Simulation of system (1)–(4) for , and for i = x, y, I. The orbit shown here is clearly of MMBO type, that is, a succession of small-amplitude slow oscillations and large-amplitude fast oscillations. The observed MMBO pattern is irregular and combines transitions of the type 41, 42, and 43.
Canard explosion at the lower fold in the van der Pol system: the slow and the fast dynamics must be as shown and the slow nullcline (shown in black), must cross the fold transversely. Four cycles are presented (in blue): headless canard in panel (a), maximal canard in panel (b), canard with head in panel (c), and relaxation oscillation in panel (d). In each panel, the left plot corresponds to a phase plane representation of the cycle together with the fast cubic nullcline S 0 and the slow linear nullcline (both in black); the right panel shows the time series of the x variable during the cycle.
Panel (a) shows the bifurcation diagram of the fast subsystem of the Hindmarsh-Rose burster, i.e., Eqs. (1) and (2) , with z acting as a parameter. All parameter values are taken to be the classical ones (Sec. I ) and I is fixed at the value corresponding to the canard explosion taking place in the full system and displayed in Fig. 4 , that is, I = 1.3278138. Panels (b1)-(b3) show snapshots of the phase portrait of the fast system for three different values of z, namely, z = 1.449 (lower homoclinic), z = 1.75 and z = 2.180 (upper homoclinic). For each value of z in between the lower and upper homoclinics, the unstable manifold of the saddle returns to a neighbourhood of the saddle and gives the leading-order location of the spike in the full system.
Spike-adding phenomenon: time series and phase portraits for the sequence of limit cycle canards observed in the transition from 0 to 1 spike in the Hindmarsh-Rose system (1)–(3) . Parameter values are those given below Eqs. (1)–(3) , with and I varying by an exponentially small amount around the value of 1.3278138026. The top plot in each panel (a) to (i) shows the time series of the x-variable, with the period normalised to 1. The bottom plot shows the limit cycle canard in the phase plane (x, z). In each frame, the behaviour of the trajectory at the end of the canard segment can be understood by looking at the phase portrait of the fast system (1) and (2) shown in Fig. 3 .
Spike-adding phenomenon: time series and phase plane projections for limit cycles along the transition from 1 spike to 2 spikes. The first spike, created during the previous spike-adding transition, remains in essentially the same place during the transition in which the second spike is generated. Along the explosion that leads to the addition of a second spike, the parameter I varies by an exponentially amount around the value of 1.3317831217.
Periodic MMBOs of system (1)–(4) , for (panel (a) for the x variable and panel (c) for the I variable) and (panels (b) and (d) for the x variable and the I variable, respectively); in all panels, hx = 5 and . In panels (c) and (d), the dashed red curve indicates the value of I corresponding to the folded node. The unlabelled panels to the side of panel (a) and panel (b) show zooms into one burst of the corresponding MMBO, with 8 LAOs (left panel) and 2 LAOs (right panel).
Periodic MMBOs of system (1)–(4) , for k = 0.45 (panel (a)) and k = 0.35 (panel (b)) and for ; in both panels hx = 5 and . Decreasing allows to find MMBOs with the correct number of SAOs as predicted by folded node theory via formula (16). In both cases, the number of LAOs is very large, about 200.
Periodic 1923-MMBOs of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are the critical manifold S 0 and the lower fold curve F –. Panel (b) is an enlargement of panel (a) near the folded node .
Periodic 12-MMBOs Γ of system (1)–(4) , obtained for k = 0.45, that is, 1/μ ≈ 6.74, and , and projected onto the (x, z, I)-space; also shown are a numerical computation of an attracting slow manifold . We highlight three orbits on : the strong canard γ s as well as two (unlabelled) secondary canards; these define three rotation sectors named si , i = 1, 2, 3. The number of SAO in a given solution depends on which sector it lies in; in particular, in sector si solutions have i SAOs.
Article metrics loading...
Full text loading...