Spike-adding canard explosion in a class of square-wave bursters (Q2022661)

The paper starts from the Morris-Lecar neuron model, also from two-time scales models of semiconductor lasers displaying sequences of quiescent phases and active bursting phases (periodic vs. aperiodic spikes), to consider the basic system \begin{gather*} \dot{v} = f_1(v,w,y;k,\varepsilon)\\ \dot{w} = f_2(v,w,y;k,\varepsilon)\\ \dot{y} = \varepsilon g(v,w,y;k,\varepsilon). \end{gather*} Here \(\varepsilon>0\) is a small parameter and \(k\) -- a bifurcation parameter; \(f_1\), \(f_2\), \(g\) are \(C^{r+1}\)-smooth functions of their arguments, \(r\geq 3\). The aforementioned system is called fast; if time \(t\) is re-scaled by \(\tau=\varepsilon t\), the corresponding slow system is obtained \[ \displaylines{\varepsilon\dot{v} = f_1(v,w,y;k,\varepsilon)\cr \varepsilon\dot{w} = f_2(v,w,y;k,\varepsilon)\cr \dot{y} = g(v,w,y;k,\varepsilon),} \] together with the layer system \[ \displaylines{\dot{v} = f_1(v,w,y;k,0)\cr \dot{w} = f_2(v,w,y;k,0)\cr \dot{y} = 0.} \] Basic notions and assumptions are defined and discussed in connection with these systems; known results are recalled. The main result of the paper is its {Theorem 2.2} ensuring existence for the basic system of a continuous one-parameter family \[ \theta \mapsto (k_{sa}(\theta,\sqrt{\varepsilon}),B(\theta,\sqrt{\varepsilon})),\ \theta\in(0,\Theta(\varepsilon)) \] of periodic orbits \(B(\theta,\sqrt{\varepsilon})\) originating at a Hopf bifurcation near the canard fold point \(F^r\). For \(\theta\in(N,N+1)\), the periodic orbit is a \(N\)-spike bursting solution.