This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equatio...This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.展开更多
We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally mo...We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.展开更多
In order to make the peak and offset of the signal meet the requirements of artificial equipment,dynamical analysis and geometric control of the laser system have become indispensable.In this paper,a locally active me...In order to make the peak and offset of the signal meet the requirements of artificial equipment,dynamical analysis and geometric control of the laser system have become indispensable.In this paper,a locally active memristor with non-volatile memory is introduced into a complex-valued Lorenz laser system.By using numerical measures,complex dynamical behaviors of the memristive laser system are uncovered.It appears the alternating appearance of quasi-periodic and chaotic oscillations.The mechanism of transformation from a quasi-periodic pattern to a chaotic one is revealed from the perspective of Hamilton energy.Interestingly,initial-values-oriented extreme multi-stability patterns are found,where the coexisting attractors have the same Lyapunov exponents.In addition,the introduction of a memristor greatly improves the complexity of the laser system.Moreover,to control the amplitude and offset of the chaotic signal,two kinds of geometric control methods including amplitude control and rotation control are designed.The results show that these two geometric control methods have revised the size and position of the chaotic signal without changing the chaotic dynamics.Finally,a digital hardware device is developed and the experiment outputs agree fairly well with those of the numerical simulations.展开更多
An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 wi...An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.展开更多
The stabilization problem of a nonuniform Timoshenko beam system With coupled locally distributed feedback controls is studied. First, by proving the uniqueness of the solution to the related ordinary differential equ...The stabilization problem of a nonuniform Timoshenko beam system With coupled locally distributed feedback controls is studied. First, by proving the uniqueness of the solution to the related ordinary differential equations, we establish the asymptotic decay of the energy corresponding to the closed loop system. Then, by virtue of piecewise multiplier method, we prove the exponential decay of the closed loop system.展开更多
Compared with input-to-state stability(ISS) in global version,the concept of local input-to-state stability(LISS) is more relevant and meaningful in practice.The key of assessing LISS properties lies in investigating ...Compared with input-to-state stability(ISS) in global version,the concept of local input-to-state stability(LISS) is more relevant and meaningful in practice.The key of assessing LISS properties lies in investigating three main ingredients,the local region of initial states,the local region of external inputs and the asymptotic gain.It is the objective of this paper to propose a numerical algorithm for estimating LISS properties on the theoretical foundation of quadratic form LISS-Lyapunov function.Given developments of linear matrix inequality(LMI) methods,this algorithm is effective and powerful.A typical power electronics based system with common DC bus is served as a demonstration for quantitative results.展开更多
文摘This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.
文摘We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.
基金Project supported by the National Natural Science Foundation of China(Grant No.61773010)Taishan Scholar Foundation of Shandong Province of China(Grant No.ts20190938)。
文摘In order to make the peak and offset of the signal meet the requirements of artificial equipment,dynamical analysis and geometric control of the laser system have become indispensable.In this paper,a locally active memristor with non-volatile memory is introduced into a complex-valued Lorenz laser system.By using numerical measures,complex dynamical behaviors of the memristive laser system are uncovered.It appears the alternating appearance of quasi-periodic and chaotic oscillations.The mechanism of transformation from a quasi-periodic pattern to a chaotic one is revealed from the perspective of Hamilton energy.Interestingly,initial-values-oriented extreme multi-stability patterns are found,where the coexisting attractors have the same Lyapunov exponents.In addition,the introduction of a memristor greatly improves the complexity of the laser system.Moreover,to control the amplitude and offset of the chaotic signal,two kinds of geometric control methods including amplitude control and rotation control are designed.The results show that these two geometric control methods have revised the size and position of the chaotic signal without changing the chaotic dynamics.Finally,a digital hardware device is developed and the experiment outputs agree fairly well with those of the numerical simulations.
基金This work was supported by National Natural Science Foundation of China （61174209, 11471034）.
文摘An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.
基金This research is partially supported by the Research Committee of the Hong Kong Polytechnic University by the Science Foundation of China Geosciences University (Beijing) (200304).
文摘The stabilization problem of a nonuniform Timoshenko beam system With coupled locally distributed feedback controls is studied. First, by proving the uniqueness of the solution to the related ordinary differential equations, we establish the asymptotic decay of the energy corresponding to the closed loop system. Then, by virtue of piecewise multiplier method, we prove the exponential decay of the closed loop system.
基金supported by the National Natural Science Foundation of China (Grant Nos 50977047 and 50907038)
文摘Compared with input-to-state stability(ISS) in global version,the concept of local input-to-state stability(LISS) is more relevant and meaningful in practice.The key of assessing LISS properties lies in investigating three main ingredients,the local region of initial states,the local region of external inputs and the asymptotic gain.It is the objective of this paper to propose a numerical algorithm for estimating LISS properties on the theoretical foundation of quadratic form LISS-Lyapunov function.Given developments of linear matrix inequality(LMI) methods,this algorithm is effective and powerful.A typical power electronics based system with common DC bus is served as a demonstration for quantitative results.
