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.

Perturbative weak KAM theory

This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.

Synchronous and Asynchronous Response in Dynamically Perturbed Proteins

We developed a Dynamic Gaussian Network Model to study perturbation and response in proteins. The model is based on the solution of the Langevin equation in the presence of noise and perturbation. A residue is perturbed periodically with a given frequency and the response of other residues is determined in terms of a storage and loss modulus of the protein. The amount of work lost upon periodic perturbation and the residues that contribute significantly to the lost work is determined. The model shows that perturbation introduces new dynamic correlations into the system with time delayed synchronous and asynchronous components. Residues whose perturbation induces large correlations in the protein and those that do not lead to correlations may be identified. The model is used to investigate the dynamic modulation of nanobodies. Despite its simplicity, the model explains several features of perturbation and response such as the role of loops and linkers in perturbation, dispersion of work of perturbation, and information transfer through preexisting pathways, all shown to be important factors in allostery.

Post-Effect on the Centre of Feet Pressure during Stance by Continuous Asymmetric Mediolateral Translations of a Supporting Platform—A Preliminary Study in Healthy Young Adults

Various diseases are associated with the impaired control of the medio-lateral (ML) position of the centre of feet pressure (CoP), and several manoeuvres have been proposed for enhancing the CoP symmetry. Here, we assessed in healthy standing subjects the feasibility and outcome of a novel protocol entailing a reaction to a continuous asymmetric ML displacement (10 cm) of the support base. The periodic perturbation consisted of a fast half-cycle (0.5 Hz) followed by a slow half-cycle (0.18 Hz). One hundred successive horizontal translation cycles were delivered in sequence. Eyes were open or closed. CoP was recorded before, after, and during the stimulation by a dynamometric platform fixed onto the translating platform. We found that the post-stimulation CoP was displaced towards the direction of the fast half-cycles. The displacement lasted several tens of seconds. Vision did not affect the amplitude or duration of the post-stimulation effect. The magnitude of post-stimulation CoP displacement was related to the perturbation-induced ML motion of CoP recorded during the stimulation. Over the successive perturbation cycles, the time-course of this motion revealed an adaptation phenomenon. Vision moderately reduced the adaptation rate. The findings support the feasibility of the administration of a simple asymmetric balance perturbation protocol in clinical settings to help patients recover the symmetry of the CoP. This protocol needs to be further validated in older populations and in patients.

Theoretical Research on Two-Phase Flow Instability in Parallel Rectangular Channels Under Periodic Perturbation

A theoretical model for Density Wave Oscillations (DWOs) flow instability in parallel rectangular channels under periodic heaving motion is established with a lumped mathematical model based on homogenous hypothesis. The parallel rectangular channels comprise of the entrance section, the heating section, the riser section and the upper- and lower plenums, which guarantee the isobaric pressure drop condition between channels and the model consists of boiling channel model, pressure drop model, parallel channel model, additional pressure drop model generated by heaving motions, the constitutive and numerical models. The effect of periodic perturbation is introduced through additional pressure drop in the momentum equation. The model is validated with experimental data of a twin-rectangular-channel flow instability experiment under static condition. Then the flow instability in parallel-rectangular-channel system is studied under periodic perturbation and the margin of flow instability and the threshold power of the system under static condition is calculated as basis condition for comparison. The effect of the amplitude and period of perturbation is analyzed analytically and the results show that the amplitude and period of perturbation shows little effect on flow instability. While when the additional pressure difference introduced by heaving motion is comparable with that under static condition, the effect of amplitude becomes stronger. And the period of perturbation strongly effects the threshold power when it is identical to that of natural period of the system, which can be explained by resonance between the perturbation and the system. And this effect is even stronger when the asymmetric heating condition is introduced.