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In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.

Constructed motions and dynamic topology are new trends in solving nonlinear systems or system interactions. In nonlinear engineering, it is significant to achieve specific complex motions to satisfy expected dynamical behaviors (e.g., nonlinear motions, singularities, bifurcations, chaos, etc.), and complex motion application and control. To achieve such expected motions and global dynamical behaviors, mapping dynamics, constructed networks, random/discontinuous dynamic theorems, etc., are applied to quantitatively determine the complex motions. These theories adopt the symbolic dynamic abstracts and topological structures with nonlinear dynamics to investigate constructed complex motions to satisfy expected dynamical behaviors.

A bifurcation tree of period-3 motions to chaos in an inverted pendulum with a periodic base movement is presented through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are also carried out. Frequency-amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Numerical simulations of periodic motions on the bifurcation are completed. The numerical and analytical results are presented for comparison. Except for period-1 motion to chaos studied before, this study focuses on other periodic motions to chaos existing in the inverted pendulum with periodic base movement. In the earthquake testing, one tests structures from 1hz to 33 hz. However, during such a frequency range, one not only can observe the period-1 motion to chaos, but one can observe period-3 motion to chaos in such an inverted pendulum with a periodic base movement. Thus, in the building design, period-3 motions to chaos should be considered, which have different dynamical behaviors from the period-1 motions to chaos.

In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2 n motions [Formula: see text]. On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.

In this paper, nonlinear piezoelectric energy harvesting induced by a Duffing oscillator is studied, and the bifurcation trees of period-1 motions to chaos for such a piezoelectric energy-harvesting system are obtained analytically. Distributed-parameter electromechanical modeling of a piezoelectric energy harvester is presented first, and the electromechanically coupled circuit equation excited by infinitely many vibration modes is developed. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates, and eventually an infinite set of algebraic equations is obtained for the complex modal vibration responses and the complex voltage responses of the energy harvester beam. One single mode case is considered in this paper, and periodic motions with bifurcation trees are obtained through an implicit discrete mapping method. The frequency–amplitude characteristics of periodic motions are obtained for the nonlinear piezoelectric energy-harvesting systems, which provide a better understanding of where and how to achieve the best energy harvesting. This study describes about how the nonlinear oscillator induces piezoelectric energy harvesting through a beam system. The nonlinear piezoelectric energy harvesting is presented through a nonlinear oscillator. Due to the nonlinear oscillator, chaotic piezoelectric energy-harvesting states can get more energy compared to the linear piezoelectric energy-harvesting system.

The book is about the global stability and bifurcation of equilibriums in polynomial functional systems. Appearing and switching bifurcations of simple and higher-order equilibriums in the...

In this paper, symmetric period-3 motions and pairs of asymmetric period-3 and period-6 motions in a periodically excited nonlinear spring pendulum system are determined semi-analytically through the implicit map method. Numerical and analytical results are illustrated for verification. The time histories of displacements and velocities are illustrated for each periodic motions; and the corresponding trajectories in phase plane are also presented.

In this paper, periodic responses in a nonlinear oscillator with electromagnetic resonant shunt tuned mass damper inerter (ERS-TMDI) are studied. Symmetric and asymmetric periodic motions of such a system are obtained through an implicit mapping method. The stability and bifurcation analysis of periodic motions are completed through eigenvalue analysis. Frequency–amplitude characteristics of periodic motions are presented by the finite Fourier series analysis for a better understanding of nonlinear responses of the nonlinear oscillator with electromagnetic resonant shunt tuned mass damper inerter. Numerical simulations of the stable periodic motions are presented to verify the semi-analytical predictions of symmetric and asymmetric periodic motions. The asymmetric periodic motions cannot be obtained easily by the traditional methods. The frequency–amplitude characteristics of symmetric and asymmetric periodic motions are very good for design of ERS-TMDI. The results presented herein are also applicable for wind-induced vibration, earthquake building vibration reduction and energy harvesting.

This paper presented the dynamics of an infinite-equilibrium system based on a quadratic oscillator. equilibria, stability and singularity of such an infinite equilibrium system are discussed through the local analysis, and numerical studies of the periodically perturbed infinite-equilibrium systems are completed. The infinite-equilibrium boundaries in infinite-equilibrium systems can be artificially designed to control motions in the corresponding non-infinite-equilibrium systems. Through this study, one can have a better understanding of dynamics of infinite-equilibrium nonlinear systems. The authors believe infinite-equilibrium systems will have extensive applications in science and engineering.