Nonlinear Random Differential Equations Research Articles

Some basic random fixed point theorems with PPF dependence and functional random differential equations

In this paper two basic random fixed point theorems with PPF dependence are proved for random operators in separable Banach spaces with different domain and range spaces. The obtained abstract results are applied to certain nonlinear functional random differential equations for proving the existence results for random solutions with PPF dependence. Mathematics subject classification (2010): 34K10, 47H10.

Existence and attractivity results for nonlinear first-order random differential equations

In this paper, the existence and attractivity results are proved for nonlinear first order ordinary random differential equations. Two examples are provided to demonstrate the realization of the abstract developed theory.

On nth-order nonlinear ordinary random differential equations

An existence result for a nonlinear nth-order ordinary random differential equation is proved under the Caratheodory condition. Two existence results for extremal random solutions are also proved in the Caratheodory case and the discontinuous case of the nonlinearity involved in the equations. Our investigations are carried out in the Banach space of continuous real-valued functions on closed bounded intervals of the real line together with the application of a random version of the Leray–Schauder principle.

Existence and attractivity results for nonlinear first order random differential equations

Existence and attractivity results for nonlinear first order random differential equations

Second order nonlinear random differential equations

In this paper, an existence result for a nonlinear second order ordinary random dif- ferential equation is proved under a Caratheodory condition. Two existence results for extremal random solutions are also proved for Caratheodory as well as discontinuous cases of the nonlin- earity involved in the equations. Our investigations are placed in the Banach space of continuous real-valued functions on closed and bounded intervals of the real line together with an application of the random version of the Leray-Schauder principle.

Transient Bimodality and Catastrophic Jumps in Innovation Diffusion

A new phenomenon in innovation diffusion that exhibits a transient bimodality familiar in physical sciences is predicted in this paper for the first time on the basis of an extended Bass model. This accounts for the population heterogeneity, with the parameters characterizing the ldquoword of mouthrdquo and ldquomass mediardquo processes as random. A theoretical framework based on a nonlinear random differential equation is thus introduced to develop a stochastic model for new product diffusion with parametric uncertainty. The analytical investigation of the model establishes the existence of transient bimodality, which manifests through a cusp catastrophe. Illustrations based on analytical and simulation studies are presented. Empirical validation of the proposed model based on well-known data sets where the Bass model is known to fit very well has been carried out.

Nonlinear functional random differential equations in Banach algebras

In this paper the existence as well as the existence of the extremal solutions for first order nonlinear perturbed functional random differential equations is proved under
 mixed Lipschitz, compactness and monotonic conditions.

Stability and Capsizing of Ships in Random Sea – a Survey

This report is a survey of methods of stochastic and nonlinear dynamics in ship stability. After a brief introduction we describe the sea as a stationary random field. We then derive the general equations of motion of a ship from ‘first principles’, specializing to the case of the equations of motion for roll, heave and sway using strip theory from which eventually the ‘archetypal’ nonlinear random differential equation for the roll motion follows. This determines in particular how and where the stochasticity of the sea enters the equation. We then analyze simple nonlinear models of ship motion by means of the theory of random dynamical systems which amounts to studying invariant measures, Lyapunov exponents, random attractors and their (random) domain of attraction and to using stochastic bifurcation theory to describe qualitative changes.

Non-Stationary Random Response of MDOF Duffing Systems

This paper presents a method to solve non-stationary random responses of nonlinear multi-degrees-of-freedom (MDOF) Duffing systems subjected to evolutionary random excitations. Specific phase-lags between the excitations can also be taken into account. The power spectral density (PSD) of the input excitations is not confined to simple white noise or filtered white noise, in fact it can also take more complicated forms. The MDOF nonlinear random differential equations are iteratively solved by means of the Equivalent Linearization Method (ELM) combined with the Pseudo Excitation Method (PEM). This combined method is easy and efficient. Two examples are given in which this method is well justified by the Monte-Carlo numerical simulations. Although only a Duffing model is dealt with in this paper for computational simplicity, the proposed method is in fact quite general, e.g. it can also deal with nonlinear hysteretic structures that will be dealt with in a separate paper.

Homogenization of a nonlinear random parabolic partial differential equation

The aim of this work is to show how to homogenize a semilinear parabolic second-order partial differential equation, whose coefficients are periodic functions of the space variable, and are perturbed by an ergodic diffusion process, the nonlinear term being highly oscillatory. Our homogenized equation is a parabolic stochastic partial differential equation.

Solutions for a system of nonlinear random integral and differential equations under weak topology

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. Then, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Ranach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.

Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive

This paper is concerned with the (non-)explosion behavior of solutions of non-linear random and stochastic differential equations.

Solutions for a system of nonlinear random integral and differential equations

In this paper, we first prove a Darbao type fixed point theorem for a system of continuous random operators with random domains. Then, by using the theorem, we give the existence criteria of solutions for a systems of nonlinear random Volterra integral equations and for the Cauchy problem of a system of nonlinear random differential equations. The existence of extremal random solutions and random comparison results for these systems of random equations are also obtained. Our theorems improve and generalize the corresponding results of Vaughn, Lakshmikantham, Lakshmikantham-Leela, De Blasi-Myjak and Ding.

Asymptotic Approximation of the Solution of a Random Boundary Value Problem Containing Small White Noise

This paper considers a nonlinear random differential equation [formula] where α(ω) is F1-measurable and w is an Rm-valued Wiener process. By introducing a weak problem, the shooting method can be used to prove the uniqueness of the Rn-valued Ft-measurable solution x(t) in the meaning of large probability. If the parameter ϵ is small, then x(t) = x0(t) + ϵx1(t) + O(ϵ2), where x0(t) is the solution with ϵ = 0 and x1(t) satisfies a linear random boundary value problem. For simplicity the discussion is given in the scalar case, but extensions to higher dimensions are readily apparent.

A note on the probability distribution of the dissipation rate in locally isotropic turbulence

A nonlinear random differential equation describing the magnitude of vorticity is given for fine vortex tube in locally isotropic turbulence. From this vortex dynamics model, the asymptotic expressions of the probability density function of the dissipation rate in locally isotropic turbulence are obtained. In particular, several physical mechanisms behind turbulence statistics are explored.

Existence and comparison results of solutions for nonlinear random integral and differential equations relative to weak topology

In this paper, we prove several existence theorems of random solutions to nonlinear random Volterra integral equations under the weak topology of Banach spaces. Then, as applications, we obtain the existence theorems of weak random solutions to random differential equations. Existence of extremal random solutions and a random comparison theorem for these random equations are also obtained. Our theorems improve and extend the corresponding results in [4,5,10,11,12].

Random directional contractors and their applications

In this paper, as the generalizations of Altman's directional contractors[4,5] and Lee and Padgett's random contractors[1,2] we introduce the concept of random directional contractors for set-valued random operators. Using the new concept and transfinite induction, we show several existence theorems to solutions of nonlinear set-valued random operator equations. Our theorems improve and generalize the corresponding results in [1,2,4,5,11]. Next, some applications of our results to nonlinear random integral and differential equations are given.

A comparison between Adomian's decomposition methods and perturbation techniques for nonlinear random differential equations

Comparison of the Adomian decomposition method and regular perturbation techniques applied to the solution of nonlinear vector random differential equations shows that the decomposition method is superior and generally applicable. Even for the cases where perturbation is applicable, the comparison shows that the decomposition method is easier to compute and supplies quantitatively reliable results.

Criteria for the existence of solutions to random integral and differential equations

In[1], we proved a general random fixed point theorem and gave some applications. In this paper, we shall give further applications of the theorem. We first obtain a random Darbo's fixed point theorem, using the theorem, we give the criteria for the existence of solutions under compactness hypotheses to nonlinear random volterra integral equations and the Cauchy problem of nonlinear random differential equations. Our theorems improve and generalize some main results of Lakshmikanthem[2], Vaughn[3,4] as well as De Blasi and Myjak[5].

Fixed point theorems of random set-valued mappings and their applications

In this paper, first we show several new random fixed point theorems for random set-valued mappings and for a system of random set-valued mappings. Then, some applications of our results are given for the existence and uniqueness of random solution for a system of nonlinear random integral and differential equations. Our theorems improved and generalize many recent findings in [4–7, 9, 11–17].