Various fascinating phenomena occur in nonlinear systems.' They are discussed generally by making use of steady solutions of :I diRerentia1 equation. The equation is usually derived from a real system by neglecting small uncertain factors-hence, it is deterministic. I f a single solution of the equation is asymptotically stable and its basin is large compared with random noise. the corresponding phenomenon turns out to be deterministic. But if a bundle of solutions containing infinitely many unstable periodic solutions is asymptotically orbitally stable. a chaotic phenomenon appears, which results from the small uncertain lactors in the real system. That is. the representative point of the physical state wanders chaoticnllq i n the bundle of solutions. Because of this characteristic. we have called the phenomenon a \"chaotically transitional process.\" '* We have long been studying this subject in connection with forced oscillatory phenomena in nonlinear electrical and electronic circuits. As analytical solutions of differential equations cannot be expected in these problems, we have been relying on analog and digital computers. Thus, both the global structure of solutions and the long-term movement of a representative point have been examined. So the question arises, Are computer solutions valid'? However, we have not entered into details of this problem, though we have found consistency between the analog and digital results. Therefore. our results may lack mathematical rigor; nevertheless, they will have an important influence on many researchers in various fields. This paper also describes the chaotically transitional processes exhibited by Duffing's equation. Special attention is directed toward the transition of the processes and the explosion of strange attractors is clarified.

Explosion of Strange Attractors Exhibited by Duffing's Equation

A triboelectric energy harvester based on a three-degree-of-freedom vibro-impact oscillator is presented. Both the dynamic model of the oscillator and the theoretical model of the oscillator-based triboelectric energy harvester are established. The dynamic response and its effect on the electrical output are considered for various mass ratios and mass spacings. The study leads to the conclusions that the symmetric mass configurations of the oscillator are more beneficial to energy harvesting than the asymmetric cases. The extent of the initial spacing between the masses influences the dynamics of the system and the electrical output by triggering grazing bifurcation. High-order periodicity is found to accompany a reduction in the electrical power. An increase in mass ratio tends to increase the electrical output, and there may exist an optimal mass ratio at which the electrical output is maximized. Chatter and sticking motion can improve the output performance dramatically, while resonance, as usual, corresponds to large amplitude response, but these large amplitudes are not optimal for triboelectric energy harvesting, and thus the maximal output does not appear around resonance. This is different from other types of vibration-based energy harvesters, such as piezoelectric and magnetoelectric energy harvesters which are usually designed to operate at resonance. In addition, chatter is found to occur at low excitation frequencies, which can help harvest energy from low-frequency ambient vibration.

/pdf/nonlinear-dynamics-and-triboelectric-energy-harvesting-from-5922kvyo5n.pdf

Nonlinear dynamics and triboelectric energy harvesting from a three-degree-of-freedom vibro-impact oscillator

This paper reports various chaotic phenomena that occur in a single-link flexible joint (SLFJ) robot manipulator. Four different cases along with subcases are considered here to show different types of chaotic behaviour in a flexible manipulator dynamics. In the first three cases, a partial state feedback as joint velocity and motor rotor velocity feedback is considered, and the resultant autonomous dynamics is considered for analyses. In the fourth case, the manipulator dynamics is considered as a non-autonomous system. The system has (1) one stable spiral and one saddle-node foci, (2) two saddle-node foci and (3) one marginally stable nature of equilibrium points. We found single- and multi-scroll chaotic orbits in these cases. However, with the motor rotor velocity feedback, the system has two unstable equilibria. One of them has an index-4 spiral repellor. In the non-autonomous case, the SLFJ robot manipulator system has an inverse crisis route to chaos and exhibits (1) transient chaos with a stable limit cycle and (2) chaotic behaviour. In all the four cases, the SLFJ manipulator dynamics exhibits coexistence of chaotic orbits, i.e. multi-stability. The various dynamical behaviours of the system are analysed using available methods like phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincare map, parameter space, bifurcation diagram, 0–1 test and frequency spectrum plot. The MATLAB simulation results support various claims made about the system. These claims are further confirmed and validated by circuit implementation using NI Multisim.

Coexistence of single- and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria

This paper attempts to find some interesting and unique properties in the dynamics of a permanent magnet synchronous motor (PMSM). When compared with the existing literature on the dynamics of a PM...

Multistability and Hidden Attractors in the Dynamics of Permanent Magnet Synchronous Motor

On bifurcation trees of period-1 to period-2 motions in a nonlinear Jeffcott rotor system

In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos.

A Semi-analytical Prediction of Periodic Motions in Duffing Oscillator Through Mapping Structures

Vibro-Impact Dynamics: Luo/Vibro

In this paper, with varying excitation amplitude, bifurcation trees of periodic motions to chaos in a periodically driven pendulum are obtained through a semi-analytical method. This method is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures are developed for specific periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved. Further, semi-analytical bifurcation trees of periodic motions to chaos are also obtained, and the corresponding eigenvalue analysis is carried out for the stability and bifurcation of the periodic motions. Finally, numerical illustrations of periodic motions on the bifurcation trees are presented in verification of the analytical prediction. Harmonic amplitude spectra are also presented for demonstrating harmonic effects on the periodic motions. The bifurcation trees of period-1 motions to chaos possess a double spiral structure. The two sets of solutions of period-
 $$2^{l}$$
 motions $$(l=0,1,2,\ldots )$$
 to chaos are based on the center around $$2m\pi $$
 and $$(2m-1)\pi (m=1,2,3,\ldots )$$
 in phase space. Other independent bifurcation trees of period-m motions to chaos are presented. Through this investigation, the motion complexity and nonlinearity of the periodically forced pendulum can be further understood.

Routes of periodic motions to chaos in a periodically forced pendulum

In this paper, periodic motions of a periodically forced, damped, Duffing oscillator with double-well potential are analytically predicted through discrete implicit mappings. The discrete implicit maps are obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions of the Duffing oscillator are predicted analytically, and the corresponding stability and bifurcation analysis of periodic motions are carried out through the eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are performed by the numerical method of the differential equation to verify the semi-analytical prediction, and the corresponding harmonic amplitudes are computed through discrete Fourier series of the analytically predicted node points of periodic motions, and the complexity of periodic motions can be measured by the harmonic amplitude. The frame work presented in this paper can provide a semi-analytical method to find periodic motions and to determine routes of periodic motions to chaos rather than numerical simulation only in nonlinear dynamical systems, and the stable and unstable periodic motions and even chaos can be predicted analytically.

Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings

It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions’ complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.

Periodic Motions to Chaos in Pendulum

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