Title 1: Bifurcations of Travelling Wave Solutions for Fully Nonlinear Water Waves with Surface Tension in the Generalized Serre-Green-Naghdi Equations
Date: 2019.12.9 9:00-9:40
Place: School of Management Meeting Room 2
Reporter: LI Jibin
Abstract: For the generalized Serre-Green-Naghdi equations with surface tension, by using the methodologies of dynamical systems and singular traveling wave theory developed Li \& Chen [2007] to their travelling wave systems, in different parameter conditions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) are obtained. More than 26 explicit exact parametric representations are given. It is interesting to find that this fully nonlinear water waves equation has coexistence of uncountably infinitely many smooth solitary wave solutions or uncountably infinitely many pseudo-peakon solutions with periodic solutions or compacton solutions. Differing from the well-known peakon solution of the Camassa-Holm equation, the generalized Serre-Green-Naghdi equations have four new forms of peakon solutions.
Title 2: Bifurcation Methods of Periodic Orbits for Piecewise Smooth Systems
Date: 2019.12.9 9:40-10:20
Place: School of Management Meeting Room 2
Reporter: HAN Maoan
Abstract: It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar smooth near Hamiltonian differential systems. In this paper, we study piecewise smooth near integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example.
Title 3: Univoque bases of real numbers: local dimension, Devil’ s staircase and isolated points
Date: 2019.12.9 10:40-11:20
Place: School of Management Meeting Room 2
Reporter: LI Wenxia
Abstract: Given a positive integer M and a real number x > 0, let U(x) be the set of all bases q ∈ (1,M +1] for which there exists a unique sequence (di) = d1d2... with each digit di∈ {0,1,...,M} satisfying x = ∑ (di/i). The sequence (di) is called a q-expansion of x. In this paper we investigate the local dimension of U(x) and prove a‘variation principle’for unique non-integer base expansions. We also determine the critical values of U(x) that when x passes the first critical value the set U(x) changes from a set with positive Hausdorff dimension to a countable set, and when x passes the second critical value the set U(x) changes from an infinite set to a singleton. Denote by U(x) the set of all unique q-expansions of x for q∈U(x). We give the Hausdorff dimension of U(x) and show that the dimensional function x→U(x) is a non-increasing Devil’s staircase. Finally, we investigate the topological structure of U(x). In contrast with x=1 that U(1) has no isolated points, we prove that for typical x > 0 the set U(x) contains isolated points. In this talk, we will introduce a class of nonlocal diffusion equations and the regular traveling waves, obtain the existence and uniqueness of the regular traveling waves for the equations, and discuss the effect of different dispersal strategies on the minimal wave speed.
Title 4: Traveling waves for a nonlocal diffusion population model
Date: 2019.12.9 11:20-12:00
Place: School of Management Meeting Room 2
Reporter: XIAO Dongmei
Abstract: In this talk we will introduce a nonlocal diffusion population model with spatio-temporal delays, and discuss the existence, uniqueness and stability of traveling waves for this model. This is based on the joint works with Zhaoquan Xu.
Title 5: Optimization by chaotic dynamics
Date: 2019.12.9 14:00-14:40
Place: School of Management Meeting Room 2
Reporter: CHEN Luonan
Abstract: We developed a new algorithm and method for optimization by chaotic dynamics, called chaotic simulated annealing (CSA). Specifically, we first constructed a neural network model by introducing a transiently chaotic dynamics. Unlike traditional neural networks only with point attractors, our transiently chaotic neural network (TCNN) has richer and more flexible dynamics, so that I can be expected to have higher ability of searching for globally optimal solutions. Bench examples of optimization problems demonstrated the high efficiency and effectiveness of CSA.
Title 6: The SIS model in switched networks
Date: 2019.12.9 16:20--17:00
Place: School of Management Meeting Room 2
Reporter: JIN Zhen
Abstract: Because of individuals’ random walk, people have different behaviors and thus have different social contact patterns. Therefore, topology of human social contact networks is time-varying, and the epidemic dynamics on networks is often subject to environmental noise and uncertainty. In this talk, we investigate dynamic characteristics of some SIS network epidemic model with Markovian switching, including individual-based network infectious disease SIS model, and mean field model based on degree distribution. An epidemic threshold is established for the extinction and permanence of the model, we shows that the epidemic propagation in switched networks is quite different from that of static networks. In addition, based on Lyapunov function method, positive recurrence and ergodicity of stochastic spreading processes are also discussed.
Title 7: Exact solutions of nonlocal Fokas-Lenells equation
Date: 2019.12.9 15:40--16:20
Place: School of Management Meeting Room 2
Reporter: ZHANG Yi
Abstract: In this work, I aim at investigating what roles the noises are playing in bioreactors, with the hope to provide some sound advices to develop proper control schemes.In this talk we propose a nonlocal Fokas-Lenells (FL) equation which can be derived from the Kaup-Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several type of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.
Title 8: Dynamics of slow-fast predator-prey model of Holling type III
Date: 2019.12.9 14:40--15:20
Place: School of Management Meeting Room 2
Reporter: ZHANG Xiang
Abstract: For a classical ratio-dependent predator-prey model of generalized Holling type III functional response, combining the geometric singular perturbation theory we obtain richer new dynamical phenomena than the existing ones, such as global stability, canard cycle, relaxation oscillation, canard explosion, cyclicity of slow-fast cycles et al.
