Chaotic dynamics of an asymmetrical gyrostat

Kuang, Jinlu; Tan, Swee Ngin; Arichandran, Kandiah; Leung, A.Y.T.

doi:10.1016/s0020-7462(00)00091-3

Cited by 52 publications

(45 citation statements)

References 15 publications

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“…It is expedient for one to solve the result for E oe = E oe (ν) and then to plot the amplitude √ E oe = R s versus the frequency parameter ν with A k , μ as family parameters. If A x = 0 and A y = 0, then the oscillations of the spherical pendulum in terms of R x (t) and R y (t) were investigated using the theory on the principal parametric resonance in details in Chin et al [21]. The stability of the formulated equilibrium points of the modulation equations is studied in the next section.…”

Section: Equilibrium Pointsmentioning

confidence: 99%

“…The theory on the bifurcation and chaos may be found in [17][18][19]. 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].…”

Section: Introductionmentioning

confidence: 99%

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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: Equilibrium Pointsmentioning

confidence: 99%

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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“…Equations (26) are called approximate Eulerian equations of the torque-free liquid-filled top rolling without sliding on the rough horizontal plane. By inspection of (26), it is noted that the acceleration of gravity g r does not affect the motion since the shape of the solid body is attached with a spherical base whose geometric centre G e coincides with the centre of mass of the entire liquid-filled body C. One may also observe from (26) that the values and direction-in-space of the principal moment of momenta at the contact point between the torque-free liquid-filled top and the rough horizontal plane are constant.…”

Section: Approximate Dynamics Of a Torque-free Liquid-filled Top Withmentioning

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%

“…[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. Certain interesting models on the nonlinear dynamics of dynamical systems may be found in Wittenburg [37], Heijden and Thompson [38], Thompson and Stewart [39], Hagedorn [40], Chen and Leung [41], and Davies and Moon [42] amongst others.…”

Section: Introductionmentioning

confidence: 99%

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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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Control and synchronization of fractional‐order chaotic satellite systems using feedback and adaptive control techniques

Kumar

MatoukAhmed

ChaudharyHarindri

et al. 2020

Adaptive Control & Signal

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In this article, we present the basic concepts of fractional calculus and control and synchronization of fractional-order chaotic satellite system . Existence and uniqueness solutions of fractional-order satellite system are discussed and local stability of the system at the equilibrium points are studied. The lowest dimension of chaotic attractor of satellite system is 2.88 which is obtained through utilization of the fractional dynamics and computational simulation. Lyapunov exponents and bifurcation diagrams are drawn to measure the existence of chaos in the system. Feedback control method for chaos control in the fractional-order satellite system is achieved. Synchronization of two identical noninteger order chaotic satellite systems are achieved through adaptive control methodology.

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Chaotic dynamics of an asymmetrical gyrostat

Cited by 52 publications

References 15 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

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

Control and synchronization of fractional‐order chaotic satellite systems using feedback and adaptive control techniques

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