Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

Huynh, Hoai Nguyen; Nguyen, Thi Phuc Tan; Chew, Lock Yue

doi:10.1016/j.cnsns.2012.06.026

Cited by 11 publications

(10 citation statements)

References 24 publications

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“…We substitute the trial solution (10) into the normalized EOM term by term and we cancel order ε n + θ (1) n = m (m th secular term)e it + other terms (12) In the method of multiple scales, the amplitudes are allowed to vary slowly so to render the series expansions uniformly valid at large time. This can be done by eliminating the secular terms that cause unbounded perturbations.…”

Section: Solving Proceduresmentioning

confidence: 99%

“…Moreover, the synchronization phenomenon of two rotating parametric pendulums attached to common elastic support under harmonic excitation has been studied theoretically and experimentally [8]. The dynamics of the two-pendulum system has been investigated numerically and experimentally, due to the exhibited rich dynamical phenomena [9,10]. For a chain of coupled pendula driven periodically at one end, Khomeriki et al [11] demonstrated experimentally the existence of a novel regime which produces an output frequency at an odd fraction of the driving frequency.…”

Section: Introductionmentioning

confidence: 99%

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Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

Bitar

Kacem

Bouhaddi

2017

International Journal of Mechanical Sciences

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The nonlinear dynamics of a weakly coupled pendulums chain is investigated under primary resonance. The coupled equations governing the nonlinear vibrations are normalized and transformed into a set of coupled complex algebraic equations using the multiple scales method coupled with standing wave decomposition. A model reduction method is proposed to calculate the dominant dynamics without significant loss of accuracy compared to the full model. The validity of the proposed semi-analytical method is verified, and its role in identifying the type of the solution branches is highlighted. The modal interactions and their effects on the nonlinear dynamics are studied in the frequency domain in order to emphasize the large number of multimode solution branches and the bifurcation topology transfer between the modal intensities. Basins of attraction analysis have been performed, showing that the distribution of the multimodal solution branches generated by all modes collectively increases by increasing the number of coupled pendulums.

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Section: Solving Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

Bitar

Kacem

Bouhaddi

2017

International Journal of Mechanical Sciences

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“…A few researchers even believed that "all chaotic responses are simply numerical noise and have nothing to do with the solutions of differential equations" [8] . Thus, due to the SDIC (or the butterfly-effect), it is indeed a challenge to accurately simulate chaotic solution of nonlinear dynamic systems in a long interval of time [10][11] .…”

Section: Introductionmentioning

confidence: 99%

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

Liao

2018

Appl. Math. Mech.-Engl. Ed.

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It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and roundoff error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e. the Clean Numerical Simulation (CNS), is briefly described, together with its applications to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems such as three-body problem, and thus should have many applications in various fields. We find that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions. In addition, it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are timedependent. Thus, our CNS results strongly suggest that one had better to be very careful on DNS results of statistically unsteady turbulent flows, al- * Corresponding author.though DNS results often agree well with experimental data when turbulent flows are in a statistical stationary state.

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“…It is well known that chaos behavior frequently occurs in some highly nonlinear systems and it has certain features including hyper sensitivity to initial values, unpredictable system states, board spectra for its Fourier transform, and fractal properties of the motion in the phase plane. About the chaos research, there are many published articles such as bifurcation analysis [1], chaos control and synchronization [2], fractional order system [3], and geometrical analysis onset of chaos [4] etc. In this study, the Chen's system is studied and will take the system state to be the chaotic random number.…”

Section: Introductionmentioning

confidence: 99%

A Modified Artificial Bee Colony Algorithm for Solving Optimization Problems

Chang

Pan

Lee

et al. 2014

Proceedings of the 2nd International Conference on Intelligent Systems and Image Processing 2014

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In this paper, we develop an improved version of artificial bee colony (ABC) algorithm for solving the numerical optimization problem.In the proposed algorithm, the uniformly random number considered in the formula of generating a new food source is replaced by the chaotic random number which is provided from the Chen's system. To show the feasibility of the proposed scheme, a function minimization problem with constrained conditions is simulated using a large number of different sets of initial conditions. It is concluded from simulation results and comparisons that the proposed method is superior to the general ABC algorithm.

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Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

Cited by 11 publications

References 24 publications

Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

A Modified Artificial Bee Colony Algorithm for Solving Optimization Problems

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