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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. [ABSTRACT FROM AUTHOR]

Due to the discontinuous physical property of the control actuators, the state space of such a dynamical system is divided into many subdomains. For each subdomain, the flow of such a system is governed by the corresponding subsystem. The state boundary between the adjacent subdomains is called the physical switching boundary. The controller is designed to switch when the subsystem of such a discontinuous dynamical system is switched in order to have the optimum control performance. Since the ambiguity and uncertainty of modeling, the mathematical expressions for describing the discontinuous physical properties of the control actuators may not be accurate. Since the nominal switching boundary where the controller really switches is not exactly the corresponding physical switching boundary, the mismatch between the subsystem and the corresponding controller will occur and it may seriously affect the control performance. Therefore, a boundary estimation algorithm is proposed to estimate the physical switching boundaries based on the model reference control and error backpropagation. The simulation results show that the adaptive sliding mode control with the boundary estimation algorithm has superior control performance and strong robustness to deal with the internal uncertainty, the external interference, and the boundary ambiguity. [ABSTRACT FROM AUTHOR]

Single-machine infinite-bus power system is a nonlinear dynamic system with state variable in the sinusoidal function. With traditional analytical approaches, it is difficult to analyze such a nonlinear system since the rotor angle difference cannot always stay in an infinitesimal small value. In this paper, a single-machine infinite-bus power system with power disturbance will be discussed. The implicit discrete maps approach will be applied to solve the periodic motions for such a power system, and the stability condition will be discussed. The analytical expressions for periodic motions for such a single-machine infinite-bus power system can be recovered with a series of Fourier functions. The bifurcation diagram for such a system will be given to show the complexity of the motions when the frequency of the disturbance varies, and 2-D parameter map for chaotic motion will be obtained by calculating the Kolmogorov-Sinai entropy density. From analytical bifurcation for period-1 and period-2 motions, the evolution process of the periodic motion to chaos can be analytically explained. [ABSTRACT FROM AUTHOR]

In this paper, analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. This nonlinear Jeffcott rotor system with two-degrees of freedom is excited periodically from the rotor eccentricity. The analytical solutions of period-m solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled, nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in such a buckled nonlinear Jeffcott rotor. [ABSTRACT FROM AUTHOR]

In this paper, period-3 motions to chaos in the periodically forced, softening Duffing oscillator are investigated analytically. The analytical solutions for period-3 and period-6 motions are approximated through the generalized harmonic balance method. The bifurcation trees of period-3 motions to chaos are presented analytically. The symmetric and asymmetric period-3 motions are found. The symmetric to asymmetric period-3 motions experience the saddle-node bifurcation. From the Hopf bifurcation of the asymmetric period-3 motion, period-6 motions are determined analytically from the bifurcation tree of period-3 motions. Such an investigation provides a complete picture of period-3 motions to period-6 motions. With such bifurcation tree, the chaotic motions relative to period-3 motions in such a softening Duffing oscillator can be determined analytically. In a similar fashion, other bifurcation trees of period-m motions to chaos can be determined analytically. [ABSTRACT FROM AUTHOR]

In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well. However, asymmetric periodic motions in the softening Duffing oscillators are not investigated analytically yet, and such asymmetric periodic motions possess much richer dynamics than the symmetric motions in the softening Duffing oscillator. For asymmetric motions, the bifurcation tree from asymmetric period-1 motions to chaos is discussed comprehensively. The corresponding, unstable and stable, asymmetric and symmetric, periodic motions in the softening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are completed. This investigation provides a better picture of complex motion in the softening Duffing oscillator. [ABSTRACT FROM AUTHOR]

In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period-k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters. [ABSTRACT FROM PUBLISHER]

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters. [ABSTRACT FROM AUTHOR]

Abstract: The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method). [Copyright &y& Elsevier]