Qualitative Analysis Of Nonlinear Systems Research Articles

Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters.
 The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters.
 The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions.
 Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions.
 Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

Analysis of the quasiperiodic response of a generalized van der Pol nonlinear system in the resonance zone

The paper addresses the description of the complex behavior of simple nonlinear systems that are excited in the neighborhood of the resonance frequency. Depending on the detuning of the excitation frequency, resonant response can vary from purely stationary to various cases of quasiperiodic or chaotic response. This type of response is characterized by regular or irregular changes of the amplitude, which, in the quasiperiodic case, represents the beating effect. The beating frequency then changes from zero in resonance to a positive value outside the resonance zone. The ratio of the energy content of quasiperiodic and stationary components decreases in the same time. Starting at a certain detuning, the quasiperiodic component fully vanishes and the stationary component absorbs the whole response energy.The motivation of this study originates from the aeroelasticity of large bridges, the tuned mass damper application, and other domains of civil engineering, where beating effects have been observed in the past. Such effects are very dangerous; hence, robust theoretical background for the design of adequate countermeasures should be developed. Nevertheless, investigations of the internal structure of a quasiperiod and its dependence on the difference between excitation frequency and eigenfrequency were conducted on a heuristic basis and an objective theoretical background is still missing.A qualitative analysis of nonlinear systems using combinations of harmonic balance, small-parameter methods, and perturbation techniques is presented in the paper. Parametric evaluations are presented along with a discussion concerning the applicability of the presented approach.

Qualitative Analysis of Nonlinear Systems by the Method of Matrix Lyapunov Functions

Qualitative Analysis of Nonlinear Systems by the Method of Matrix Lyapunov Functions

Instabilities and chaos in nonlinear dynamic systems

AbstractIncreasing computational power in the last decade has led to a breakthrough in the understanding of nonlinear dynamic systems. Deterministic chaos, a distinct mode of behavior qualitatively different from linear modes, can arise in systems with instabilities constrained by nonlinearities.This article overviews this new field. It is observed that deterministic chaos is associated with the presence of a basic instability, often associated with a negative loop with high‐gain and significant delay, that allows small random fluctuations to be amplified and eventually dominate the behavior of the system.The mathematical tools of qualitative analysis of nonlinear systems are applied to familiar models of urban dynamics. Spatial disaggregation in an urban migration model is shown to yield a route to chaos through period‐doubling. Alternative routes to chaos are discussed. Examples from the natural and social sciences are used to demonstrate the applicability of the tools and the ubiquity of such complex nonlinear dynamic phenomena.