Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function

Zigen Song; Jian Xu; Bin Zhen; Zigen Song; Jian Xu; Bin Zhen

doi:10.3934/mbe.2019320

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6406-6425. doi: 10.3934/mbe.2019320

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Research article

Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function

1.
College of Information Technology, Shanghai Ocean University, Shanghai, 201306, P.R. China
2.
School of Aerospace and Mechanics Engineering, Tongji University, Shanghai 200092, P.R. China
3.
School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China

Received: 20 April 2019 Accepted: 03 July 2019 Published: 11 July 2019

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.
- inertial neuron system,
- nonmonotonic activation function,
- multistability,
- attractor merging crisis,
- period-doubling bifurcation,
- transient chaos
Citation: Zigen Song, Jian Xu, Bin Zhen. Mixed-coexistence of periodic orbits and chaotic attractors in an inertial neural system with a nonmonotonic activation function[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6406-6425. doi: 10.3934/mbe.2019320

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Abstract

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.

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