Research On Well Posedness Of Solutions For Some Uncertain Dynamical Systems

The theory of Fuzzy differential equations is important tool to study mathematical models with uncertain or subjective information.By solving fuzzy differential equations,practical problems with uncertainties from the fields of physics,control theory and neural networks can be solved.In particular,many physical phenomena are closely related to the periodic solutions or the period-doubling solutions of fuzzy differential equations.However,due to the particularity of the subtraction operation on the fuzzy numbers,solving a fuzzy differential equation is different from solving an ordinary differential equation in the real field.In order to solve fuzzy differential equations,the commonly methodologies are: the method based on the Zadeh expansion principle,that is,the solutions of fuzzy differential equations is extended from solutions of the ordinary differential equation with uncertain parameters or initial values by applying the Zadeh expansion principle;the method based on the H derivatives and Bede’s generalized derivatives,that is,solving the ordinary differential equation in the fuzzy number space by the corresponding derivatives;the method based on the theory of differential inclusions,that is,by taking the level set of the fuzzy differential equation and transforming it to solve the corresponding problem of differential inclusions,and then let the solution set of the differential inclusions constitutes the level set of the solution to original fuzzy differential equation.In recent years,the differential inclusion method has gradually become an important tool for solving fuzzy differential equations.Since the fuzzy differential equation is solved by the Zadeh expansion principle,the calculation is complicated.When the fuzzy differential equation is solved based on the H derivatives,the support set of the solution will increase.Therefore,the two-point boundary value problem of the fuzzy differential equation often has no solution.In particular,the periodic problem of fuzzy differential equations has no solution in the sense of H derivatives.When fuzzy differential equations are solved based on Bede’s generalized derivatives,the solutions are obtained in pairs.The support set of one solution will increaseing,while another is decreasing.For the periodic problem of fuzzy differential equations,a switching point is needed,and the derivative of periodic solution with different differential properties on two sides of the switching point under Bede’s generalized derivatives.This result leads to certain limitations in practical engineering applications.But the differential inclusions method is effective in solving the two-point boundary value problem of fuzzy differential equations,especially the periodic problems.Based on differential inclusions method,we can not only discuss the existence of solutions to periodic problems,but also discuss the stability of its solutions.In this paper,we study the fuzzy differential equations under differential inclusions(DI-type fuzzy differential equations)i.e.,uncertain dynamical systems,such as the periodic problem and the period-doubling problem of semi-linear uncertain dynamical systems,and the related problems of universal oscillator uncertain dynamical systems.The main content includes the following aspects:The first part studies the periodic problems and period-doubling problems of onedimensional first-order semi-linear uncertain dynamical systems.Since the support set of the solution to the fuzzy differential equation is increasing in the sense of H derivatives,there is no solution to the periodic problems.Under the Bede’s generalized derivative,the solutions appearing in pairs also have limitations.In this paper,the differential inclusions method is used to study the periodic problem and the period-doubling problem of uncertain dynamical systems.Based on the differential inclusions method,the Green function and the concept of big solution are introduced to study the properties of solutions to the periodic problem and the period-doubling problem for the first-order semi-linear uncertain dynamical system.The second part studies the structural stability of one-dimensional first-order semilinear uncertain dynamical systems.Since there is no solution to the fuzzy periodic problem in the sense of H derivatives,it is impossible to discuss its structural stability.Therefore,this paper studies the structural stability of semi-linear uncertain dynamical systems under the differential inclusions method.On the basis of the existence of the solution and the large solution of the semi-linear uncertain dynamical system,the metric defined by the support function,the Dini theorem and the convergence theorem in the differential inclusion theory are used to discuss the structural stability of the big solution and solution under some perturbations on coefficient,the forcing function or coefficient and the forcing function.The third part studies the periodic problem of n-dimensional first-order semi-linear uncertain dynamical systems.The periodic problems of semi-linear uncertain dynamical systems have many applications in the field of physics.Since the n-dimensional fuzzy number cannot be represented by the new parameter method,it is impossible to use the big solution to discuss the boundedness of solution sets.In this paper,we use the theory of differential inclusion,functional analysis,Sobolev space theory and set-value analysis to study the boundedness of solution sets.The existence and uniqueness of the periodic solutions of the semi-linear uncertain dynamical systems are discussed.When the forcing function has a specific perturbation,the structural stability problem of the periodic solution is discussed by using the support function,the Dini Theorem and the convergence Theorem in the differential inclusion theory.The fourth part studies the two-point boundary value of the universal oscillator uncertain dynamical systems.Universal oscillator uncertain dynamical systems are widely present in physical practical problems with uncertainties.However,due to H derivatives method to solve the two point boundary value problem often has no solution,this paper uses the differential inclusions method to study the two point boundary value problems.According to the relationship of the coefficients,universal oscillator uncertain dynamical systems can be considered as three different damping system probelms: underdamped uncertain dynamical systems,critically damped uncertain dynamical systems and overdamped uncertain dynamical systems.Based on the differential inclusion method,Green function and boundary value constraints,the existence and uniqueness of the solutions of the above three damping systems are discussed respectively.When the forcing function does not contain the damping term,this problem is studied by introducing the concept of big solution.In general,this paper uses the method of differential inclusion to study several types of uncertain dynamic systems.The periodic problems and period-doubling problems of several kinds of semi-linear uncertain dynamical systems are discussed.The two-point boundary value problems of the universal oscillator uncertain dynamical systems are studied.