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With the help of the variable substitution and applying the fixed point theorem, we derive the sufficient conditions which guarantee the existence of the positive almost periodic solutions for a class of Lotka-Volterra type system. The main results improve and generalize the former corresponding results.

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter κ. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.

The existence of approximate inertial manifold using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation has a good localization property of the numerical solution distinguishably.

The traveling wave soliton solutions and pair soliton solution to a class of new completely integrable shallow water equation, Camassa-Holm equation are studied. The concept of concave or convex peaked soliton and smooth soliton were introduced. And the research shows that the traveling wave solution of that equation possesses concave and convex peaked soliton and smooth soliton solutions with the peakson. Simultaneously by applying Backlund transformation the new pair soliton solutions to this class of equation are given.

The existence of local attractors in thin 2D domains for the weakly damped forced KdV equation, whose principal operator is a non-self adjoint and non-sectorial one is given.

The time estimate of the blow-up of the weakly damped forced KdV equation in thin 2D domains is given.

The numerical analysis of the approximate inertial manifold in weakly damped forced KdV equation is given. The results of numerical analysis under five models is the same as that of nonlinear spectral analysis.

In the paper what is studied is the wild solution of the induced form under the spline wavelet basis in weakly damped forced KdV equation.

It is presented that there exists approximate inertial manifolds in weakly damped forced KdV equation with periodic boundarý conditions. The approximate inertial manifolds provide approximant of the attractor by finite dimensional smooth manifolds which are explicitly defined. And the concept leads to new numerical schemes which are well adapted to the longtime behavior of dynamical system.

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

In the present paper we study the maximum dissipative extension of Schrodinger operator, introduce the generalized indefinite metric space and get the representation of maximum dissipative extension of Schrodinger operator in natural boundary space, make preparation for the further study longtime chaotic behavior of infinite dimension dynamic system in nonlinear Schrodinger equation.

In the paper, we study longtime dynamic behavior of dissipative soliton equation: existence of attractor, geometrical structure of attractor, dynamic behavior under the parametric perturbation of dissipative soliton equation, estimate of fractal dimension of attractor.

We study the following nonlinear equationdx(t)/dt=x(t)[a(t)-b(t)xα(t)-f(t,x(t))]+g(t), by using fixed point theorem, the sufficient conditions of the existence of a unique positive almost periodic solution for above system are obtained, by using the theories of stability, the sufficient conditions which guarantee the stability of the unique positive almost periodic solution are derived.

By constructing two new Riccati equations and using the generalized Riccati method, we simplified the form and enriched the general results.Using Mathematica software, exact solutions of the variable coefficient combined KdV equation with forced term are obtained, including many kinds of solitary-wave-like solutions,guasi-periodical solutions and solitary wave solutions with variable speed.

Study one type of nonlinear equations,namely generalized Camassa-Holm equation C(n):u\-t+ku\-x+β\-1uxxt+β\-2(un+1)x+β\-3u\-x(u\+n)xx+β\- 4u(u\+n)xxx=0.Obtain abundant exact solutions by four ansasz,particularly when k≠0,we obt ain compacton solutions;while whenk=0,we obtain floating compacton solution s.At last,we also study other forms of fully nonlinear generalized Camassa-Holm equation,and their compacton solutions are governed by nonlinear equations.