Change of the Most Probable Escape Path in a Fast-Slow Insect Outbreak System Under Different Noise Amplitude Ratios

In the present paper, noise-induced escape from the domain of attraction of a stable fixed point of a fast-slow insect outbreak system is investigated. According to Dannenberg's theory(Dannenberg PH, Neu JC, 2014)[1], different noise amplitude ratios μ lead to the change of the Most Probable Escape Path(MPEP). Therefore, the research emphasis of this paper is to extend their study and discuss the changes of the MPEPs in more detail. Firstly, the case for μ=1, wherein the MPEP almost traces out the critical manifold, is considered. Via projecting the full system onto the critical manifold, a reduced system is obtained and the quasi-potential of the full system can be partly evaluated by that of this reduced system. In order to test the accuracy of the computed MPEP, a new relaxation method is then presented. Then, as μ converges to zero, an improved analytical method is given, through which a better approximation for the MPEP at the turning point is obtained. And then, in the case that the value of μ is moderate, wherein the MPEP will peel off the critical manifold, to determine the changing point of the MPEP on the critical manifold, an effective numerical algorithm is given. In brief, in this paper, a complete investigation on the structural changes of the MPEPs of a fast-slow insect outbreak system under different values of μ is given, and the results of the numerical simulations match well with the analytical ones.

Controlling Canard Cycles

Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

Finite-size transitions in complex membranes

ABSTRACTThe lipid raft hypothesis postulates that cell membranes possess some degree of lateral organization. The last decade has seen a large amount of experimental evidence for rafts. Yet, the underlying mechanism remains elusive. One hypothesis that supports rafts relies on the membrane to lie near a critical point. While supported by experimental evidence, the role of regulation is unclear. Using both a lattice model and molecular dynamics simulations, we show that lipid regulation of a many-component membrane can lead to critical behavior over a large temperature range. Across this range, the membrane displays a critical composition due to finite-size effects. This mechanism provides a rationale as to how cells tune their composition without the need for specific sensing mechanisms. It is robust and reproduces important experimentally verified biological trends: membrane-demixing temperature closely follows cell growth temperature, and the composition evolves along a critical manifold. The simplicity of the mechanism provides a strong argument in favor of the critical membrane hypothesis.SIGNIFICANCEWe show that biological regulation of a large amount of phospholipids in membranes naturally leads to a critical composition for finite-size systems. This suggests that regulating a system near a critical point is trivial for cells. These effects vanish logarithmically and therefore can be present in micron-sized systems.

Switching in Cerebellar Stellate Cell Excitability in Response to a Pair of Inhibitory/Excitatory Presynaptic Inputs: A Dynamical System Perspective

Cerebellar stellate cells form inhibitory synapses with Purkinje cells, the sole output of the cerebellum. Upon stimulation by a pair of varying inhibitory and fixed excitatory presynaptic inputs, these cells do not respond to excitation (i.e., do not generate an action potential) when the magnitude of the inhibition is within a given range, but they do respond outside this range. We previously used a revised Hodgkin–Huxley type of model to study the nonmonotonic first-spike latency of these cells and their temporal increase in excitability in whole cell configuration (termed run-up). Here, we recompute these latency profiles using the same model by adapting an efficient computational technique, the two-point boundary value problem, that is combined with the continuation method. We then extend the study to investigate how switching in responsiveness, upon stimulation with presynaptic inputs, manifests itself in the context of run-up. A three-dimensional reduced model is initially derived from the original six-dimensional model and then analyzed to demonstrate that both models exhibit type 1 excitability possessing a saddle-node on an invariant cycle (SNIC) bifurcation when varying the amplitude of [Formula: see text]. Using slow-fast analysis, we show that the original model possesses three equilibria lying at the intersection of the critical manifold of the fast subsystem and the nullcline of the slow variable [Formula: see text] (the inactivation of the A-type K[Formula: see text] channel), the middle equilibrium is of saddle type with two-dimensional stable manifold (computed from the reduced model) acting as a boundary between the responsive and non-responsive regimes, and the (ghost of) SNIC is formed when the [Formula: see text]-nullcline is (nearly) tangential to the critical manifold. We also show that the slow dynamics associated with (the ghost of) the SNIC and the lower stable branch of the critical manifold are responsible for generating the nonmonotonic first-spike latency. These results thus provide important insight into the complex dynamics of stellate cells.

Mathematical Simulation of Using a Combination of Mesh and Porous Materials as a Phase Separator

A combination of mesh and porous materials featuring bulk capillary properties is used as a phase separator in in-tank capillary inlet devices. These bulk capillary properties ensure a non-zero fluid flow into the interior of the in-tank inlet device after critical pressure has been reached. This quality makes it possible to reduce residual propellant volume in spacecraft engine tanks. We developed a mathematical simulation of an in-tank capillary inlet device comprising a phase separator made of a combination of mesh and porous materials. We represented a combination of mesh and porous materials as an array of "closely packed" transverse and longitudinal capillaries. Our mathematical simulation describes the operation of an in-tank inlet device after the critical manifold pressure has been reached. The fluid enters the interior of the in-tank inlet device one portion at a time. We determined the volume and arrival frequency of these portions and estimated the residual propellant volume in the propellant tank.

Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation

Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.