In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem { x ˙ + A ( t , x ) + B x = f ( t ) a.e. t ∈ R , x ( t + T ) = − x ( t ) , where A ( t , x ) is a nonlinear map and B is a bounded linear operator from R N to R N . Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W 1 , 2 ( I , R N ) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.

Decision-making for autonomous maneuvering in dynamic, uncertain, and nonlinear environments represents a challenging frontier problem. Deep deterministic policy gradient (DDPG) is an effective method to solve such problems, but it is found that complex strategies require extensive computation and time in the learning process. To address this issue, we propose a node clustering (NC) method, inspired by grid clustering, integrated into the DDPG algorithm for the learning of complex strategies. In the NC method, the node membership degree is defined according to the specific characteristics of the maneuvering decision-making problem, and error handling strategies are designed to reduce the number of transitions in the replay database effectively, ensuring that the most typical transitions are retained. Then, combining NC and DDPG, an autonomous learning and decision-making algorithm of maneuvering is designed. The algorithm flow and the pseudo-code of the algorithm are given. Finally, the NC_DDPG algorithm is applied to a typical short-range air combat maneuvering decision problem for verification. The results show that the NC_DDPG algorithm significantly accelerates the autonomous learning and decision-making process under both balanced and disadvantageous conditions, taking only about 77% of the time required by Vector DDPG. The scale of NC impacts learning speed; the simulation results across five scales indicate that smaller clustering scales significantly increase learning time, despite a high degree of randomness. Compared with Twin Delayed DDPG (TD3), NC_DDPG consumes only 0.58% of the time of traditional TD3. After applying the NC method to TD3, NC_DDPG requires approximately 20–30% of the time of NC_TD3.

We analyze a three species predator-prey chain model with stochastic perturbation. First, we show that this system has a unique positive solution and itspth moment is bounded. Then, we deduce conditions that the system is persistent in time average. After that, conditions for the system going to be extinction in probability are established. At last, numerical simulations are carried out to support our results.

This paper concerns with the stability of the orbits for nearly integrable Hamiltonian systems. Based on Nekehoroshev's original works in [14], we present the definition of quasi-effective stability and prove a theorem on quasi-effective stability under the Rüssmann's non-degeneracy. Our result gives a relation between KAM theorem and effective stability. A rapidly converging iteration procedure with two parameters is designed.

This paper is concerned with the stability of the orbits for a nearly integrable volume-preserving mapping. We prove that the nearly integrable volume-preserving mapping possesses quasi-effective stability under the classical KAM-type nondegeneracy, that is, there is an open subset of the phase space whose measure is nearly full, such that the considered mapping is effective stable on this subset. This announces a connection between the Nekhoroshev theory and KAM theory.

This paper deals with the stability of the orbits for time-dependent nearly integrable Hamiltonian systems. Under the classical non-degeneracy in KAM theory we prove that the considered system possesses quasi-effective stability. Our result generalized the works in [F. Z. Cong, J. L. Hong, H. T. Li, Dyn. Syst. Ser. B, \(\bf 21\) (2016), 67--80] to time-dependent system and gave a connection between KAM theorem and effective stability.