Leonov G. A.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 40-62
We present an overview of development of the direct Lyapunov method in the global analysis of chaotic systems and describe three directions in which the Lyapunov functions are applied: in the methods of localization of global attractors, where the estimates of dissipativity in a sense of Levinson are obtained, in the problems of existence of homoclinic trajectories, and in the estimation of the dimension of attractors. The efficiency of construction of Lyapunov-type functions is demonstrated. In particular, the Lyapunov dimension formula is proved for the attractors of the Lorentz system.
On the problem of estimation of the number of cycles in two-dimensional quadratic systems from the viewpoint of nonlinear mechanics
Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 48–57
Two-dimensional quadratic systems are considered as a Liénard equation with certain special nonlinearities. Theorems on the existence or absence of cycles are given.