The paper presents a model of fractal parametric oscillator. Showing that the solution of such a model exists and is unique. A study of the solution with the aid of diagrams Stratton-Ince. The regions of instability, which can occur parametric resonance. It is suggested that this solution can be any signal, including acoustic.
A mathematical model is proposed for describing a wide class of radiating or memory oscillators. As a basic equation in this model is an integro-dierential equation of Voltaire type with dierence kernels memory functions, which were chosen by power functions. This choice is due, on the one hand, to broad applications of power law and fractal properties of processes in nature, and on the other hand it makes it possible to apply the mathematical apparatus of fractional calculus. Next, the model integro-dierential equation was written in terms of derivatives of fractional Gerasimov Caputo orders. Using approximations of operators of fractional orders, a non-local explicit nite-dierence scheme was compiled that gives a numerical solution to the proposed model. With the help of lemmas and theorems, the conditions for stability and convergence of the resulting scheme are formulated. Examples of the work of a numerical algorithm for some hereditary oscillators such as Dung, Airy and others are given, their oscillograms and phase trajectories are constructed.
The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.
In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.
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