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
In this paper, Melnikov functions which appear in the study of limit cycles of a perturbedplanar Hamiltonian system are studied. There are two main contributions here. The first oneis related to the explicit formulae ...In this paper, Melnikov functions which appear in the study of limit cycles of a perturbedplanar Hamiltonian system are studied. There are two main contributions here. The first oneis related to the explicit formulae for these functions: a new method is developed to achievethe goal for the second one (Theorem A). the authors also discover a close relation betweenMelnikov functions and focal quantities (Theorem B). This relation is useful in both judgingwhen a Melnikov function is identically zero and simplifying the computation of a Melnikovfunction (see 5). Despite these results, this paper also includes other related results, e.g. someestimations of the upper bound for the number of limit cycles in a perturbed Hamiltoniansystem.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
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
This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0....This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.展开更多
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
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In this paper, Melnikov functions which appear in the study of limit cycles of a perturbedplanar Hamiltonian system are studied. There are two main contributions here. The first oneis related to the explicit formulae for these functions: a new method is developed to achievethe goal for the second one (Theorem A). the authors also discover a close relation betweenMelnikov functions and focal quantities (Theorem B). This relation is useful in both judgingwhen a Melnikov function is identically zero and simplifying the computation of a Melnikovfunction (see 5). Despite these results, this paper also includes other related results, e.g. someestimations of the upper bound for the number of limit cycles in a perturbed Hamiltoniansystem.
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
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.
基金supported by the Natural Science Foundation of Ningxia(2022AAC05044)the National Natural Science Foundation of China(12161069)。
文摘This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.
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]).
