Learning in a double-phase cobweb model

The idea of fuzzy systems as models for the complex or incomplete processes is not a new idea. The determination of a satisfactory model for an imprecise, complex or incomplete process is basically a matter of approximation. Namely, determining an admissible model best deserving the particular physical phenomena. We suggest that one of the ways to represent imprecise or complex dynamical processes in a fuzzy environment is via fuzzy differential equations and some analysis of their typical solution. In particular, total differential equations of first and second order and partial differential equations of second order are considered. They are unusual in that the coefficients appearing in the equation are "fuzzy".

We present a multi-frequency phase control able to preserve a periodic behaviour within a chaotic window as well as to re-excite chaotic behaviour when it is destroyed. The validity of this non feedback method has been shown in the cobweb model with adaptive price expectations as well as in the quadratic map near an interior crisis. A crucial role is played by the phase of the applied periodic perturbations

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

We show that under suitable conditions the covariance of a Gaussian reciprocal process satisfies a self-adjoint linear differential equation of second order. We also give a revised definition of a linear stochastic differential equation of second order and necessary and sufficient conditions for the existence of solutions of such equations with Dirichlet boundary conditions. We close with a series of examples of the theory applied to the scalar stationary Gaussian Reciprocal processes which have been completely classified.

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

59 - Exact Second Order Equations

This paper aims to study price dynamics in two different continuous time cobweb models with delays close to [Hommes, 1994]. In both cases, the stationary equilibrium may be not representative of the long-term dynamics of the model, since it is possible to observe endogenous and persistent fluctuations (supercritical Hopf bifurcations) even if a deterministic context without external shocks is considered. In the model in which markets are in equilibrium every time, we show that the existence of time delays in the expectations formation mechanism may cause chaotic dynamics similar to those obtained in [Hommes, 1994] in a discrete time context. From a mathematical point of view, we apply the Poincaré–Lindstedt perturbation method to study the local dynamic properties of the models. In addition, several numerical experiments are used to investigate global properties of the systems.

Second-order differential equations are one of the most widely studied classes of differential equations in mathematics, physical science, and engineering. One sure reason is that Newton’s second law of motion is expressed as a law that involves acceleration of a particle, which is the second derivative of position. Thus, general one-dimensional mechanical systems are governed naturally by second-order equations.

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

Using the Riccati transformation techniques, we establish some new oscillation criteria for the second-order nonlinear difference equation Some comparison between our theorems and the previously known results in special cases are indicated. Some examples are given to illustrate the relevance of our results.

Fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second-order differential equations. There are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second-order differential equations. These are (i) fractional differential operators are nonlocal operators, and (ii) the adjoint of a fractional differential operator is not the negative of itself. The wellposedness of a Galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously. Many subsequent works were carried out to extend the analysis to other numerical methods. A constant diffusivity coefficient has been assumed in all these works. In this paper we present a counterexample which shows that the Galerkin weak formulation loses coercivity in the context of variable-coefficient conservative fractional elliptic differential equations. Hence, the previous results cannot be extended to variable-coefficient conservative fractional elliptic differential equations. We adopt an alternative approach to prove the existence and uniqueness of the classical solution to the variable-coefficient conservative fractional elliptic differential equation and characterize the solution in terms of the classical solutions to second-order elliptic differential equations. Furthermore, we derive a Petrov--Galerkin weak formulation to the fractional elliptic differential equation. We prove that the bilinear form of the Petrov--Galerkin weak formulation is weakly coercive and so the weak formulation has a unique weak solution and is well posed. Finally, we outline potential application of these results in the development of numerical methods for variable-coefficient conservative fractional elliptic differential equations.

In this paper, the distributions of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations are studied. By means of inequality techniques, specific function sequences and non-increasing solutions for corresponding first-order difference inequality, some new estimates for the distribution of the zeros of oscillatory solutions are presented, which extend and improve some well-known results.

The paper investigates the oscillation, zero-convergence, and solutions of second-order neutral delay difference equations containing three nonlinear delayed terms with different growth rates. By using positivity and monotonicity conditions on an auxiliary function along with divergence-type conditions on the coefficient sequences of the neutral and delayed terms, the paper establishes new criteria that guarantee oscillation or convergence of all solutions. These novel findings extend and enhance several of the existing oscillation criteria established by the literature. Suggestions for further investigation are included with illustrative examples.

Oscillation theorems for second-order nonlinear difference equations

The limit circle and limit point criteria for second-order linear difference equations