arXiv subjects

Jose-Luis Lopez

Publications and source records attributed to Jose-Luis Lopez.

11 recordsLinked to original sources

arXiv2014

Random Market Models with an H-Theorem

In this communication, some economic models given by functional mappings are addressed. These are models for random markets where agents trade by pairs and exchange their money in a random and conservative way. They display the exponential wealth distribution as asymptotic equilibrium, independently of the effectiveness of the transactions and of the limitation of the total wealth. The entropy increases with time in these models and the existence of an H-theorem is computationally checked. Also, it is shown that any small perturbation of the models equations make them to lose the exponential distribution as an equilibrium solution.

q-fin.TR↗

arXiv2011

Exponential wealth distribution in a random market. A rigorous explanation

In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way.

q-fin.GN↗

arXiv2011

Exponential wealth distribution: a new approach from functional iteration theory

Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space. Also, several multi-agent economic models based on mappings, with random, deterministic or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution. An alternative approach to this problem in the framework of iterations in the space of distributions has been recently presented. Concretely, the new iteration given by $ f_{n+1}(x) = \int\int_{u+v>x}{f_n(u)f_n(v)\over u+v} dudv.$. It is found that the exponential distribution is a stable fixed point of the former functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution (or by extension, other kind of distributions) is asymptotically obtained in different multi-agent economic models.

nlin.AO↗

arXiv2010

Equilibrium distributions and relaxation times in gas-like economic models: an analytical derivation

A step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDF) of well known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, e.g., allowing the calculation of its relaxation times. Their equilibrium PDFs can also be calculated by finding its stationary solutions. This gives as a result an integro-differential equation, which can be solved analytically in some cases and numerically in others. This should provide some guidance into the type of probability density functions that can be derived from particular economic agent exchange rules, or for that matter, any other kinetic model of gases with particular collision physics.

q-fin.GN↗

arXiv2008

Formulas for the amplitude of the van der Pol limit cycle

The limit cycle of the van der Pol oscillator, $\ddot{x}+ ε(x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $ε$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.

nlin.AO↗

arXiv2008

The homotopy analysis method and the Lienard equation

In this work, Lienard equations are considered. The limit cycles of these systems are studied by applying the homotopy analysis method. The amplitude and frequency obtained with this methodology are in good agreement with those calculated by computational methods. This puts in evidence that the homotopy analysis method is an useful tool to solve nonlinear differential equations.

nlin.PS↗

arXiv2006

The Limit Cycles of Lienard Equations in the Weakly Nonlinear Regime

Liénard equations of the form $\ddot{x}+εf(x)\dot{x}+x=0$, with $f(x)$ an even function, are considered in the weakly nonlinear regime ($ε\to 0$). 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 ${\mathcal O}(ε^8)$ approximation for the amplitude of the van der Pol limit cycle is explicitly obtained.

nlin.AO↗

arXiv2006

Approximating the Amplitude and Form of Limit Cycles in the Weakly Nonlinear Regime of Lienard Systems

Liénard equations, $\ddot{x}+εf(x)\dot{x}+x=0$, with $f(x)$ an even continuous function are considered. In the weakly nonlinear regime ($ε\to 0$), the number and an order zero in $ε$ approximation of the amplitude of limit cycles present in this type of systems can be obtained by applying a methodology recently proposed by the authors [López-Ruiz R, López JL. Bifurcation curves of limit cycles in some Liénard systems. Int J Bifurcat Chaos 2000; 10:971-980]. In the present work, that method is carried forward to higher orders in $ε$ and is embedded in a general recursive algorithm capable to approximate the form of the limit cycles and to correct their amplitudes as an expansion in powers of $ε$. Several examples showing the application of this scheme are given.

nlin.AO↗

arXiv2002

Number and Amplitude of Limit Cycles emerging from {\it Topologically Equivalent} Perturbed Centers

We consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+ε(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\int_{H=h}fdy-gdx$, associated to each particular problem. We obtain the information on the bifurcation curves of the limit cycles by solving explicitly the equation $M(h)=0$ in each case.

nlin.PS↗

arXiv2002

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

Lienard systems of the form $\ddot{x}+εf(x)\dot{x}+x=0$, with f(x) an even function, are studied in the strongly nonlinear regime ($ε\to\infty$). 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.

nlin.CD↗

arXiv2002

Bifurcation Curves of Limit Cycles in some Lienard Systems

Lienard systems of the form $\ddot{x}+εf(x)\dot{x}+x=0$, with f(x) an even continous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak ($ε\to 0$) and in the strongly ($ε\to\infty$) nonlinear regime in some examples. The number of limit cycles does not increase when $ε$ increases from zero to infinity in all the cases analyzed.

nlin.PS↗