| Description: |
When studying dynamic systems, it often becomes necessary to quantify the degree of chaotization of the dynamic regime. The presence of a positive Lyapunov exponent in the system indicates a rapid divergence over time of any two close trajectories and sensitivity to the values of the initial conditions. At the same time, the chaotic attractor geometry can be complex, contain both divergent and convergent trajectories. For such attractors, the Bennettin algorithm will give an underestimated small value. This is due to the fact that the section of convergence of the trajectories is included in the average value with a minus sign. But by definition, a strange attractor, like any attractor of a dissipative dynamical system, is limited in phase space, so the distance between the representing points on it will necessarily be either large or small. A modification of the Bennettin algorithm is proposed, which makes it possible to take into account both the divergence and the convergance of the trajectory on the attractor of a dissipative dynamical system. The method is applied to a strange attractor in the cross-waves hydrodynamic system. Estimation of the Lyapunov exponent based on the classical Benettin algorithm does not take into account the convergence and divergence of trajectories on a chaotic attractor. The article proposes a universal algorithm, the accuracy of which does not depend on this and other features of chaotic attractors, such as their shape, size, and structure, as well as their number and location in phase space. The new algorithm makes it possible to increase the accuracy of calculations and expand the field of application of the method. The application of the proposed modification to the model of excitation of cross-shaped waves on the free surface of a liquid in a rectangular channel of finite length and depth made it possible to more accurately identify the dynamic regime. |