We carry out numerical and theoretical investigations on the global unstable invariant set(manifold)of a saddlenode limit cycle in a leech heart interneuron model.The corresponding global bifurcation is accompanied by...We carry out numerical and theoretical investigations on the global unstable invariant set(manifold)of a saddlenode limit cycle in a leech heart interneuron model.The corresponding global bifurcation is accompanied by an explosion of secondary bifurcations of limit cycles and the emergence of loop-shaped bifurcation structures.The dynamical behaviors of the trajectories of the invariant set are very complicated and can only be partially explained by existing theories.展开更多
A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confi...A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confined within linear response theory. The calculating experience suggests that the proposed technique is simple and efficient in implementation, and the comparison with stochastic simulation shows that the first three orders of susceptibilities calculated by the proposed technique have high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated, and another observed phenomenon of stochastic resonance in the systems induced by the location of a single periodic orbit is disclosed and explained.展开更多
Mechanism of period-adding cascades with chaos in a reduced leech neuron model is suggested as the bifurcation of a saddle-node limit cycle with homoclinic orbits satisfying the ＂small lobe condition＂, instead of th...Mechanism of period-adding cascades with chaos in a reduced leech neuron model is suggested as the bifurcation of a saddle-node limit cycle with homoclinic orbits satisfying the ＂small lobe condition＂, instead of the blue-sky catastrophe. In every spiking adding, the new spike emerges at the end of the spiking phase of the bursters.展开更多
This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, ...This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.展开更多
A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itseff in an alternat...A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itseff in an alternation between the oscillation phase following attracting the limit cycle branch and resting phase following a repelling fixed point branch in a reduced leech neuron model Via periodic-chaotic alternating of infinite times, the number of windings within a canards bursting can approach infinity at a Gavrilov-Shilnikov homoclinic tangency bifurcation of a simple saddle limit cycle.展开更多
Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attr...Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable in- variant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make gen- eralized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.展开更多
In this paper,a new type of stabilized finite element method is discussed for Oseen equations based on the local L^(2)projection stabilized technique for the velocity field.Velocity and pressure are approximated by tw...In this paper,a new type of stabilized finite element method is discussed for Oseen equations based on the local L^(2)projection stabilized technique for the velocity field.Velocity and pressure are approximated by two kinds of mixed finite element spaces,P^(2)_( l)-P_(1),(l=1,2).A main advantage of the proposed method lies in that,all the computations are performed at the same element level,without the need of nested meshes or the projection of the gradient of velocity onto a coarse level.Stability and convergence are proved for two kinds of stabilized schemes.Numerical experiments confirm the theoretical results.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos 10432010 and 10872155.
文摘We carry out numerical and theoretical investigations on the global unstable invariant set(manifold)of a saddlenode limit cycle in a leech heart interneuron model.The corresponding global bifurcation is accompanied by an explosion of secondary bifurcations of limit cycles and the emergence of loop-shaped bifurcation structures.The dynamical behaviors of the trajectories of the invariant set are very complicated and can only be partially explained by existing theories.
基金Project supported by BH foundation of XJTU and the major program of the National Natural Science Foundation of China（Grant No 10432010）.
文摘A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confined within linear response theory. The calculating experience suggests that the proposed technique is simple and efficient in implementation, and the comparison with stochastic simulation shows that the first three orders of susceptibilities calculated by the proposed technique have high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated, and another observed phenomenon of stochastic resonance in the systems induced by the location of a single periodic orbit is disclosed and explained.
基金Supported by the National Natural Science Foundation of China under Grant No 10432010.
文摘Mechanism of period-adding cascades with chaos in a reduced leech neuron model is suggested as the bifurcation of a saddle-node limit cycle with homoclinic orbits satisfying the ＂small lobe condition＂, instead of the blue-sky catastrophe. In every spiking adding, the new spike emerges at the end of the spiking phase of the bursters.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10432010 and 10472086)
文摘This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.
基金Supported by Natural Science Foundation of China under Grant No 10432010.
文摘A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itseff in an alternation between the oscillation phase following attracting the limit cycle branch and resting phase following a repelling fixed point branch in a reduced leech neuron model Via periodic-chaotic alternating of infinite times, the number of windings within a canards bursting can approach infinity at a Gavrilov-Shilnikov homoclinic tangency bifurcation of a simple saddle limit cycle.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10432010, 10502039)
文摘Three main parts of generalized cell mapping are improved for global analysis. A simple method, which is not based on the theory of digraphs, is presented to locate complete self-cycling sets that corre- spond to attractors and unstable invariant sets involving saddle, unstable periodic orbit and chaotic saddle. Refinement for complete self-cycling sets is developed to locate attractors and unstable in- variant sets with high degree of accuracy, which can start with a coarse cell structure. A nonuniformly interior-and-boundary sampling technique is used to make the refinement robust. For homeomorphic dissipative dynamical systems, a controlled boundary sampling technique is presented to make gen- eralized cell mapping method with refinement extremely accurate to obtain invariant sets. Recursive laws of group absorption probability and expected absorption time are introduced into generalized cell mapping, and then an optimal order for quantitative analysis of transient cells is established, which leads to the minimal computational work. The improved method is applied to four examples to show its effectiveness in global analysis of dynamical systems.
基金This work is supported by NSF of China(Nos.11071184,11271273,11371275,41674141)NSF of Shanxi Province(No.2012011015-6)+3 种基金STIP of Higher Education Institutions in Shanxi(No.20111121)Young Scholars Development Fund of SWPU(No.201599010041)Young Science and Technology Innovation Team of SWPU(No.2015CXTD07)Key Program of SiChuan Provincial Department of Education(No.16ZA0066).
文摘In this paper,a new type of stabilized finite element method is discussed for Oseen equations based on the local L^(2)projection stabilized technique for the velocity field.Velocity and pressure are approximated by two kinds of mixed finite element spaces,P^(2)_( l)-P_(1),(l=1,2).A main advantage of the proposed method lies in that,all the computations are performed at the same element level,without the need of nested meshes or the projection of the gradient of velocity onto a coarse level.Stability and convergence are proved for two kinds of stabilized schemes.Numerical experiments confirm the theoretical results.
