By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by ...By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.展开更多
In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the tw...In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the two Melnikov functions,we give a general method to obtain the number of limit cycles near the cuspidal loop.As an application,we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.展开更多
The research of homoclinic and heteroclinic bifurcations for high-dimension systems havedrawn much attention. It is often difficult to study such bifurcation problems ofhigh-dimension. We considered the
In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsil...In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).展开更多
The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's...The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's function. The results can be applied to the construction of phase portraits in the bifurcation set of codimension two bifurcations of flows with doublezero eigenvalues.展开更多
It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-d...It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.展开更多
Using the method of multi-parameter perturbation theory and qualitative analysis,a cubic system perturbed by degree four are investigated in this paper. After systematic analysis,it is found that the studied system ca...Using the method of multi-parameter perturbation theory and qualitative analysis,a cubic system perturbed by degree four are investigated in this paper. After systematic analysis,it is found that the studied system can have nine limit cycles with their distributions are obtained.展开更多
In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cy...In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.展开更多
Single-pulse chaos are studied for a functionally graded materials rectangular plate. By means of the global perturbation method, explicit conditions for the existence of a SiZnikov-type homoclinic orbit are obtained ...Single-pulse chaos are studied for a functionally graded materials rectangular plate. By means of the global perturbation method, explicit conditions for the existence of a SiZnikov-type homoclinic orbit are obtained for this sys- tem, which suggests that chaos are likely to take place. Then, numerical simulations are given to test the analytical predic- tions. And from our analysis, when the chaotic motion oc- curs, there are a quasi-period motion in a two-dimensional subspace and chaos in another two-dimensional supplemen- tary subspace.展开更多
基金supported by the National Natural Science Foundation of China (Grant 11172199)
文摘By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.
基金supported by National Natural Science Foundation of China(Grant No.11971145)supported by National Natural Science Foundation of China(Grant No.11931016)the National Key R&D Program of China(Grant No.2022YFA1005900)。
文摘In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the two Melnikov functions,we give a general method to obtain the number of limit cycles near the cuspidal loop.As an application,we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.
文摘The research of homoclinic and heteroclinic bifurcations for high-dimension systems havedrawn much attention. It is often difficult to study such bifurcation problems ofhigh-dimension. We considered the
文摘In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).
基金The project is supported by the National Natural Science Foundation of China
文摘The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's function. The results can be applied to the construction of phase portraits in the bifurcation set of codimension two bifurcations of flows with doublezero eigenvalues.
文摘It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.
基金supported by the Natural Science Foundation of Shandong Province,China（No.ZR2010AZ003）
文摘Using the method of multi-parameter perturbation theory and qualitative analysis,a cubic system perturbed by degree four are investigated in this paper. After systematic analysis,it is found that the studied system can have nine limit cycles with their distributions are obtained.
文摘In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.
基金supported by the National Natural Science Foundation of China(11172125,11202095 and 11201226)Natural Science Foundation of Henan,China(2009B110009,B2008-56 and 649106)
文摘Single-pulse chaos are studied for a functionally graded materials rectangular plate. By means of the global perturbation method, explicit conditions for the existence of a SiZnikov-type homoclinic orbit are obtained for this sys- tem, which suggests that chaos are likely to take place. Then, numerical simulations are given to test the analytical predic- tions. And from our analysis, when the chaotic motion oc- curs, there are a quasi-period motion in a two-dimensional subspace and chaos in another two-dimensional supplemen- tary subspace.
