Slow Passage Research Articles

Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster

This article concerns the phenomenon of Mixed-Mode Bursting Oscillations (MMBOs). These are solutions of fast-slow systems of ordinary differential equations that exhibit both small-amplitude oscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs). The name MMBO is given in analogy to Mixed-Mode Oscillations, which consist of alternating SAOs and LAOs, without the LAOs being organized into burst events. In this article, we show how MMBOs are created naturally in systems that have a spike-adding bifurcation or spike-adding mechanism, and in which the dynamics of one (or more) of the slow variables causes the system to pass slowly through that bifurcation. Canards are central to the dynamics of MMBOs, and their role in shaping the MMBOs is two-fold: saddle-type canards are involved in the spike-adding mechanism of the underlying burster and permit one to understand the number of LAOs in each burst event, and folded-node canards arise due to the slow passage effect and control the number of SAOs. The analysis is carried out for a prototypical fourth-order system of this type, which consists of the third-order Hindmarsh-Rose system, known to have the spike-adding mechanism, and in which one of the key bifurcation parameters also varies slowly. We also include a discussion of the MMBO phenomenon for the Morris-Lecar-Terman system. Finally, we discuss the role of the MMBOs to a biological modeling of secreting neurons.

Direct monitoring of the strand passage reaction of DNA topoisomerase II triggers checkpoint activation.

By necessity, the ancient activity of type II topoisomerases co-evolved with the double-helical structure of DNA, at least in organisms with circular genomes. In humans, the strand passage reaction of DNA topoisomerase II (Topo II) is the target of several major classes of cancer drugs which both poison Topo II and activate cell cycle checkpoint controls. It is important to know the cellular effects of molecules that target Topo II, but the mechanisms of checkpoint activation that respond to Topo II dysfunction are not well understood. Here, we provide evidence that a checkpoint mechanism monitors the strand passage reaction of Topo II. In contrast, cells do not become checkpoint arrested in the presence of the aberrant DNA topologies, such as hyper-catenation, that arise in the absence of Topo II activity. An overall reduction in Topo II activity (i.e. slow strand passage cycles) does not activate the checkpoint, but specific defects in the T-segment transit step of the strand passage reaction do induce a cell cycle delay. Furthermore, the cell cycle delay depends on the divergent and catalytically inert C-terminal region of Topo II, indicating that transmission of a checkpoint signal may occur via the C-terminus. Other, well characterized, mitotic checkpoints detect DNA lesions or monitor unattached kinetochores; these defects arise via failures in a variety of cell processes. In contrast, we have described the first example of a distinct category of checkpoint mechanism that monitors the catalytic cycle of a single specific enzyme in order to determine when chromosome segregation can proceed faithfully.

Why does Brain Metabolism not Favor Burning of Fatty Acids to Provide Energy? - Reflections on Disadvantages of the Use of Free Fatty Acids as Fuel for Brain

It is puzzling that hydrogen-rich fatty acids are used only poorly as fuel in the brain. The long-standing belief that a slow passage of fatty acids across the blood-brain barrier might be the reason. However, this has been corrected by experimental results. Otherwise, accumulated nonesterified fatty acids or their activated derivatives could exert detrimental activities on mitochondria, which might trigger the mitochondrial route of apoptosis. Here, we draw attention to three particular problems: (1) ATP generation linked to β-oxidation of fatty acids demands more oxygen than glucose, thereby enhancing the risk for neurons to become hypoxic; (2) β-oxidation of fatty acids generates superoxide, which, taken together with the poor anti-oxidative defense in neurons, causes severe oxidative stress; (3) the rate of ATP generation based on adipose tissue-derived fatty acids is slower than that using blood glucose as fuel. Thus, in periods of extended continuous and rapid neuronal firing, fatty acid oxidation cannot guarantee rapid ATP generation in neurons. We conjecture that the disadvantages connected with using fatty acids as fuel have created evolutionary pressure on lowering the expression of the β-oxidation enzyme equipment in brain mitochondria to avoid extensive fatty acid oxidation and to favor glucose oxidation in brain.

A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells

The electrical activity of endocrine pituitary cells is mediated by a plethora of ionic currents and establishing the role of a single channel type is difficult. Experimental observations have shown however that fast-activating voltage- and calcium-dependent potassium (BK) current tends to promote bursting in pituitary cells. This burst promoting effect requires fast activation of the BK current, otherwise it is inhibitory to bursting. In this work, we analyze a pituitary cell model in order to answer the question of why the BK activation must be fast to promote bursting. We also examine how the interplay between the activation rate and conductance of the BK current shapes the bursting activity. We use the multiple timescale structure of the model to our advantage and employ geometric singular perturbation theory to demonstrate the origin of the bursting behaviour. In particular, we show that the bursting can arise from either canard dynamics or slow passage through a dynamic Hopf bifurcation. We then compare our theoretical predictions with experimental data using the dynamic clamp technique and find that the data is consistent with a burst mechanism due to a slow passage through a Hopf.

Nonlinear energy transfer in classical and quantum systems

In this paper we investigate the effect of slowly-varying parameters on the energy transfer in a weakly coupled system. For definiteness, we consider a system of two nonlinear oscillators, in which the directly excited first oscillator with constant parameter is attached to the oscillator with slowly time-varying frequency. It is proved that the equations of the slow passage through resonance in this system are identical to the equations of nonlinear Landau-Zener (LZ) tunneling. Three types of dynamical behavior are distinguished, namely, quasilinear, moderately nonlinear, and strongly nonlinear ones. Quasilinear systems exhibit a gradual energy transfer from the excited to the attached oscillator, while moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator. In strongly nonlinear systems, the transition from the energy localization to strong energy exchange between the oscillators is revealed. Explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested. Correctness of the constructed approximations is confirmed by numerical results. The results presented in this paper, in addition to providing an analytical framework for understanding the transient dynamics, suggest an approximate procedure for solving the nonlinear LZ problem with arbitrary initial conditions over a finite time-interval.

Decision making in noisy bistable systems with time-dependent asymmetry

Our work draws special attention to the importance of the effects of time-dependent parameters on decision making in bistable systems. Here, we extend previous studies of the mechanism known as speed-dependent cellular decision making in genetic circuits by performing an analytical treatment of the canonical supercritical pitchfork bifurcation problem with an additional time-dependent asymmetry and control parameter. This model has an analogous behavior to the genetic switch. In the presence of transient asymmetries and fluctuations, slow passage through the critical region in both systems increases substantially the probability of specific decision outcomes. We also study the relevance for attractor selection of reaching maximum values for the external asymmetry before and after the critical region. Overall, maximum asymmetries should be reached at an instant where the position of the critical point allows for compensation of the detrimental effects of noise in retaining memory of the transient asymmetries.

On how whales avoid decompression sickness and why they sometimes strand

Whales are unique in that the supply of blood to the brain is not by the internal carotid arteries, but by way of thoracic and intra-vertebral arterial retia. We found in the harbor porpoise (Phocoena phocoena) that these retia split up into smaller anastomosing vessels and thin-walled sinusoid structures that are embedded in fat. The solubility of nitrogen is at least six times larger in fat than in water, and we suggest that nitrogen in supersaturated blood will be absorbed in the fat, by diffusion, during the very slow passage of the blood through the arterial retia. Formation of nitrogen bubbles that may reach the brain is thereby avoided. We also suggest that mass stranding of whales may be due to disturbances to their normal dive profiles, resulting in extra release of nitrogen that may overburden the nitrogen 'trap' and allow bubbles to reach the brain and cause abnormal behavior.

Auto-balancing of a rotor with an orthotropic elastic shaft

The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor.

Postoperative Swallow Study as a Predictor of Intermediate Weight Loss after Sleeve Gastrectomy

Laparoscopic sleeve gastrectomy (LSG) is an accepted bariatric procedure. Swallow studies (SS) after LSG are not uniform and display different patterns with regard to contrast passage through the gastric sleeve. The impact of immediate postoperative contrast transit time on weight loss has not been studied. The influence of immediate fluid tolerance on weight loss after LSG is herein reported. Ninety-nine patients after LSG were included. There were 67 females, mean age 41 (range 17-67), mean BMI 44.4 (range 37-75). A routine SS was performed on postoperative day (POD) 1. Pattern of contrast transit was noted. Patients were followed-up in our bariatric clinic. Percent excess weight loss was significantly lower in the patients with rapid contrast passage (Group 1, n = 50) than those with delayed passage (Group 2, n = 49). Group 1 achieved 62, 58, and 53% at 1, 2, and 3years, respectively, while Group 2 attained 69, 74, and 75% at the same time points (p = 0.05, 0.001, and 0.04, respectively). Group 1 patients displayed a negative weight loss trend after 1year whereas Group 2 patients plateaued after 2years. Tolerance of fluid intake after LSG is crucial for patient recovery and discharge. Distinct radiologic appearance on POD 1 helps predict this behavior. Mid-term weight loss after LSG appears to be dependent on immediate postoperative contrast transit time, whereas patients with slow contrast passage tend to lose more weight. Long-term follow-up will reveal whether this finding will hold true.

Restreindre l’ingestion du jeune lapin : de nouvelles stratégies pour renforcer sa santé digestive et améliorer son efficacité alimentaire

La restriction temporaire de l'ingestion du lapin après son sevrage est maintenant une stratégie d'alimentation couramment employée en cuniculture. L'objet de cette synthèse est de présenter les différents effets d'une limitation de l'ingestion sur la santé, la physiologie digestive et l'efficacité alimentaire du lapin en croissance. Même si une restriction alimentaire conduit à une croissance plus lente, ces stratégies sont désormais utilisées par plus de 85% des éleveurs français. En effet, elles permettent de réduire les risques de mortalité et de morbidité post-sevrage par troubles digestifs, par exemple par l'entéropathie épizootique du lapin. De plus, la conversion alimentaire est améliorée, plus particulièrement lorsque les lapins sont de nouveau alimentés librement, en raison d'une importante croissance compensatrice. Cette meilleure efficacité alimentaire est associée à un transit plus lent et à une meilleure digestion, bien qu'on observe des interactions avec la composition chimique de l'aliment. La physiologie digestive est par ailleurs peu modifiée, en particulier la morphométrie de la muqueuse intestinale, l’activité fermentaire et le microbiote caecal. La qualité de la viande est peu affectée par la restriction alimentaire, alors qu'on observe une baisse de l'état d'engraissement des carcasses et une légère dégradation du rendement à l'abattage. Les effets d'une limitation de l'ingestion sur le comportement et le bien-être animal sont discutés, sachant que le jeune lapin présente une adaptation très rapide aux stratégies de restriction. Ainsi, les stratégies de restriction peuvent améliorer la rentabilité de l'atelier cunicole, mais elles doivent être adaptées à chaque situation d'élevage.

Slow passage through multiple bifurcation points

The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations. The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp--a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly. We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter. These delay and memory effects are independent of the ramp rate, for small enough ramp rates. The slow passage through the torus break-up bifurcation is qualitatively different. It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate. This is typical of transient behavior. We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used. The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.

Zircon fission-track dating of granites from the Vepor-Gemer Belt (Western Carpathians): constraints for the Early Alpine exhumation history

*Corresponding author We present new zircon fission-track (ZFT) data from Variscan granitoid bodies in the Veporic (footwall unit) and Gemeric (hangingwall unit) thick-skinned nappe sheets of the Central Western Carpathians. All samples show Late Cretaceous to earliest Paleogene cooling ZFT ages, which contribute to constraining the low-temperature exhumation history of the Vepor–Gemer Belt. Four granite samples from the western part of the Gemericum near the contact with the underlying Veporicum provided central ZFT ages between 70.4 ± 5.4 and 74.7 ± 5.6 Ma. One sample from this area shows an older age of 87.7 ± 5.9 Ma, possibly owing to its higher structural position. One remoter sample from the SE part of the Gemeric Unit has 61.7 ± 3.4 Ma central ZFT age, which probably reflects exhumation associated with a younger compressional tectonic event in that area. One sample from the centre of the Veporic metamorphic core complex yielded a cooling age 64.9 ± 4.8 Ma. However, most of these samples exhibit an internal age scatter pointing to complex cooling and exhumation history influenced by a slow passage through the zircon partial annealing zone and/or reheating brought about by the Cretaceous Rochovce granite intrusion. In spite of this, the acquired ages generally match the exhumation trend of the Veporic metamorphic core complex.

A slow pushed front in a Lotka–Volterra competition model

We study invasion speeds in the Lotka–Volterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization. The main challenge arises from the slow passage through a saddle-node bifurcation. From a perspective of linear versus nonlinear speed selection, this front provides an interesting example as the propagation speed is slower than the linear spreading speed. However, our front shares many characteristics with pushed fronts that arise when the influence of nonlinearity leads to faster than linear speeds of propagation. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of as a branch pole. Using the pointwise Green's function, we show that this pole poses no a priori obstacle to marginal stability of the nonlinear travelling front, thus explaining how nonlinear systems can exhibit slower spreading that their linearization in a robust fashion.

Investigation of electrode material – Redox couple systems for reverse electrodialysis processes. Part I: Iron redox couples

The performances of electrodialysis (ED) and reverse electrodialysis (RED) processes depend on several factors, including the nature of the electrode material and of the redox couple adopted to make possible the conversion between electric power and chemical potential. In this paper, the possible utilization of iron-based redox couples (FeCl3/FeCl2, hexacyanoferrate(III)/hexacyanoferrate(II) and Fe(III)-EDTA/Fe(II)-EDTA) on graphite and DSA electrodes for RED processes was studied by a detailed experimental investigation.The hexacyanoferrate(III)/hexacyanoferrate(II) system was stable for long time (more than 12days) in the absence of light and oxygen at high redox couple concentrations and low current densities both at compact graphite and DSA electrodes. Perfluorinated Nafion cationic membranes were found to be impermeable to the components of the redox couple. Fe(II)-EDTA exhibited a limited electrochemical stability in long term electrolyses at all adopted operative conditions, which discourages the use of the Fe(III)-EDTA/Fe(II)-EDTA for RED applications. The FeCl3/FeCl2 system was, on the other hand, stable for long times (more than 12days) at acidic pH at compact graphite electrodes. Selemion anionic membranes allowed to confine the redox couple in the electrode compartments with very slow passage of protons to the side compartment (“dilute or concentrated compartment”).

Speed-Dependent Cellular Decision Making in Nonequilibrium Genetic Circuits

Despite being governed by the principles of nonequilibrium transitions, gene expression dynamics underlying cell fate decision is poorly understood. In particular, the effect of signaling speed on cellular decision making is still unclear. Here we show that the decision between alternative cell fates, in a structurally symmetric circuit, can be biased depending on the speed at which the system is forced to go through the decision point. The circuit consists of two mutually inhibiting and self-activating genes, forced by two external signals with identical stationary values but different transient times. Under these conditions, slow passage through the decision point leads to a consistently biased decision due to the transient signaling asymmetry, whereas fast passage reduces and eventually eliminates the switch imbalance. The effect is robust to noise and shows that dynamic bifurcations, well known in nonequilibrium physics, are important for the control of genetic circuits.

Geochemistry, origin and correlation of crude oils in Lower Cretaceous sedimentary sequences of the southern Mesopotamian Basin, southern Iraq

Thirty one crude oil samples from Lower Cretaceous reservoirs in southern Iraq were analyzed using bulk property and molecular methods to determine their maturity and biomarker characteristics, as well as to obtain information on their respective source rocks. All the oils are unaltered, non-biodegraded, have high sulfur content and API gravity is in the range for light to heavy oil (19–40° API). They are characterized by low Pr/Ph values, even/odd predominance and front-end biased n-alkane distributions. Based on these parameters the oils were generated and expelled from a marine carbonate source rock bearing Type II-S kerogen. Compositional similarities of hopane and sterane biomarkers with those from potential source rocks allowed identification of the Upper Jurassic–Lower Cretaceous Sulaiy and Yamama carbonate succession as the effective source beds. A similar composition of normal and isoprenoid hydrocarbons among the oils suggests an origin from a common source rock. However, biomarker maturity ratios indicate a wide range of maturity. This appears to result from the type of burial history of the source rock, characterized by a slow passage through the liquid window interval during an extended period of geologic time.

Classical analog of quasilinear Landau-Zener tunneling

In this paper we develop an analytical framework to study the effect of nonlinearity on irreversible energy transfer in a system of two weakly coupled oscillators with time-dependent parameters, with special attention to an analogy between classical energy transfer and nonadiabatic quantum tunneling. For preciseness, we suppose that a linear oscillator with constant parameters is excited by an initial impulse but a coupled quasilinear oscillator with slowly varying parameters is initially at rest. It is shown that the equations of the slow passage through resonance in this system are identical to quasilinear equations of nonadiabatic Landau-Zener tunneling. Due to revealed equivalence, a recently found analogy between irreversible energy transfer in a classical linear system and conventional linear Landau-Zener tunneling can be extended to quasilinear systems. An explicit analytical solution of the quasilinear problem is found with the help of an iteration procedure, wherein the linear solution is chosen as an initial approximation. Correctness of the constructed approximations is confirmed by numerical simulations. The results presented in this paper, in addition to providing an analytical framework for understanding the transient dynamics of coupled oscillators, suggest an approximate procedure for solving the quasilinear Landau-Zener equations with arbitrary initial conditions over a finite time interval.

Vibraphone Solo VI

- 290 Published by Calabrese Brothers Music,LLC 2153 Vermont Avenue •Toms River,NJ 08755 www.CalabreseBrothersMusic.com Calabrese Brothers Music,LLC Vibraphone Solo VI David Saperstein (2011) Copyright © 2011 Calabrese Brothers Music,LLC All Rights Reserved. Printed in U.S.A. - 291 Vibraphone Solo VI – David Saperstein (2011) Performance Notes: 1. Accidentals, once introduced into a measure, apply to one register only, and remain in effect until cancelled by a natural or until the bar line. Additional accidentals are sometimes added as a precaution. 2. Four mallets are required, and the choice of specific types of mallets is left to the performer, in accordance with individual preferences and the acoustical conditions of the performance space. 3. With the exception of when the pedal clears the sound, let all notes vibrate throughout. 4. The duration of the work is approximately 6’26” Program Notes: Vibraphone Solo VI was written in June 2011 in response to a call for scores from Robert Morris and Benjamin Boretz, the editors of Perspectives of New Music, for works dedicated to the memory of Milton Babbitt, who passed away at the beginning of 2011. The score bears the dedication, “In Memory of Milton Babbitt, Composer, Author, Teacher, Friend.” In this work, a similar process takes place as in Vibraphone Solo II, where there is a duality between long, slowly moving passages and short sections of fastpaced music. In Vibraphone Solo II, the slow passages win out in the end, but in this piece, it is difficult to say which element reigns supreme. - David Saperstein - 292 Copyright © David Saperstein. Member ASCAP. All Rights Reserved. Printed in U.S.A. Vibraphone f sempre mf p * N mp mf f ff q = 60 Vib. pp sempre ff * senza pedal f 3 Vib. mp mf f fpp ff * 6 Vib. senza pedal mp pp ff * 9 Vib. pp sempre ff * mp pp p mp mf 14 G motor off l.v. David Saperstein (2011) Vibraphone Solo VI To the memory of Milton Babbitt (1916-2011) Composer, Author, Teacher, Friend G 0 l.v. G ff _ G mf Start tremolo slowly gradually increasing to full speed. motor on (slow vibrato) G Engraved by Peter Jarvis PeterMJarvis@gmail.com l.v. + 5 5 5 = = 5 ! 4 =: 5 5 3 5 5 5 3 5 5 5 5 3 5 5 5 5 3 5 5 5 5 E ! ? + 5 5 5 5 5 5 5 5 E 5 E 5 E 5 ! = ? ? 3 5 $ = 4 3 5 5 $ = 4 + 5 5 @ + 5 5 5 5 E $ = 4 + 5 5 5 5 ! E @ 3 = 4 ? M M M ' 4 = 5 5 5 5 E ? 3 5 5 5 5 = 4 ? M M M ' M M M ' M M M ' + 5 E 5 E 5 E 5 5 E 5 ! 5 5 ! 5 5 5 ! = ? 3 5 ! $ = 4 3 5 5 ! $ 3 5 5 ! 3 5 5 $ 3 5 5 - 293 Vib . f fp f * 16 Vib. senza pedal pp 19 Vib. (senza pedal) f f mp q = 120 22 Vib. pp mp f 24 Vib. p cresc. poco a poco 26 Vib. (cresc. poco a poco) f f * 29 G Start tremolo slowly gradually increasing to full speed. _ G 0 G G G 0 G _ _ _ _ _ _ 4 + 5 5 5 5 + = ? = M M M 5 : M M M B M M M ' 4 5 $ =: 4 + 5 5 + = ? + 5 5 + = 4 3 5 5 5 5 5 5 5 5 5 E 5 5 4 4 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 4 4 3 5 5 5 5 5 5 E 5 5 ! 5 5 5 ? 5 5 5 E 5 ! 5 5 ! 5 5 3 5 5 5 5 5 5 5 5 5 5 E 5 5 5 5 ! 5 E 5 5 E 5 ! 5 E 5 ! 5 E 5 ! 5 5 ! 5 E 5 5 5 ! 5 E 5 5 ! 4 5 5 5 5 5 5 Vibraphone Solo VI - David Saperstein 3 - 294 Vib . mf N 31 Vib. mp 35 Vib. p N 39 Vib. p N f p...

Slow passage through resonance

The slow passage problem through a resonance is considered. As a model problem, we consider a damped harmonically forced oscillator whose forcing frequency is slowly ramped linearly in time. The setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem by an amount that is the difference between the Hopf value and the initial value of the parameter. In sharp contrast, in the resonance problem there is an early onset of resonance, setting in when the ramped forcing frequency is midway between its initial value and the natural frequency for resonance in the unforced problem; we term this value the jump frequency. Numerically, we find that the jump frequency is independent of the system's damping coefficient, and so we also consider the undamped problem, which is analytically tractable. The analysis of the undamped problem confirms the numerical results found in the damped problem that the maximal amplitude obtained at the jump frequency scales as A~ε(-1/2), ε being the ramp rate, and that the jump frequency is midway between the initial frequency at the start of the ramp and the natural frequency of the unforced problem.

Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation

This paper investigates the emergence of mixed-mode oscillations (MMOs) in the forced Van der Pol’s equation. It is found that the MMOs studied here can be classified as a slow passage through canard explosion, which is different from the usual fast-slow bursters. We first consider the external forcing as a control parameter and study its influence on the Van der Pol’s equation with constant forcing (VPCF). Then we briefly discuss the famous canard phenomenon in VPCF. The results of these analysis, together with the “transformed phase diagram,” are applied to the forced Van der Pol’s equation, which shows that the canard explosion and the external forcing plays an important role in the generation of MMOs, that is, the MMOs are created since the external forcing slowly and periodically visits the rest and spiking areas of VPCF.