We consider a self-organized system with a hierarchy structure to allow multiple leaders in the highest rank, and with free-will. In the model, we use both Cucker-Smale and Motsch-Tadmor functions for the pair influence of agents, and we derive suffcient conditions for such a system to converge to a flock, where agents ultimately move in the same velocity. We provide examples to show our suffcient conditions are sharp, and we numerically observe that such a self-organized system may have agents moving in different (final) velocities but maintain finite distance from each other due to the free-will.

One fundamental step towards grasping the global dynamic structure of a population system involves characterizing the convergence behavior (specifically, how to characterize the convergence behavior). This paper focuses on the neutral functional differential equations arising from population dynamics. With the help of monotonicity techniques and functional methods, we analyze the subtle relations of both the $\omega$-limited set and special point. Meanwhile, we prove that every bounded solution converges to a constant vector, as t tends to positive infinity. Our results correlate with the findings from earlier publications, and our proof yields an improved Haddock conjecture.

In this study, the stable dynamics of a kind of high-order cellular neural networks accompanying $ D $ operators and mixed delays are analyzed. The global existence of bounded positive solutions is substantiated by applying some novel differential inequality analyses. Meanwhile, by exploiting Lyapunov function method, some sufficient criteria are gained to validate the positiveness and globally exponential stability of pseudo almost periodic solutions on the addressed networks. In addition, computer simulations are produced to test the derived analytical findings.

This work investigates a class of inertial Cohen-Grossberg neural networks incorporating distributed delays using a direct mathematical analysis approach. First, by exploiting Lyapunov functional method, one can verify that the solutions of the addressed system are exponentially attractive to each other. Second, with the help of differential inequality techniques, the existence and global exponential stability of the anti-periodic solution are gained. Lastly, a simulation example is provided to illustrate the theoretical findings.

We study intelligence control systems and propose a new cascade flocking model with feedback. Compared to the one-way nature of past flocking models, our model adds a feedback mechanism, which means that the followers can have an influence on the direct leader's action. We demonstrate that these models can form a flock under specific conditions. This makes the flocking model more suitable for realistic applications.