The Cauchy problem for the Riccati equation with variable power memory and non-constant coeffcients

The Cauchy problem for the Riccati equation with non-constant coefficients and taking into account variable power memory is proposed. Power memory is defined by the operator of a fractional derivative of a variable order generalizing the Gerasimov-Caputo derivative. In work with the help of numerical methods: the Newton method and the explicit finitedifference scheme, the solution of the proposed Cauchy problem is found, and also their calculation accuracy is determined using the Runge rule. It is shown that both methods can be used to solve the proposed Cauchy problem, but Newton’s method converges faster. Further in this work, the calculated curves and phase trajectories were constructed for a different choice of the fractional order function of the differentiation operator. It is assumed that the proposed model can be used in describing economic cyclical processes.