This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with quadratic term and delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics. Bifurcation diagrams are obtained numerically or analytically from the mathematical model, and the parameter regions of different behaviors are clarified. The neural system exhibits a unique rest point or three ones by employing the saddle-node bifurcation, when strong coupling is applied in the system. Also the trivial rest point shows transcritical bifurcation with one of the new rest points. The asymptotic stability and possible Hopf and Bautin bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Fold cycle, torus, fold-torus, and big homoclinic bifurcations of limit cycles, together with $1:1$ and $1:4$ resonances are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf, Hopf-transcritical, and double-zero bifurcation points can be obtained from the intersection of different bifurcation branches. Various patterns of multistability have been observed, both for small and large values of delay. The system may exhibit synchronous and anti-phase periodic solutions, which occur due to Hopf, fold cycle and torus bifurcations. By increasing the range of parameter $τ$ other branches of fold and torus bifurcations appear, which lead to creation of more stable periodic solutions.
Experimental observation of chaotic and multistable dynamics in a Duffing--Holmes--type analog circuit: antiperiodicity and attractor-coexistence signatures
Nonlinear optical response of a two-dimensional quantum-dot supercrystal: Emerging multistability, periodic and aperiodic self-oscillations, chaos, and transient chaos
Bifurcation Structure of Double Hysteresis and Isolated Branches in a Parametric Duffing Nanoresonator
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；TUM Center for Quantum Engineering (ZQE)（慕尼黑工业大学量子工程中心）；Ain Shams University（艾因夏姆斯大学）
