A General Framework for Dynamic Complex Networks

Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

“…Then it remain to compare the solution V (x 1 , ..., x n ) of equation ( 9) and a similar solution of inequality (10). From here it follows the boundedness of solution x(t) of system (7) for any initial condition x 0 ∈ R n .This completes the proof of case (a1) for strictly odd functions.…”

“…Denote by S the ball centered at the origin whose surface passes through point T u ∈ S. In this case H ⊂ S. Therefore, by virtue of ( 12) and according to LaSalle's Theorem [25], there is a moment T s > T u such that x(T s ) ∈ H. Again we get that solution x(t) of system (7) starting at S belongs to S. In addition, x(t) is attracted to the boundary of H as t → +∞. Thus, it is bounded.…”

“…We define the applicant for the role of the Lyapunov function for system (7) the real function V (x 1 , ..., x n ) by the following rule:…”

“…Further, from the definition of function V (x 1 , ..., x n ) and system (7) it follows that Vt (x 1 (t), ..., x n (t)) = h 1 (x 1 , γ 1 + 1)…”

“…Now we can apply Theorem 4.1 [25] to equation (7). Then all the ideas that were used in the proof of Theorem 4.1 [25] can be directly carried over to the proof of Theorem 3.…”