In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues an...In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.展开更多
To simulate a multivariate density with multi_hump, Markov chainMonte Carlo method encounters the obstacle of escaping from one hump to another, since it usually takes extraordinately long time and then becomes practi...To simulate a multivariate density with multi_hump, Markov chainMonte Carlo method encounters the obstacle of escaping from one hump to another, since it usually takes extraordinately long time and then becomes practically impossible to perform. To overcome these difficulties, a reversible scheme to generate a Markov chain, in terms of which the simulated density may be successful in rather general cases of practically avoiding being trapped in local humps, was suggested.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12171171)Natural Science Foundation of Fujian Province of China(Grant Nos.2022J01303 and 2023J01121)the Scientific Research Funds of Huaqiao University。
文摘In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.
文摘To simulate a multivariate density with multi_hump, Markov chainMonte Carlo method encounters the obstacle of escaping from one hump to another, since it usually takes extraordinately long time and then becomes practically impossible to perform. To overcome these difficulties, a reversible scheme to generate a Markov chain, in terms of which the simulated density may be successful in rather general cases of practically avoiding being trapped in local humps, was suggested.
