Shannon information, LMC complexity and Renyi entropies: a straightforward approach

The LMC complexity, an indicator of complexity based on a probabilistic description, is revisited. A straightforward approach allows us to establish the time evolution of this indicator in a near-equilibrium situation and gives us a new insight for interpreting the LMC complexity for a general non equilibrium system. Its relationship with the Renyi entropies is also explained. One of the advantages of this indicator is that its calculation does not require a considerable computational effort in many cases of physical and biological interest.Comment: 11 pages, 0 figure

An approach to the problem of generating irreducible polynomials over the finite field GF(2) and its relationship with the problem of periodicity on the space of binary sequences

A method for generating irreducible polynomials of degree n over the finite field GF(2) is proposed. The irreducible polynomials are found by solving a system of equations that brings the information on the internal properties of the splitting field GF(2^n) . Also, the choice of a primitive normal basis allows us to build up a natural representation of GF(2^n) in the space of n-binary sequences. Illustrative examples are given for the lowest orders.Comment: 22 pages, 6 tables, 0 figure

Complex Systems with Trivial Dynamics

In this communication, complex systems with a near trivial dynamics are addressed. First, under the hypothesis of equiprobability in the asymptotic equilibrium, it is shown that the (hyper) planar geometry of an $N$-dimensional multi-agent economic system implies the exponential (Boltzmann-Gibss) wealth distribution and that the spherical geometry of a gas of particles implies the Gaussian (Maxwellian) distribution of velocities. Moreover, two non-linear models are proposed to explain the decay of these statistical systems from an out-of-equilibrium situation toward their asymptotic equilibrium states.Comment: 9 gaes, 0 figures; Contributed Talk to ECCS'12 (European Conference of Complex Systems, Brussels, September, 2012

Symmetry induced Dynamics in four-dimensional Models deriving from the van der Pol Equation

Different models of self-excited oscillators which are four-dimensional extensions of the van der Pol system are reported. Their symmetries are analyzed. Three of them were introduced to model the release of vortices behind circular cylinders with a possible transition from a symmetric to an antisymmetric Benard-von Karman vortex street. The fourth reported self-excited oscillator is a new model which implements the breaking of the inversion symmetry. It presents the phenomenon of second harmonic generation in a natural way. The parallelism with second harmonic generation in nonlinear optics is discussed. There is also a small region in the parameter space where the dynamics of this system is quasiperiodic or chaotic.Comment: 14 pages, 0 figure

Complexity in some Physical Systems

The LMC-complexity introduced by Lopez-Ruiz, Mancini and Calbet [Phys. Lett. A 209, 321-326 (1995)] is calculated for different physical situations: one instance of classical statistical mechanics, normal and exponential distributions, and a simplified laser model. We stand out the specific value of the population inversion for which the laser presents maximun complexity.Comment: 12 pages, 0 figures; Published in Int. Journal of Bifurcation and Chaos, vol. 11, 2669-2673 (2001

A Binary Approach to the Lorenz Model

The order of orbit generation in one-dimensional Lorenz-like maps is presented within a two letter symbolics scheme. This order is derived from the natural order of a set of fractions associated to the binary sequences. Its relation to the universal sequence of unimodal maps is explained.Comment: 12 pages, 0 figure

Bifurcation Curves of Limit Cycles in some Lienard Systems

Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even continous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak ($\epsilon\to 0$) and in the strongly ($\epsilon\to\infty$) nonlinear regime in some examples. The number of limit cycles does not increase when $\epsilon$ increases from zero to infinity in all the cases analyzed.Comment: 25 pages, 0 figures. Published in Int. Journal of Bifurcation and Chaos, vol. 10, 971-980 (2001

The Limit Cycles of Lienard Equations in the Weakly Nonlinear Regime

Li\'enard equations of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even function, are considered in the weakly nonlinear regime ($\epsilon\to 0$). A perturbative algorithm for obtaining the number, amplitude and shape of the limit cycles of these systems is given. The validity of this algorithm is shown and several examples illustrating its application are given. In particular, an ${\mathcal O}(\epsilon^8)$ approximation for the amplitude of the van der Pol limit cycle is explicitly obtained.Comment: 17 text-pages, 1 table, 6 figure

The Limit Cycles of Lienard Equations in the Strongly Non-Linear Regime

Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even function, are studied in the strongly nonlinear regime ($\epsilon\to\infty$). A method for obtaining the number, amplitude and loci of the limit cycles of these equations is derived. The accuracy of this method is checked in several examples. Lins-Melo-Pugh conjecture for the polynomial case is true in this regime.Comment: 22 pages, 0 figures. Published in Chaos, Solitons and Fractals, vol. 11, 747-756 (2001

Shape of Traveling Densities with Extremum Statistical Complexity

In this paper, we analyze the behavior of statistical complexity in several systems where two identical densities that travel in opposite direction cross each other. Besides the crossing between two Gaussian, rectangular and triangular densities studied in a previous work, we also investigate in detail the crossing between two exponential and two gamma distributions. For all these cases, the shape of the total density presenting an extreme value in complexity is found.Comment: 11 pages, 14 figure