Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

Purpose This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs. Design/methodology/approach The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory. Findings Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur. Originality/value By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – II

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [ Katsanikas & Wiggins, 2021 ]. Similar to that method, for an [Formula: see text] degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a [Formula: see text] dimensional geometrical object in the energy surface of a [Formula: see text] dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.

The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with Three or More Degrees of Freedom – I

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a ([Formula: see text])-dimensional object in the ([Formula: see text])-dimensional energy surface. We present our methods using benchmark examples for two and three degrees of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end, we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.