Homoclinic orbits of the Kovalevskaya top with perturbations

Kuang, Jinlu; Leung, A.Y.T.

doi:10.1002/zamm.200310165

Cited by 6 publications

(9 citation statements)

References 31 publications

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“…The application of Melnikov's techniques [20] for the chaotic oscillations of the gyrostat may be found in Kuang et al [21,22] as suggested by Wiggins and Shaw [23]. The homoclinic chaos of the disturbed Kovalevskaya top was investigated via a MHM integral in Kuang and Leung [24]. The standard spherical pendulum may be regarded as a special case of the gyrostat.…”

Section: Introductionmentioning

confidence: 99%

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

Leung

Kuang

2006

Nonlinear Dyn

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The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov-Holmes-Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge-Kutta algorithms for certain cases to verify the analysis.

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Section: Introductionmentioning

confidence: 99%

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

Leung

Kuang

2006

Nonlinear Dyn

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“…A nonlinear dynamical system with negative linear stiffness as shown in Table 2 is resulted. When S > S critical the numerical simulation shows that the values of α in Equation (14) are positive, yielding a nonlinear dynamical system with positive stiffness as in Table 2. From the relationship between the linear stiffness α and the quantity S in Equation (14), one knows that the linear stiffness α increases as the quantity S increases linearly.…”

Section: Reduction To the Disturbed Hamiltonian Equations With Two Dementioning

confidence: 90%

“…The linear stiffness α in Equation (14) becomes negative and the nonlinear stiffness β in Equation (14) is always positive. Since the gyroscopic coefficient is of the order 10 −5 , we can regard it as a small parameter.…”

Section: Symmetric Periodic Orbits Of the Undisturbed Nanoresonatormentioning

confidence: 98%

Chaotic flexural oscillations of a spinning nanoresonator

Kuang

Leung

2007

Nonlinear Dyn

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The paper investigates the chaotic flexural oscillations of the spinning nanoresonator. The influence of cubic nonlinearity arising from the van der Waals interactions between two neighboring layers of carbon nanotubes on the structural oscillations of the system is considered. The integraldifferential equations describing the flexural displacements of the nanoresonator are discretized into two coupled Duffing-type equations using the GalerkinRitz procedures. The linear stiffness can be either positive or negative, depending on the amplitudes of the linear trap rigidity arising from both the van der Waals interactions and the axial tensile loads. The chaotic flexural oscillations of the appropriately excited spinning nanoresonator are predicted theoretically. Using the Nayfeh-Mook multiscale perturbation algorithms, the coupled Duffing-type equations with linear positive stiffness may be transformed into autonomous equations of slowly modulated amplitudes whose equilibrium points and chaotic dynamics are investigated numerically. The potential chaotic oscillations of the elastic nanoresonator can be determined by the Melnikov-Holmes-Marsden (MHM) integral associated with the homoclinic/heteroclinic solutions of the disturbed Hamiltonian systems with linear negative stiffness. The findings are validated through the Poincare sections and Lyapunov exponents.

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“…The periodic solutions of the torque-free liquid-filled top rotating about a fixed point can also be obtained from the elliptic integral theory and the details for derivation are omitted here for brevity. The constructed heteroclinic orbits in Equations (28) and (29) of the torque-free symmetrical liquid-filled top rolling without sliding on the rough horizontal plane will be exploited to investigate the effect of small disturbances on its motions in the following sections. Figures 3 and 4 show that the heteroclinic orbits of the symmetrical liquid-filled top rolling without sliding on the rough horizontal plane, computed from the relevant physical parameters given in Equations (91)-(93) have the properties lim t→±∞¯ k (t) = kp and lim t→±∞¯ k (t) = kp where the hyperbolic equilibriums kp and kp satisfy Equations (62).…”

Section: Heteroclinic Solutions To a Torque-free Symmetrical Liquid-fmentioning

confidence: 99%

“…Applications of the MHM integrals on a variety of mechanical systems have been carried out by Kozlov [5], Kuang et al. [24,[26][27][28][29], Leung and Kuang [30], Mielke and Holmes [31], Tong et al [32,33], Koiller [34], and Ziglin [35] amongst many others. Based on the work of Wiggins and Shaw [36], Kuang et al [24,26] obtained the Melnikov integral of chaotic rotational dynamics of the gyrostat under the action of small perturbation torques, in the form of damping torques plus periodic moments.…”

Section: Introductionmentioning

confidence: 99%

On the Chaotic Instability of a Nonsliding Liquid-Filled Top with a Small Spheroidal Base via Melnikov-Holmes-Marsden Integrals

2006

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Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.

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Homoclinic orbits of the Kovalevskaya top with perturbations

Cited by 6 publications

References 31 publications

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

Chaotic flexural oscillations of a spinning nanoresonator

On the Chaotic Instability of a Nonsliding Liquid-Filled Top with a Small Spheroidal Base via Melnikov-Holmes-Marsden Integrals

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