Bounded And Unbounded Solutions For Asymmetric Oscillators

Duffing equation is an important planar Hamiltonian system. It has important physical significance. The boundedness and unboundedness of solutions, the existence and multiplicity of periodic solutions for Duffing equation are active research topics for many years.It is well-known that Duffing equation has the following form x"+f(x)=p(t), p(t+2Ï€)=p(t), where p(t) is defined in R, f(x) is a continuous function in R.J. Moser and L. Markus in1962and1969proposed to study Duffing equation independently. It has been widely investigated and many results have been obtained for the existence and multiplicity of periodic solutions by various methods, such as the critical point theory, the phase plane technique and the continuation method based on a degree theory and so on. Littlewood asked whether or not all the solutions of Duffing equation are bounded all the time, i.e., whether there are resonances that might cause the amplitude of the oscillation to increase without bound, here and in what follows, a solution x(t) is bounded if it exists for all t E R andFrom then on, this problem is always called Littlewood boundedness problem.The situation is very different when the function f(x) has linear growth at infinity and, in particular, the following limits exist and are finite:Ifa≠b, the equation is called an asymmetric equation. A simple second-order differ-ential equation with asymmetric nonlinearities x"+ax+-bx-=p(t), where a and b are positive constants with a≠b, x+=max{x,0},x-=max{-x,0}, p(t) is a2Ï€-periodic continuous function, was firstly studied by Dancer and Fucik in their investigations of boundary value problems. Lately, this class of equations also called equations with "jumping nonlinearities". The interest for such equations has been motivated in particular by models of suspension bridges. This kind of equation also has already been considered in the engineering literature as a model of oscillator with "stops". The boundedness of all the solutions, the existence of periodic solutions or unbounded solutions have been widely studied in the literature.The known results for linear equations on the existence of periodic solutions al-ready shed some light on the boundedness problem. Actually, the second theorem of Massera implies that if it has no periodic solutions then every solution must be un-bounded. In1998, R. Ortega proved the different case for the asymmetric oscialltors. He proved that the existence of2Ï€-periodic function p(t) such that all the solutions of this equation with large initial conditions are unbounded. That is to say, the periodic and unbounded solutions can coexist when a≠b under the resonance case, i.e.1/√a+1/√b∈Q.For the asymmetric oscillator, we always consider two cases:(1) the resonance case:(2) the nonresonance case:The thesis mainly focus on the boundedness and unboundedness of solutions, the existence of periodic solutions for the second order Duffing equation with asymmetric oscillators. We also consider whether or not the unbounded and periodic solutions can coexistence for the asymmetric oscillators.In the thesis, we mainly consider the boundedness and unboundedness of solu-tions, the existence of periodic solutions for the asymmetric oscillators. We consider whether or not the unbounded and periodic solutions can coexistence for the asymmet-ric oscillators?The thesis consists of three parts. The first part considers to study the existence of periodic solutions and unbounded solutions for the Lienard perturbed asymmetric os-cillator with damping; the second part is devoted to consider the unbounded solutions of more general asymmetric oscillator with Rayleigh equation; the third part is to inves- tigate the boundedness and unboundedness of solutions for the asymmetric oscillator x"+ax+-bx-+g(x)=p(t), where x+=max{x,0},x-=max{-x,0}, a and b are two positive constants, p(t) is a2Ï€-periodic smooth function and g(x) satisfies lim|x|â†'+∞x-1g(x)=0. This is the most important part in this thesis. Our main results are the following:We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions with large initial values. Both these results allow the function g(x) be oscillatory without asymptotic limits at infinity or be unbounded. So our results are almost necessary and sufficient for the boundedness of all the solutions of above asymmetric equation.The first part of this thesis consists of Chapter2. We study the existence of pe-riodic solutions and unbounded solutions for the perturbed asymmetric oscillator with damping x"+f(x)x’+ax+-bx-+g(x)=p(t), f(x) is a continuous function and p(t) is a2Ï€-periodic continuous function, g(x) is locally Lipschitz continuous and bounded. The bounded and unbounded solutions have been widely studied for the equation by many authors. Unlike many existing results in the literature where the function g(x) is required to be a bounded function with asymptotic limits, here we allow g(x) oscillatory without asymptotic limits. That is, g(x) is bounded and has generalized limits, i.e., limxâ†'±∞(1/x)∫0xg(s)ds=G±exist and are finite. We discuss the existence of unbounded solutions under two classes of conditions:the resonance case1/√a+1/√b∈Q and the nonresonance case1/√a+1/√b(?)Q. The main results are Theorems2.1-2.3.The second part of the thesis consists of Chapter3. In this chapter, we consider the unboundedness of solutions for the asymmetric equation x"+ax+-bx-+ψ(x)ψ(x’)+f(x)+g(x’)=p(t),where f(x) is locally Lipschitz continuous, bounded and admits generalized limits at infinity, ψ(x),ψ(x),g(x) and p(t) are continuous functions, and p(t) is a2Ï€-periodic function. The unboundedness problem for such equations with asymmetric nonlineari-ties depending on the derivatives has been relatively little research. So we discuss the 2existence of unbounded solutions under two classes of conditions:the resonance case1/√2+1/√b=∈Q and the nonresonance case The main results are Theorems3.1-3.4.Chapter4is the third part. In this chapter, we consider the boundedness and unboundedness of solutions for the asymmetric oscillator x"+ax+-bx-+g(x)=p(t).Firstly, we establish some sharp sufficient conditions concerning the boundedness of all the solutions. By using the expression of time map Ï„(h) given in the Poincare map and the variant twist theorem given by Ma Shiwang and Wu Jianhong, and estimating strictly, we get that all the solutions are bounded. Unlike many existing results in the literature where the function g(x) is required to be a bounded function with asymptotic limits, here we allow g(x) be unbounded or oscillatory without asymptotic limits. See the main results:Theorem4.1, Theorem4.2and Theorem4.3. Then we study the the unbounded solutions of abstract planar mapAt last, by using this abstract result, we get some sharp sufficient conditions on the exis-tence of unbounded solutions with large initial values under the condition of unbounded function of g(x). The main results are Theorems4.4-4.5.