New Canards Bursting and Canards Periodic-Chaotic Sequence

A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itseff in an alternation between the oscillation phase following attracting the limit cycle branch and resting phase following a repelling fixed point branch in a reduced leech neuron model Via periodic-chaotic alternating of infinite times, the number of windings within a canards bursting can approach infinity at a Gavrilov-Shilnikov homoclinic tangency bifurcation of a simple saddle limit cycle.     

Bifurcation of a Saddle-Node Limit Cycle with Homoclinic Orbits Satisfying the Small Lobe Condition in a Leech Neuron Model

Mechanism of period-adding cascades with chaos in a reduced leech neuron model is suggested as the bifurcation of a saddle-node limit cycle with homoclinic orbits satisfying the ＂small lobe condition＂, instead of the blue-sky catastrophe. In every spiking adding, the new spike emerges at the end of the spiking phase of the bursters.