Abstract
Abstract In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.
Highlights
Many applied problems represented by dynamical systems have many difficulties to obtain their solutions in explicit forms definitely when these problems are modelled bynonlinear systems, cf. [5,7,12]
The directions of progress to analyze qualitatively for obtaining the main mathematical properties become must. Most of these properties are stuck in the presence of periodic solutions, the bifurcations and the route to the chaotic behaviours or more less based on their dynamics as well
Special type of nonlinear ordinary differential equations called the Rayleigh–Duffing equation is qualitatively studied by seeking the stability of solutions and existence of periodic solutions via the fixed point method and the second method of Lyapunov
Summary
Many applied problems represented by dynamical systems have many difficulties to obtain their solutions in explicit forms definitely when these problems are modelled by (non-autonomous)nonlinear systems, cf. [5,7,12]. The directions of progress to analyze qualitatively for obtaining the main mathematical properties become must Most of these properties are stuck in the presence of periodic solutions, the bifurcations and the route to the chaotic behaviours or more less based on their dynamics as well. Many literatures have been discussed on the qualitative analysis of Eq. and its periodic solutions by different techniques, cf [1, 8,9,10, 14, 23, 24, 26] Motivated by this argument, in this work, we discuss the stability and the existence of periodic solutions as well as the transition to chaos for very specific type of non-autonomous nonlinear ODEs, so-called the Rayleigh–. The theoretical results are applied to the tackled application besides the study of bifurcation and transition to chaos using the perturbed form of the differential equation.
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