College of Information Technology, Shanghai Ocean University, Shanghai, 201306, P.R. China.
School of Aerospace and Mechanics Engineering, Tongji University, Shanghai 200092, P.R. China.
Math Biosci Eng. 2019 Jul 11;16(6):6406-6425. doi: 10.3934/mbe.2019320.
In this paper, we construct an inertial two-neuron system with a non-monotonic activation function. Theoretical analysis and numerical simulation are employed to illustrate the complex dynamics. It is found that the neural system exhibits the mixed coexistence with periodic orbits and chaotic attractors. To this end, the equilibria and their stability are analyzed. The system parameters are divided into some regions with the different number of equilibria by the static bifurcation curve. Then, employing some numerical simulations, including the phase portraits, Lyapunov exponents, bifurcation diagrams, and the sensitive dependence to initial values, we find that the system generates two coexisting single-scroll chaotic attractors via the period-doubling bifurcation. Further, the single-scroll chaos will evolve into the double-scroll chaotic attractor. Finally, to view the global evolutions of dynamical behavior, we employ the combined bifurcation diagrams including equilibrium points and periodic orbits. Many types of multistability are presented, such as the bistable periodic orbits, multistable periodic orbits, and multistable chaotic attractors with multi-periodic orbits. The phase portraits and attractor basins are shown to verify the coexisting attractors. Additionally, transient chaos in neural system is observed by phase portraits and time histories.
在本文中，我们构建了一个具有非单调激活函数的惯性双神经元系统。通过理论分析和数值模拟，说明了其复杂的动力学行为。结果表明，该神经网络系统表现出混合共存的周期轨道和混沌吸引子。为此，分析了平衡点及其稳定性。通过静态分岔曲线，将系统参数分为具有不同平衡点数量的几个区域。然后，通过一些数值模拟，包括相图、李雅普诺夫指数、分岔图和对初始值的敏感依赖性，我们发现系统通过倍周期分岔产生了两个共存的单涡旋混沌吸引子。进一步，单涡旋混沌将演变成双涡旋混沌吸引子。最后，为了观察动力学行为的全局演化，我们采用了包括平衡点和周期轨道的组合分岔图。呈现了多种类型的多稳定性，如双稳周期轨道、多稳周期轨道和多稳混沌吸引子与多周期轨道。通过相图和吸引子盆地验证了共存吸引子的存在。此外，还通过相图和时间历程观察到了神经网络系统中的暂态混沌。
