Quintic Hamiltonian System Research Articles

Bifurcation of limit cycles for a class of quintic Hamiltonian systems with quartic perturbed terms

Bifurcation of limit cycles for a class of quintic Hamiltonian systems with quartic perturbed terms

Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop

In this paper, we study the cyclicity of periodic annulus and Hopf cyclicity in perturbing a quintic Hamiltonian system. The undamped system is hyper-elliptic, non-symmetric with a degenerate heteroclinic loop, which connects a hyperbolic saddle to a nilpotent saddle. We rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated Hamiltonian system. This result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4. When the smooth polynomial damping term has degree n, first, a transformation based on the involution of the Hamiltonian is introduced, and then we analyze the coefficients involved in the bifurcation function to show that the Hopf cyclicity is [2n+13]. Further, for piecewise smooth polynomial damping with a switching manifold at the y-axis, we consider the damping terms to have degrees l and n, respectively, and prove that the Hopf cyclicity of the origin is [3l+2n+43] ([3n+2l+43]) when l≥n (n≥l).

The Number of Zeros of Abelian Integral for a Quintic Hamiltonian Systems With a Rose-Figure Loop

In this article, we consider the near-Hamiltonian system where a < 0, 0 < |ε| << 1, f (x, y) and g(x, y) are polynomials of degree n ( n = 2 m , m ≥ 3, m ∈ N ). The number of isolated zeros of the corresponding Abelian integral for h ∈ is estimated.

Bifurcations of limit cycles in equivariant quintic planar vector fields

In this paper, we obtain 23 limit cycles for a Z3-equivariant near-Hamiltonian system of degree 5 which is the perturbation of a Z6-equivariant quintic Hamiltonian system. The configuration of these limit cycles is new and different from the configuration obtained by H.S.Y. Chan, K.W. Chung and J. Li, where the unperturbed system is a Z3-equivariant quintic Hamiltonian system. Our unperturbed system is different from the unperturbed systems studied by Y. Wu and M. Han. The limit cycles are obtained by Poincaré–Pontryagin theorem and Poincaré–Bendixson theorem.

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper, we consider Lienard systems of the form $$\frac{{dx}} {{dt}} = y, \frac{{dy}} {{dt}} = - \left( {x + bx^3 - x^5 } \right) + \varepsilon \left( {\alpha + \beta x^2 + \gamma x^4 } \right)y,$$ where b ∈ ℝ, 0 < |∈| ≪ 1, (α, β, γ) ∈ D ∈ ℝ3 and D is bounded. We prove that for |b| ≫ 1 (b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals Open image in new window is 2 (counting the multiplicity) and this upper bound is a sharp one.

The cyclicity of period annuli of a class of quintic Hamiltonian systems

In this paper, we prove that the cyclicity of period annuli for system dxdt=y,dydt=−(ax+bx3+cx5) under perturbations of polynomials with degree n is not more than 54n−13 (taking into account the multiplicity), where a,b,c∈R with c≠0 such that the unperturbed systems have at least one center.

Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop

In this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic Hamiltonian vector field X H = y ∂ ∂ x − x 3 ( x 2 − 1 ) ∂ ∂ y under small perturbations of the form ε ( α + β x 2 + γ x 4 ) y ∂ ∂ y , where 0 < ∣ ε ∣ ≪ 1 and α , β , γ are real constants. Using Picard–Fuchs equations for related abelian integrals, asymptotic expansion of these integrals about critical level curves of H , and some geometric properties of the curves defined by ratios of two especial integrals, we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two.

Bifurcations of limit cycles in a perturbed quintic Hamiltonian system with six double homoclinic loops

This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation. With the aid of numeric integral computation provided by Mathematica 4.1, at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation, Hopf bifurcation and qualitative analysis. The four configurations of 45 limit cycles of the system are also shown. The results obtained are useful to the study of the weakened 16th Hilbert Problem.

Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (II)

In this paper we consider Lieńard equations of the form { x ̇ = y , y ̇ = − ( x − 2 x 3 + x 5 ) − ε ( α + β x 2 + γ x 4 ) y where 0 < | ε | ≪ 1 , ( α , β , γ ) ∈ Λ ⊂ R 3 and Λ is bounded. We prove that the least upper bound for the number of zeros of the related Abelian integrals I ( h ) = ∮ Γ h ( α + β x 2 + γ x 4 ) y d x for h ∈ ( 1 / 6 , ∞ ) is three and for h ∈ ( 0 , ∞ ) is four (counted with multiplicity) for all parameters α , β and γ . This implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for ε = 0 outside an eye-figure loop is less than or equal to three.

Detection function method and its application to a class of quintic Hamiltonian systems with quintic perturbations

By using the bifurcation theory of planar dynamical systems and the method of detection functions to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system. It is shown that under two different sets of controlled parameters, the given system has at least 23 limit cycles having two different configurations of the compound eyes, respectively.

Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop

In this paper we consider Lieńard equations of the form { x ̇ = y , y ̇ = − ( x − 2 x 3 + x 5 ) − ε ( α + β x 2 + γ x 4 ) y , where 0 < | ε | ≪ 1 , ( α , β , γ ) ∈ Λ ⊂ R 3 and Λ is bounded. We prove that the least upper bound for the number of zeros of the related Abelian integrals I ( h ) = ∮ Γ h ( α + β x 2 + γ x 4 ) y d x is 2 (taking into account their multiplicities) for h ∈ ( 0 , 1 / 6 ) and this upper bound is a sharp one. This implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for ε = 0 inside an eye-figure loop is less than or equal to 2.

Bifurcations of limit cycles for a perturbed quintic Hamiltonian system with four infinite singular points

By using the theory of bifurcations of dynamical systems and the method of detection function to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system with 25 finite singular points and four infinite singular points. From the detection functions for the perturbed system, we prove that under different determined parameter condition, the given system has at least 22 and 20 limit cycles and the configurations of compound eyes are also obtained.

Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form x ˙ = y , y ˙ = P ( x ) + y Q ( x , y ) , with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis.

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

Abstract This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H (6) ⩾ 35 = 6 2 − 1, where H (6) is the Hilbert number for sixth-degree polynomial systems.

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

This paper deals with Liénard equations of the form x ̇ =y, y ̇ =P(x)+yQ(x,y) , with P and Q polynomial of degree 5 and 4, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight-loop. It is proved that the Hopf cyclicity is two, and it is also given by the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinic loop for quintic system with quintic perturbations by using the methods of bifurcation theory and qualitative analysis.

The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

AbstractThe Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

Detection function method and its application to a perturbed quintic Hamiltonian system

We employ the methods of both qualitative analysis and numerical exploration to investigate the bifurcation of limit cycles of a perturbed quintic Hamiltonian system. With the help of the detection functions for the perturbed Hamiltonian system, we study the perturbation of a quintic Hamiltonian system with 25 finite singular points and 4 infinite singular points. We first classify the phase portraits of the unperturbed system and categorize the closed orbits, then obtain the detection functions for the perturbed system, from which the detection curves and the number and the distribution of limit cycles are obtained. As application examples, we numerically compute the detection curves and draw the distribution of limit cycles. It seems that our methods are very effective on the study of the bifurcation of limit cycles of quintic Hamiltonian systems.