DCDS
Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system
Ling-Hao Zhang Wei Wang

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

keywords: Bifurcation hyperbolic function energy balance method strongly nonlinear
DCDS
Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain
Wei Wang Anthony Roberts
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
keywords: averaging overlapping finite elements interelement coupling. Stochastic reaction-diffusion equation
DCDS-B
Limit behavior of nonlinear stochastic wave equations with singular perturbation
Wei Wang Yan Lv
Dynamical behavior of the following nonlinear stochastic damped wave equations

$ \nu $utt$+u_t=$Δ$u+f(u)+$ε$\dot{W}$

on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu, $ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation

$u_t=$Δ$u+f(u).$

is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.

keywords: stationary measure Tightness measure attractor. stochastic partial differential equations
DCDS
Closed trajectories on symmetric convex Hamiltonian energy surfaces
Wei Wang
In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin, where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sigma$.
keywords: Hamiltonian systems. closed characteristics Compact convex hypersurfaces
DCDS-S
On fuzzy filters of Heyting-algebras
Wei Wang Xiao-Long Xin
The concept of fuzzy filter of Heyting-algebras was introduced and some important properties were discussed. Some special kinds of fuzzy filters were defined and we prove that fuzzy Boolean filter is equivelent to fuzzy implicative filter in Heyting-algebras. And the relation among the fuzzy filters were proposed.
keywords: Heyting-algebras positive implicative implicative fuzzy filter ultra fuzzy (prime obstinate) filters. Boolean
DCDS
Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations
Boling Guo Yan Lv Wei Wang
This paper derives a Schrödinger approximation for weakly dissipative stochastic Klein--Gordon--Schrödinger equations with a singular perturbation and scaled small noises on a bounded domain. Detail uniform estimates are given to pass the limit as perturbation and noise disappear. Approximation in two different spaces are considered. Furthermore a large deviation principe of solutions is derived by weak convergence approach.
keywords: large deviation principle weak convergence. Yukawa coupling Klein--Gordon--Schrödinger equations
JIMO
Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model
Linyi Qian Wei Wang Rongming Wang
This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give an example for single unit-linked insurance contracts with guarantee to display the specific locally risk-minimizing hedging strategy.
keywords: Unit-linked life insurance regime switching locally risk-minimizing strategy. Lévy process
DCDS-B
Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity
Hao Yu Wei Wang Sining Zheng

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system $u_t=Δ u-\nabla (u^α \nabla v)$, $v_t=Δ v-v+u$ under non-flux boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^{n}$. The case of $α≥ \max\{1,\frac{2}{n}\}$ with $n≥1$ was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case $α∈(\frac{2}{3},1)$ that if $\|u_0\|_{L^\frac{nα}{2}(Ω)}$ and $\|\nabla v_0\|_{L^{nα}(Ω)}$ are small enough with $n≥q3$, then the solutions are globally bounded with both $u$ and $v$ decaying to the same constant steady state $\bar{u}_0=\frac{1}{|Ω|}∈t_Ω u_0(x) dx$ exponentially in the $L^∞$-norm as $t? ∞$. Moreover, the above conclusions still hold for all $α≥q2$ and $n≥q1$, provided $\|u_0\|_{L^{nα-n}(Ω)}$ and $\|\nabla v_0\|_{L^{∞}(Ω)}$ sufficiently small.

keywords: Keller-Segel system boundedness nonlinear sensitivity
JIMO
Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model
Wei Wang Linyi Qian Xiaonan Su
In this paper, we consider pricing and hedging of catastrophe equity put options under a Markov-modulated jump diffusion process with a Markov switching compensator. We assume that the risk free interest rate, the appreciation rate and the volatility of the risky asset depend on a finite-state Markov chain. We investigate the pricing of catastrophe equity put options and obtain the explicit pricing formulas. A numerical analysis is provided to illustrate the effect of regime switching on the price of catastrophe equity put options. In the end, since the market which we consider is not complete, we also provide an optimal hedging strategy by using the local risk minimization method.
keywords: option pricing. Markov-modulated Local risk minimization
DCDS-B
Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity
Wei Wang Yan Li Hao Yu

In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity:$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$, $v_t=k\Delta v-v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}$ ($n\ge 2$), where $\chi, \, k>0$. It is shown that the solution is globally bounded provided $0<\chi<\frac{-(k-1)+\sqrt{(k-1)^2+\frac{8k}{n}}}{2}$. This result removes the additional restriction of $n \le 8 $ in Zhao, Zheng [15] for the global boundedness of solutions.

keywords: Chemotaxis global boundedness singular sensitivity

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