Phase shift in experimental trajectory scaling functions

For one dimensional maps the trajectory scaling functions is invariant under coordinate transformations and can be used to compute any ergodic average. It is the most stringent test between theory and experiment, but so far it has proven difficult to extract from experimental data. It is shown that the main difficulty is a dephasing of the experimental orbit which can be corrected by reconstructing the dynamics from several time series. From the reconstructed dynamics the scaling function can be accurately extracted.Comment: CYCLER Paper 93mar008. LaTeX, LAUR-92-3053. Replaced with a version with all figure

Multifractal Analysis of Dimensions and Entropies

The theory of dynamical systems has undergone a dramatical revolution in the 20th century. The beauty and power of the theory of dynamical systems is that it links together different areas of mathematics and physics. In the last 30 years a great deal of attention was dedicated to a statistical description of strange attractors. This led to the development of notions of various dimensions and entropies, which can be associated to the attractor, dynamical system or invariant measure. In this paper we review these notions and discuss relations between those, among which the most prominent is the so-called multifractal formalism

Investigation of the complex dynamics and regime control in Pierce diode with the delay feedback

In this paper the dynamics of Pierce diode with overcritical current under the influence of delay feedback is investigated. The system without feedback demonstrates complex behaviour including chaotic regimes. The possibility of oscillation regime control depending on the delay feedback parameter values is shown. Also the paper describes construction of a finite-dimensional model of electron beam behaviour, which is based on the Galerkin approximation by linear modes expansion. The dynamics of the model is close to the one given by the distributed model.Comment: 18 pages, 6 figures, published in Int. J. Electronics. 91, 1 (2004) 1-1

A Tool to Recover Scalar Time-Delay Systems from Experimental Time Series

We propose a method that is able to analyze chaotic time series, gained from exp erimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form $dy(t)/dt = h(y(t),y(t-\tau_0))$, the delay time $\tau_0$ and the functi on $h$ can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators

Oscillations and secondary bifurcations in nonlinear magnetoconvection

Complicated bifurcation structures that appear in nonlinear systems governed by partial differential equations (PDEs) can be explained by studying appropriate low-order amplitude equations. We demonstrate the power of this approach by considering compressible magnetoconvection. Numerical experiments reveal a transition from a regime with a subcritical Hopf bifurcation from the static solution, to one where finite-amplitude oscillations persist although there is no Hopf bifurcation from the static solution. This transition is associated with a codimension-two bifurcation with a pair of zero eigenvalues. We show that the bifurcation pattern found for the PDEs is indeed predicted by the second-order normal form equation (with cubic nonlinearities) for a Takens-Bogdanov bifurcation with Z2 symmetry. We then extend this equation by adding quintic nonlinearities and analyse the resulting system. Its predictions provide a qualitatively accurate description of solutions of the full PDEs over a wider range of parameter values. Replacing the reflecting (Z2) lateral boundary conditions with periodic [O(2)] boundaries allows stable travelling wave and modulated wave solutions to appear; they could be described by a third-order system

Transmission of Information in Active Networks

Shannon's Capacity Theorem is the main concept behind the Theory of Communication. It says that if the amount of information contained in a signal is smaller than the channel capacity of a physical media of communication, it can be transmitted with arbitrarily small probability of error. This theorem is usually applicable to ideal channels of communication in which the information to be transmitted does not alter the passive characteristics of the channel that basically tries to reproduce the source of information. For an {\it active channel}, a network formed by elements that are dynamical systems (such as neurons, chaotic or periodic oscillators), it is unclear if such theorem is applicable, once an active channel can adapt to the input of a signal, altering its capacity. To shed light into this matter, we show, among other results, how to calculate the information capacity of an active channel of communication. Then, we show that the {\it channel capacity} depends on whether the active channel is self-excitable or not and that, contrary to a current belief, desynchronization can provide an environment in which large amounts of information can be transmitted in a channel that is self-excitable. An interesting case of a self-excitable active channel is a network of electrically connected Hindmarsh-Rose chaotic neurons.Comment: 15 pages, 5 figures. submitted for publication. to appear in Phys. Rev.

Observable Optimal State Points of Sub-additive Potentials

For a sequence of sub-additive potentials, Dai [Optimal state points of the sub-additive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573] gave a method of choosing state points with negative growth rates for an ergodic dynamical system. This paper generalizes Dai's result to the non-ergodic case, and proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.Comment: 16 pages. This work was reported in the summer school in Nanjing University. In this second version we have included some changes suggested by the referee. The final version will appear in Discrete and Continuous Dynamical Systems- Series A - A.I.M. Sciences and will be available at http://aimsciences.org/journals/homeAllIssue.jsp?journalID=

Properties making a chaotic system a good Pseudo Random Number Generator

We discuss two properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai entropy and high-dimensionality. We propose the multi dimensional Anosov symplectic (cat) map as a Pseudo Random Number Generator. We show what chaotic features of this map are useful for generating Pseudo Random Numbers and investigate numerically which of them survive in the discrete version of the map. Testing and comparisons with other generators are performed.Comment: 10 pages, 3 figures, new version, title changed and minor correction

Characteristics of a Delayed System with Time-dependent Delay Time

The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.Comment: 4 pages, 6 figures (to be published in Phys. Rev. E