On a nonlinear model with complicated dynamics (Q1348535)

In the theory of nonlinear dynamical systems the idea of searching for mathematical models of the most simple form and structure of the equations on the one hand, and such that the models have a wide range of regimes, regular and chaotic, on the other hand, is considered to be very important. There are several dynamical systems described by ordinary differential equations (i.e. Lorenz system, Rössler system, etc.) that have been investigated by many researchers and their behavior along scale time is well known. For example, a so-called ``strange attractor'' stochastically or chaotically corresponds in a phase space to a dynamical chaos, generating by dissipative differential systems of order \( n \geq 3 \). As regular attractors of such systems might be points (equilibrium positions), limit circles and tori of dimension \(d \geq 2 \). An abstract mathematical model of a vibrating process with dissipative and inertia excitating terms that is analogous to the above-mentioned Lorenz and Rössler dynamical models is suggested by the author. The new model is simpler and has only one (bifurcation) parameter \(A>0\): \[ \dot X = 1 + AYZ, \qquad \dot Y = X-Y, \qquad \dot Z = 1-XY . \tag{1} \] The results of computer investigations of the system (1) are presented.