Abstract
<jats:p>The paper presents mathematical modeling of the alternation process in the spatiotemporal chaotic dynamics of discrete-continuous systems with existing structural-evolutionary restructuring. Usually, the motion of a single particle of matter in a certain wave packet field is modeled by a second-order differential equation with a regular (non-random) right-hand side. The study establishes the conditions under which the motion of a particle becomes stochastic and it begins to accelerate. As the particle's speed increases, the time it takes to travel a certain distance decreases. This means that the degree of adiabaticity of the perturbation increases, and the emergence of a chaos boundary can be expected due to the weak influence of the wave field on the dynamics of a high-energy particle. It can also be expected that in this case, chaos should be very weak in relation to the regular component of motion. In addition, the chaotic component of motion should have heavy temporal and spatial scales against the background of high-frequency regular motion. In turbulence theory, this type of motion is commonly referred to as intermittent motion. Alternation is usually understood as the spatially and temporally chaotic dynamics of a system with a fairly well-defined spatial and temporal structure. In this work, the process of alternation is investigated for dissipative systems. In the vicinity of a stable limit cycle, the dynamics of a (discrete-continuous) system is mainly determined by the spectrum of the cycle. It is shown that if the cycle has only one period, the Fourier spectrum of the system in the vicinity of the cycle has a form close to the delta-function, with a maximum at the cycle frequency. If there is very weak chaos in the vicinity of the cycle, the spectrum of this system is also close to the delta--form. The temporal evolution can be clearly represented as randomly “stitched” long sections of regular oscillations. The existence of sufficiently large sections of regular motion already implies a significant degree of dynamic regularity. The spatial alternation in the motion of a continuous medium looks similar. It has been established that the phenomenon of alternation is associated with the property of multifractality, i.e., with the inhomogeneous distribution of singular or fractal properties in space and time. This paper studies the phenomenon of Hamiltonian alternation in the problem of particle motion in a wave packet field. The nature of the phenomenon described is universal and inherent in many problems that, at first glance, appear to be unrelated. Hamiltonian alternation covers a significant class of diverse physical and mathematical problems, including those considered in this study, in which the phase change is inversely proportional to an arbitrary (any) degree of action. There are a number of additional possibilities for the emergence of weak chaos. Keywords: intermittent, spatio-temporal evolution, chaotic dynamics, discrete-continuous systems, structural-evolutionary restructuring.</jats:p>