Analytical Solutions Of Periodic Motions Research Articles

Dynamics and Chaos of Convective Fluid Flow

In this paper, a mathematical model of fluid flows in a convective thermal system is developed, and a five-dimensional dynamical system is developed for the investigation of the convective fluid dynamics. The analytical solutions of periodic motions to chaos of the convective fluid flows are developed for steady-state vortex flows, and the corresponding stability and bifurcations of periodic motions in the five-dimensional dynamical system are studied. The harmonic frequency-amplitude characteristics for periodic flows are obtained, which provide energy distribution in the parameter space. Analytical homoclinic orbits for the convective fluid flow systems are developed for the asymptotic convection through the infinite-many homoclinic orbits in the five-dimensional dynamical system. The dynamics of fluid flows in the convective thermal systems are revealed, and one can use such methodology to predict atmospheric and oceanic phenomena through thermal convections.

Period-3 motions to chaos in a periodically forced nonlinear-spring pendulum.

In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.

Experimental and analytical periodic motions in a first-order nonlinear circuit system

In this paper, the analytical solution of a first-order nonlinear circuit is obtained through the generalized harmonic method. The analytical solutions of periodic motions are firstly expressed by finite Fourier series. Both the stability and bifurcations of the nonlinear system with a sinusoidal power-supply are studied by the eigenvalues of the system matrix. In order to verify the analytical solution, the researchers compared the spectrums and waveforms obtained from the analytical model, the simulation model and the circuit. The comparison of the spectrum shows the generalized harmonic balance method can describe the amplitude and phase of the nonlinear circuit more precisely. The comparison of the waveform indicates the output waveform from the simulation model have a good agreement with the results obtained from the analytical model. In the real circuit system, little errors on the output voltage amplitude and response speed maybe due to the nonlinear elements in the circuit and intrinsic voltage attenuation of the chips.

Analytical solutions of periodic motions in a first-order quadratic nonlinear system

In this paper, analytical solutions of period-1 motion of a first-order quadratic nonlinear system with both parametric excitation and external excitation are investigated through the generalized harmonic balance method. The analytical solutions of the system are obtained via solving the coefficients of all harmonic terms at the equilibrium position. The precision of analytical solutions is guaranteed via convergence study of harmonic balance terms. Stability analysis is carried out via eigenvalue analysis. The analytical solutions are different from the perturbation analysis solutions. Moreover, the trajectories of periodic motions obtained from analytical solutions can better explain the dynamics of the system. To verify the accuracy of analytical solutions, numerical simulation are performed and the simulation results are compared with the analytical solutions. The harmonic spectra are also presented to illustrate of the contribution of each harmonic term on a periodic motion.

Bifurcation Trees of Period-1 Motions to Chaos of a Nonlinear Cable Galloping

In this paper, period-m motions on the bifurcation trees of peiod-1 to chaos for nonlinear cable galloping are studied analytically, and the analytical solutions of the period-m motions in the form of the finite Fourier series are obtained through the generalized harmonic balance method, and the corresponding stability and bifurcation analyses of the period-m motions in the galloping system of nonlinear cable are carried out. The bifurcation trees of period-m motions to chaos are presented through harmonic frequency-amplitudes. Numerical illustrations of trajectories and amplitude spectra are given for periodic motions in nonlinear cables. From such analytical solutions of periodic motions to chaos, galloping phenomenon in flow-induced vibration can be further understood.

Analytical solutions of periodic motions in 1-dimensional nonlinear systems

In this paper, analytical solutions of periodic motions in a 1-D nonlinear dynamical system are obtained through the generalized harmonic balance method with prescribed-computational accuracy. From this method, the 1-D dynamical system is transformed to a nonlinear dynamical system of coefficients in the Fourier series. The analytical solutions of periodic motions are obtained by equilibriums of the coefficient dynamical systems, and the corresponding stability and bifurcations of periodic motions are completed via the eigenvalue analysis. The frequency-amplitude characteristics of periodic motions are analyzed through the different order harmonic terms in the Fourier series, and the corresponding quantity levels of harmonic amplitudes are determined. From such frequency-amplitude characteristics, the nonlinearity, singularity and complexity of periodic motions in the 1-D nonlinear systems can be discussed. Displacements and trajectories of periodic motions are illustrated for a better understanding of periodic motions in the 1-D nonlinear dynamical systems. From this study, the periodic motions in the 1-dimensional dynamical systems possess similar behaviors of periodic motions in the van der Pol oscillator.

Periodic motions and limit cycles of linear cable galloping

In this paper, analytical galloping dynamics of linear cables with the uniform airflow of the wind is discussed through a two-degree-of-freedom nonlinear oscillator. The nonlinearity in the two-degree-of-freedom oscillator is only from aero-dynamic forces caused by the uniform airflow. The analytical solutions of periodic motions for linear cable galloping are discussed from the analytical method with the finite Fourier series. The corresponding stability and bifurcation of the periodic motions of the linear cable galloping are determined through the two-degree-of-freedom nonlinear system. Frequency-amplitude analysis of periodic motions and limit cycles of linear cable galloping are presented. Numerical illustrations of trajectories and amplitude spectrums are given for galloping motions of linear cables. From such analytical solutions, galloping phenomenon in flow-induced vibration can be further understood. The galloping dynamics of linear cables is similar to the dynamics of the van der Pol oscillator.

Further analytic solutions for periodic motions in the Duffing oscillator

Harmonic balance method is employed for analytic periodic motions in the Duffing oscillator. Traditionally, one used the perturbation method to discuss the nonlinearity of periodic motions in the Duffing oscillator. For decades, one cannot achieve the appropriate analytical solutions of periodic motions. The harmonic balance method with prescribed-computational accuracy will be used, for a better understanding of nonlinear characteristics of periodic motions in the Duffing oscillator. In this method, the finite Fourier series with coefficient varying with time is adopted to approximately describe periodic motions in nonlinear dynamical system. The dynamical system of the coefficients of the Fourier series is obtained. The equilibrium of such a system gives the constant coefficients in the Fourier series. Thus, the periodic solution is obtained and the stability and bifurcation of the periodic motions can be determined. The analytic periodic motions should be further revisited, since the Duffing oscillator is extensively applied in engineering, and the comprehensive investigation of such periodic motions in the Duffing oscillator is necessary and significant. In this paper, the further analytic solutions of periodic motion in the Duffing oscillator will be investigated, and frequency amplitude characteristics will be discussed, and numerical simulation of periodic motions will be presented. Some new roads from periodic motions to chaos based on generalized harmonic balance method are found, such as independent period-2 motions to chaos and independent period-4 motions to chaos.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

On Analytical Routes to Chaos in Nonlinear Systems

In this paper, the analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems are reviewed. Briefly discussed are the traditional analytical methods including the Lagrange stand form, perturbation methods, and method of averaging. A brief literature survey of approximate methods in application is completed, and the weakness of current existing approximate methods is also discussed. Based on the generalized harmonic balance, the analytical solutions of periodic motions in nonlinear dynamical systems with/without time-delay are reviewed, and the analytical solutions for period-m motion to quasi-periodic motion are discussed. The analytical bifurcation trees of period-1 motion to chaos are presented as an application.

Bifurcation Trees of Period-m Motions to Chaos in a Time-Delayed, Quadratic Nonlinear Oscillator under a Periodic Excitation

In this paper, analytical solutions of periodic motions in a periodically excited, time-delayed, quadratic nonlinear oscillator are obtained through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The analytical bifurcation tree of period-1 motion to chaos in such a time-delayed, quadratic oscillator is presented through period-1 to period-8 motion. Numerical illustrations of stable and unstable periodic motions are given by numerical and analytical solutions. Compared to dynamical systems without time-delay, the timedelayed dynamical systems possess different periodic motions and the bifurcation trees of periodic motions to chaos are also distinguishing.

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

On Periodic Motions in a Parametric Hardening Duffing Oscillator

In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay

In this paper, the analytical dynamics of periodic flows to chaos in nonlinear dynamical systems is presented from the ideas of Luo (Continuous dynamical systems, Higher Education Press/L&H Scientific, Beijing/Glen Carbon, 2012)). The analytical solutions of periodic flows and chaos in autonomous systems are discussed through the generalized harmonic balance method, and the analytical dynamics of periodically forced nonlinear dynamical systems is presented as well. The analytical solutions of periodic motions in free and periodically forced vibration systems are presented. The similar ideas are extended to time-delayed nonlinear systems. The analytical solutions of periodic flows to chaos for time-delayed, nonlinear systems with/without periodic excitations are presented, and time-delayed, nonlinear vibration systems will be also discussed. The analytical solutions of periodic flows and chaos are independent of small parameters, which are different from the traditional perturbation methods. The methodology presented herein will provide the accurate analytical solutions of periodic motions to chaos in dynamical systems with/without time-delay. This approach can handle nonlinear dynamical systems with either single time-delay or multiple time-delays.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

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.

Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance

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.