On the number of nonstationary bounded trajectories of a class of autonomous systems on the plane

• Published in: Differential equations 2008-08, Vol.44 (8), p.1082-1087
• Main Author: Naimov, A. N.
• Format: Article
• Language: English
• Subjects: Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266108080053

We study the number of nonstationary bounded trajectories of autonomous systems of the form z′ = , z = x + iy ∈ C, where P n ( z ) is a polynomial of degree n with complex coefficients that has k distinct roots, n, k > 1. We prove that the number N of nonstationary bounded trajectories of this system satisfies the following assertions (Theorem 1): (a) N = n + k − N + , N + = N − , n + 1 ≤ N + ≤ n + k , where N + and N − are the numbers of system trajectories unbounded as t → +∞ and t → −∞, respectively; (b) if some r distinct roots , ..., of the polynomial P n satisfy the relations V n +1 ( ) = ··· = V n +1 ( ), where V n +1 is the imaginary part of the indeterminate integral of P n , then N ≥ + ··· + + r − n − 1; (c) if k = 2, then the conditions N = 1 and V n +1 ( c 1 ) = V n +1 ( c 2 ) are equivalent. For n = k = 3, we derive a formula for the number of nonstationary bounded trajectories (Theorem 2).