministic polarization chaos from a laser diode
Fifty years after the invention of the laser diode and fourty years after the
report of the butterfly effect - i.e. the unpredictability of deterministic
chaos, it is said that a laser diode behaves like a damped nonlinear
oscillator. Hence no chaos can be generated unless with additional forcing or
parameter modulation. Here we report the first counter-example of a
free-running laser diode generating chaos. The underlying physics is a
nonlinear coupling between two elliptically polarized modes in a
vertical-cavity surface-emitting laser. We identify chaos in experimental
time-series and show theoretically the bifurcations leading to single- and
double-scroll attractors with characteristics similar to Lorenz chaos. The
reported polarization chaos resembles at first sight a noise-driven mode
hopping but shows opposite statistical properties. Our findings open up new
research areas that combine the high speed performances of microcavity lasers
with controllable and integrated sources of optical chaos.Comment: 13 pages, 5 figure
E&F Chaos: a user friendly software package for nonlinear economic dynamics
The use of nonlinear dynamic models in economics and finance has expanded rapidly in the last two decades. Numerical simulation is crucial in the investigation of nonlinear systems. E&F Chaos is an easy-to-use and freely available software package for simulation of nonlinear dynamic models to investigate stability of steady states and the presence of periodic orbits and chaos by standard numerical simulation techniques such as time series, phase plots, bifurcation diagrams, Lyapunov exponent plots, basin boundary plots and graphical analysis. The package contains many well-known nonlinear models, including applications in economics and finance, and is easy to use for non-specialists. New models and extensions or variations are easy to implement within the software package without the use of a compiler or other software. The software is demonstrated by investigating the dynamical behavior of some simple examples of the familiar cobweb model, including an extension with heterogeneous agents and asynchronous updating of strategies. Simulations with the E&F chaos software quickly provide information about local and global dynamics and easily lead to challenging questions for further mathematical analysis.
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