The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-P...The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.展开更多
Consider an initial-boundary problem vt - ux=0,u, + （）x + f（u） = （）x,θt+ux=（）ux=（）x+ （E） v（x,0） = v0（x）,u（x,0） = u0（x）,θ（0,x） = θ0（x）, （I） u（t,0） = u（t,1） = θx（t,0） = θx（t,1） （J...Consider an initial-boundary problem vt - ux=0,u, + （）x + f（u） = （）x,θt+ux=（）ux=（）x+ （E） v（x,0） = v0（x）,u（x,0） = u0（x）,θ（0,x） = θ0（x）, （I） u（t,0） = u（t,1） = θx（t,0） = θx（t,1） （J） Sufficient and necessary conditions for （E）, （I） and （J） to have asymptotic stability of the gobal smooth solution are given by means of the elemental L2 energy method.展开更多
We consider the following Hamiltonian system: q″(t)+V′(q(t))=0 q ∈C^2(R,R^n\{0}) (HS) where V ∈C^2(R^n\{0},R)is an even function.By looking for closed geodesics,we prove that (HS)has a nonconstant periodic solutio...We consider the following Hamiltonian system: q″(t)+V′(q(t))=0 q ∈C^2(R,R^n\{0}) (HS) where V ∈C^2(R^n\{0},R)is an even function.By looking for closed geodesics,we prove that (HS)has a nonconstant periodic solution of prescribed energy under suitable assumptions.Our main assumption is related with the strong force condition of Gordon.展开更多
In this paper we analyze globally the behavior of the solutions of a class of cooperative systems. Our main results is that every orbit of the cooperative system (3.1) either approaches the equilibrium (0, 0, 0), or i...In this paper we analyze globally the behavior of the solutions of a class of cooperative systems. Our main results is that every orbit of the cooperative system (3.1) either approaches the equilibrium (0, 0, 0), or is unbounded, ast→+∞.展开更多
Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a...Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a nonattracting closed orbit. Smith<sup class='a-plus-plus'>[3]</sup> considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine<sup class='a-plus-plus'>[4]</sup> is given. The stability problem of closed orbits raised in [5] and [6] is resolved.展开更多
基金Project Supported by The National Natural Science Foundation of China
文摘The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.
文摘Consider an initial-boundary problem vt - ux=0,u, + （）x + f（u） = （）x,θt+ux=（）ux=（）x+ （E） v（x,0） = v0（x）,u（x,0） = u0（x）,θ（0,x） = θ0（x）, （I） u（t,0） = u（t,1） = θx（t,0） = θx（t,1） （J） Sufficient and necessary conditions for （E）, （I） and （J） to have asymptotic stability of the gobal smooth solution are given by means of the elemental L2 energy method.
文摘We consider the following Hamiltonian system: q″(t)+V′(q(t))=0 q ∈C^2(R,R^n\{0}) (HS) where V ∈C^2(R^n\{0},R)is an even function.By looking for closed geodesics,we prove that (HS)has a nonconstant periodic solution of prescribed energy under suitable assumptions.Our main assumption is related with the strong force condition of Gordon.
基金This is a part of my Master thesis under the direction of Professor Li Bingxi.
文摘In this paper we analyze globally the behavior of the solutions of a class of cooperative systems. Our main results is that every orbit of the cooperative system (3.1) either approaches the equilibrium (0, 0, 0), or is unbounded, ast→+∞.
文摘Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a nonattracting closed orbit. Smith<sup class='a-plus-plus'>[3]</sup> considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine<sup class='a-plus-plus'>[4]</sup> is given. The stability problem of closed orbits raised in [5] and [6] is resolved.
