Richard H. Rand

Cornell University | CU · Department of Mathematics and Department of MAE

The problem of two van der Pol oscillators coupled by velocity delay terms was studied by Wirkus and Rand in 2002. The small-epsilon analysis resulted in a slow flow which contained delay terms. To simplify the analysis, Wirkus and Rand followed a common procedure of replacing the delay terms by non-delayed terms, a step said to be valid for small...

This paper investigates the dynamics of a delay limit cycle oscillator under periodic external forcing. The system exhibits quasiperiodic motion outside of a resonance region where it has periodic motion at the frequency of the forcer for strong enough forcing. By perturbation methods and bifurcation theory, we show that this resonance region is as...

Delay-coupled oscillators exhibit unique phenomena that are not present in systems without delayed coupling. In this paper, we experimentally demonstrate mutual synchronisation of two free-running micromechanical oscillators, coupled via light with a total delay 139 ns which is approximately four and a half times the mechanical oscillation time per...

This paper involves the dynamics of a delay limit cycle oscillator being driven by a time-varying perturbation in the delay:ẋ=−x(t−T(t))−ϵx3 with delay T(t)=π2+ϵk+ϵcosωt. This delay is chosen to periodically cross the stability boundary for the x=0 equilibrium in the constant-delay system. For most of parameter space, the system is non-resonant, le...

Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The game-theoretic aspect of the model lies in the...

This paper concerns the dynamics of the following nonlinear differential-delay equation: x˙ = -x(t-T)-x3+αx in which T is the delay and α is a coefficient of self-feedback. Using numerical integration, continuation programs and bifurcation theory, we show that this system exhibits a wide range of dynamical phenomena, including Hopf and pitchfork bi...

We investigate the dynamics of three-strategy (rock-paper-scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval \(T\) . Taking \(T\) as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an anal...

Nonlinear oscillators with hardening and softening cubic Duffing nonlinearity are considered. Such classical conservative oscillators are known to have an amplitude-dependent period. In this work, we design oscillators with the Duffing-type restoring force but an amplitude-independent period. We present their Lagrangians, equations of motion, conse...

Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscillators, where each is driven by the same independent third oscillator. The presence of numerous bifurcation curves defines parameter regions with 2, 4, or 6 solutions corresponding to phase locking. In all cases, only one solution is stable. Elsewhere, ph...

Frequency-locking and other phenomena emerging from nonlinear interactions between mechanical oscillators are of scientific and technological importance. However, existing schemes to observe such behaviour are not scalable over distance. We demonstrate a scheme to couple two independent mechanical oscillators, separated in frequency by 80kHz and si...

An optically thin MEMS beam suspended above a substrate and illuminated with a CW laser forms an interferometer, coupling out-of-plane deflection of the beam to absorption within it. In turn, laser absorption creates thermal stresses which drive further deflection. This coupling of motion to thermal stresses can cause limit cycle oscillations in wh...

In this work we propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena and offer sample conservative systems for which the backbone curve is a straight vertical line.

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the "replicator" equation) which involves a combination of game theory and differential equations. In this paper we appl...

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by x.i=xifix-ϕ,i = 1,…, n, where xi is the frequency, or relative abundance, of strategy i, fi is its fitness, and ϕ=∑i=1nxifi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A, where...

We propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena by designing an oscillator for which the backbone curve is a straight vertical line. In particular we consider the class of conservative oscillators of the form: Display Formulax..+xx.2+fx=0 and we choose f(x) so that the frequency o...

We investigate a phenomenon observed in systems of the form dx/dt=a1tx+a2tydy/dt=a3tx+a4ty where ait=Pi+εQicos2t where Pi, Qi and ε are given constants, and where it is assumed that when ε = 0 this system exhibits a pair of linearly independent solutions of period 2π. Since the driver cos2t has period π, we have the ingredients for a 2:1 subharmoni...

The nonlinear dynamics of micromechanical oscillators are explored experimentally. Devices consist of singly and doubly supported Si beams, 200 nm thick and 35 μm long. When illuminated within a laser interference field, devices self-oscillate in their first bending mode due to feedback between laser heating and device displacement. Compressive pre...

This work is concerned with nonlinear oscillators that have a fixed, amplitude-independent frequency. This characteristic, known as isochronicity/isochrony, is achieved by establishing the equivalence between the Lagrangian of the simple harmonic oscillator and the Lagrangian of conservative oscillators with a position-dependent coefficient of the...

The nonlinear dynamics of a transverse galloping blunt body oscillator is analyzed with respect to its geometric shape and size. The oscillator's equation of motion is studied using an approximation for the lateral aerodynamic force that is a polynomial function of the angle of attack. The harmonic balance method is used to solve the nonlinear diff...

Nanoscale resonators whose motion is measured through laser interferometry are known to exhibit stable limit cycle mo-tion. Motion of the resonator through the interference field mod-ulates the amount of light absorbed by the resonator and hence the temperature field within it. The resulting coupling of motion and thermal stresses can lead to self...

In this work, a differential delay equation (DDE) with a cubic nonlinearity is analyzed as two parameters are varied by means of a center manifold reduction. This reduction is applied directly to the case where the system undergoes a Hopf-Hopf bifurcation. This procedure replaces the original DDE with four first-order ODEs, an approximation valid i...

We investigate the dynamics of a simple pendulum coupled to a horizontal mass–spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass–spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dyna...

Nonlinear parametric excitation refers to the nonlinear analysis of a system of ordinary differential equations with periodic coefficients. In contrast to linear parametric excitation, which offers determinations of the stability of equilibria, nonlinear parametric excitation has as its goal the structure of the phase space, as given by a portrait...

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators: 1 2 where D α x is the order α derivative of x(t), and 0<α<1. We use a two-variable perturbation method on the system’s corresponding variational equations to derive expressions for the transition curves separating regions...

A delay differential equation (DDE) which exhibits a double Hopf or Hopf-Hopf bifurcation [1] is studied using both Lindstedt's method and center manifold reduction. Results are checked by comparison with a numerical continuation program (DDEBIFTOOL). This work has application to the dynamics of two interacting microbubbles.

We revisit the two degrees of freedom model of the thin elastica presented by Cusumano and Moon (1995) [3]. We observe that for the corresponding experimental system (Cusumano and Moon, 1995 [3]), the ratio of the two natural frequencies of the system was ≈44≈44 which can be considered to be of O(1/ε)O(1/ε), where ε⪡1ε⪡1. The presence of such a vas...

A micro-scale resonator suspended over a substrate and illuminated with a continuous wave (CW) laser forms an interferometer which couples deflection of the resonator to light absorption. In turn, absorption creates temperature and thermal stress fields which feedback into the motion. Experiments have shown that this coupling can lead to limit cycl...

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient...

We investigate a problem in evolutionary game theory based on replicator equations with periodic coefficients. This approach to evolution combines classical game theory with differential equations. The RPS (Rock-Paper-Scissors) system studied has application to the population biology of lizards and to bacterial dynamics. The presence of periodic co...

A system of three coupled limit cycle oscillators with vastly different frequencies is studied. The three oscillators, when uncoupled, have the frequencies ω 1=O(1), ω 2=O(1/ε) and ω 3=O(1/ε 2), respectively, where ε≪1. The method of direct partition of motion (DPM) is extended to study the leading order dynamics of the considered autonomous system...

A new model of coupled oscillators is proposed and investigated. All phase variables and parameters are integer-valued. The model is shown to exhibit two types of motions, those which involve periodic phase differences, and those which involve drift. Traditional dynamical concepts such as stability, bifurcation and chaos are examined for this class...

Periodic motions in DDE (Differential-Delay Equations) are typically 4 created in Hopf bifurcations. In this chapter we examine this process from several 5 points of view. Firstly we use Lindstedt's perturbation method to derive the Hopf Bi-6 furcation Formula, which determines the stability of the periodic motion. Then we 7 use the Two Variable Ex...

After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE X-n + (delta + epsilon cost)x + cD(alpha)x = 0 where D(x)x is the order 7 derivative of x(t), where 0 < alpha < 1. We use the method of harmonic balance and obtain both a lowest...

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that...

We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theo...

This paper investigates the dynamics of a new model of two coupled relaxation oscillators. The model replaces the usual DDE (differential-delay equation) formulation with a discrete-time approach with jumps. Existence, bifurcation and stability of in-phase periodic motions is studied. Simple periodic motions, which involve exactly two jumps per per...

A one-dimensional, steady-state, constant temperature model of diffusion and absorption of CO//2 in the intercellular air spaces of a leaf is presented. The model includes two geometrically distinct regions of the leaf interior, corresponding to palisade and spongy mesophyll tissue, respectively. Sun, shade, and intermediate light leaves are modele...

We investigate the existence, stability and bifurcation of phase-locked motions in a ring network consisting of phase-only oscillators arranged in multiple simple rings (sub-rings) which are themselves arranged in a single large ring. In the case of networks with three or four sub-rings, we give approximate expressions for critical coupling coeffic...

An investigation of the nonlinear dynamics of a heart model is presented. The model compartmentalizes the heart into one part that beats autonomously (the x oscillator), representing the pacemaker or SA node, and a second part that beats only if excited by a signal originating outside itself (the y oscillator), representing typical cardiac tissue....

This work concerns linear parametrically excited systems that involve multiple resonances. The property of such systems is that if the parameters are fixed and lie inside a resonance tongue, the motion becomes unbounded as time goes to infinity. In this work we consider what happens when the parameters are not fixed, but rather are constrained to v...

The sinoatrial (SA) node is a group of self-oscillatory cells in the heart which beat rhythmically and initiate electric potentials, producing a wave of contraction that travels through the heart resulting in the circulation of blood. The SA node is an inhomogeneous collection of cells which have varying intrinsic frequencies. Experimental measurem...

The dynamics of a ring of three identical relaxation oscillators is shown to exhibit a variety of periodic motions, including clockwise and counter-clockwise wave-like modes, and a synchronous mode in which all three oscillators are in phase. The model involves individual oscillators which exhibit sudden jumps, modeling the relaxation oscillations...

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine whi...

We investigate the stability of the in-phase mode in a system of two delay-coupled bubble oscillators. The bubble oscillator model is based on a 1956 paper by Keller and Kolodner. Delay coupling is due to the time it takes for a signal to travel from one bubble to another through the liquid medium that surrounds them. Using techniques from the theo...

We study the dynamics of a system of four coupled phase-only oscillators. This system is analyzed using phase difference variables in a phase space that has the topology of a three-dimensional torus. The system is shown to exhibit numerous phase-locked motions. The qualitative dynamics are shown to depend upon a parameter representing coupling stre...

The dynamics of an autonomous conservative three degree of freedom system which exhibits autoparametric quasiperiodic excitation is investigated. The system is a generalization of a classical system known as the “particle in the plane”. The system exhibits a motion, the z=0 mode, whose stability is governed by a linear second order ODE with quasipe...

We analyze a model of gene transcription and protein synthesis which has been previously presented in the biological literature. The model takes the form of an ODE (ordinary differential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. Linear analysis gives a critical t...

Optically actuated radio frequency microelectromechanical system (MEMS) devices are seen to self-oscillate or vibrate under illumination of sufficient strength (Aubin, Pandey, Zehnder, Rand, Craighead, Zalalutdinov, Parpia (Appl. Phys. Lett. 83, 3281–3283, 2003)). These oscillations can be frequency locked to a periodic forcing, applied through an...

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated in the presence of vertical quasiperiodic forcing having two incommensurate frequencies omega1 and omega2. By means of a Galerkin projection truncated to the first order, the governing linear system corresponding to the onset of convection is reduced to...

We investigate the dynamics of a delayed nonlinear Mathieu equation: [(x)\ddot]+(d+eacost)x +egx3=ebx(t-T)\ddot{x}+(\delta+\varepsilon\alpha\,\cos t)x +\varepsilon\gamma x^3=\varepsilon\beta x(t-T) in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The me...

Instabilities associated with 2:1 and 4:1 resonances of two models for the parametric forcing of a strictly nonlinear oscillator are analyzed. The first model involves a nonlinear Mathieu equation and the second one is described by a 2 degree of freedom Hamiltonian system in which the forcing is introduced by the coupling. Using averaging with elli...

It is shown that a transformation of time can allow the periodic solution of a strongly nonlinear oscillator to be written as a simple cosine function. This enables the stability of strongly nonlinear normal modes in multidegree of freedom systems to be investigated by standard procedures such as harmonic balance.

We study the dynamics of a thermo-mechanical model for a forced disc shaped, micromechanical limit cycle oscillator. The forcing can be accomplished either parametrically, by modulating the laser beam incident on the oscillator, or non-parametrically, using inertial driving. The system exhibits both 2:1 and 1:1 resonances, as well as quasiperiodic...

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using Lindstedt’s perturbation method.

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expan...

This paper deals with a class of conservative nonlinear oscillators of the form $$\ddot x(t)+f(x(t))=0$$, where f(x) is analytic. A transformation of time from t to a new time coordinate τ is defined such that periodic solutions can be expressed in the form x(τ) = A 0+A 1 cos 2τ. We refer to this process of trigonometric simplification as trigonome...

The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the u...

We analyze a model of gene transcription and protein syn- thesis which has been previously presented in the biological lit- erature. The model takes the form of an ODE (ordinary differ- ential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. The delay is assumed to depe...

Thin, planar, radio frequency microelectromechanical systems (MEMS) resonators have been shown to self-oscillate in the absence of external forcing when illuminated by a direct current (dc) laser of sufficient amplitude. In the presence of external forcing of sufficient strength and close enough in frequency to that of the unforced oscillation, the...

We use center manifold theory to analyze a model of gene transcription and protein synthesis which consists of an ordinary differential equation (ODE) coupled to a delay differential equation (DDE). The analysis involves reformulating the problem as an operator differential equation which acts on function space, with the result that an infinite dim...

We investigate the damped cubic nonlinear quasiperiodic Mathieu equation \fracd2xdt2+(d+ecost+emcoswt)x+emc\fracdxdt+emgx3=0 \frac{d^2x}{dt^2}+(\delta+\varepsilon \cos t+\varepsilon \mu \cos\omega t)x+\varepsilon \mu c\frac{dx}{dt}+\varepsilon \mu \gamma x^3=0 in the vicinity of the principal 2:2:1 resonance. By using a double perturbation method...

A radio frequency (RF) micromechanical shell-type resonator with a resistive thermal actuator is shown to perform as a highly linear, broadband mixer and a high-quality factor post-translation (intermediate frequency) filter. The resistor is capable of frequency translation of RF carrier signals as high as 1.5 GHz to the intermediate frequency of 1...

Coexistence phenomenon refers to the absence of expected tongues of instability in parametrically excited systems. In this paper we obtain sufficient conditions for coexistence to occur in the generalized Ince equation The results are applied to the stability of motion of a non-linear normal mode, the x-mode, in a class of conservative two degree o...

This paper presents a size-structured dynamical model of plant growth. The model takes the form of a partial differential-integral equation and includes the effects of self- shading by leaves. Closed form solutions are presented for the equilibrium size density distribution. Analytic conditions are derived for community persistence, and the self-th...

We investigate the dynamics of a system consisting of a simple harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurca...

We present a small $ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 $ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 in the neighborhood of the point = 0.25 and = 0.5. We use multiple scales including terms of O(2) with three time scales. We obtain an asymptotic expansion for an associated instability region....

We present a small ε perturbation analysis of the quasiperiodic Mathieu equation: x¨+(δ+εcost+εcosωt)x=0 in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales including terms of O(ε2) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows go...

In this paper we investigate the dynamics of a Mathieu-van der Pol equation, which is forced both parametrically and nonparametrically. It is shown that the steady state response can consist of either 1:1 frequency locking, or 2:1 subharmonic locking, or quasiperiodic motion. The system displays hysteresis when the forcing frequency is slowly varie...

In this work we study a system of three van der Pol oscillators, x, y and w, coupled as follows: ẍ − ε(1 − x ²)ẋ + x = εμ(w − x) ÿ − ε(1 − y ²)ẏ + y = εμ(w − y) ẅ − ε(1 − w ²)ẇ + p ²w = εμ(x − w) + εμ(y − w) Here the x and y oscillators are identical, and are not directly coupled to each other, but rather are coupled via the w oscillator. We invest...

Limit cycle, or self-oscillations, can occur in a variety of NEMS devices illuminated within an interference field. As the device moves within the field, the quantity of light absorbed and hence the resulting thermal stresses changes, resulting in a feedback loop that can lead to limit cycle oscillations. Examples of devices that exhibit such behav...

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We...

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z:We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ϵ>0 and k>0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a...

This paper concerns the quadratically-damped Mathieu equation: x + (+ cos t)x + ˙ x| ˙ x| = 0: Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In – parameter space, this secondary bifurcation appears as a curve which emanates from one o...

In this paper we examine the dynamics of two time-delay coupled relaxation oscillators of the van der Pol type. By integrating the governing differential-delay equations numerically, we find the various phase-locked mo-tions including the in-phase and out-of-phase modes. Our computations reveal that depending on the strength of coupling (α) and the...

A study of the flow characteristics in vascular tissues of plants was made. Results of research in this field are evaluated. The upward flow of water and minerals is driven by evaporation at the leaves. An introduction is made of the concepts that are unique to the fluid mechanics of plants.

We demonstrate synchronization of laser-induced self-sustained vibrations of radio-frequency micromechanical resonators by applying a small pilot signal either as an inertial drive at the natural frequency of the resonator or by modulating the stiffness of the oscillator at double the natural frequency. By sweeping the pilot signal frequency, we de...

Abstract We examine how species richness and species-specific plant density (number of species and number of individuals per species, respectively) vary within community size frequency distributions and across latitude. Communities from Asia, Africa, Europe, and North, Central and South America were studied (60°4′N–41°4′S latitude) using the Gentry...

We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions. We use the method of multiple scales to obtain a sl...

We show that explicit mathematical and biological relationships exist among the scaling exponents and the allometric constants (α and β, respectively) of log - log linear tree-community size frequency distributions, plant density N T, and minimum, maximum and average stem diameters (D min, D max, and D̄, respectively). As individuals grow in size a...

Carruba et al. (2003) previously studied a layer of chaos for satellites in Kozai resonance, i.e., satellites whose argument of pericenter, instead of circulating from 0 to 360 degrees, librates around +/- 90 degrees. By performing numerical simulations with test particles covering the orbital space surrounding S/2000S5, a satellite of Saturn curre...

In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation \fracd2 xdt2 + ( d+ ecost + emcos( 1 + eD )t )x = 0,\frac{{d^2 x}}{{dt^2 }} + \left( {\delta + \varepsilon \cos t + \varepsilon \mu \cos \left( {1 + \varepsilon \Delta } \right)t} \right)x = 0, using two successive perturbatio...

Abstract We examine how species richness and species-specific plant density (number of species and number of individuals per species, respectively) vary within community size frequency distributions and across latitude. Communities from Asia, Africa, Europe, and North, Central and South America were studied (60o4'N-41o4'S latitude) using the Gentry...

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomenon of coexistence. The differential equation studied governs the stability of a mode of vibration in an unforced conservative two degree of freedom system used to model thefree vibrations of a thin elastica....

Self-sustained mechanical vibrations of a disc-type microfabricated resonator were experimentally observed when a continuous wave (CW) laser beam was focused on the periphery of the disc (for a 40 μm diameter resonator, natural frequency 0.89MHz, the laser power above a 250 W threshold was required). A theoretical model for self-oscillatory behavio...

A model of a strip of cardiac tissue consisting of a one-dimensional chain of cardiac units is derived in the form of a non-linear partial difference equation. Perturbation analysis is performed on this equation, and it is shown that regular perturbations are inadequate due to the appearance of secular terms. A singular perturbation procedure known...

We investigate the dynamics of a system of twovan der Pol oscillators with delayed velocity coupling.We use the method of averaging to reduce the problem to the studyof a slow-flow in three dimensions.We study the steady state solutions of this slow-flow, with specialattention given to the bifurcations accompanying their change innumber and stabili...

In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation:d2xdt2+(δ+ϵcost)x+ϵAx3+Bx2dxdt+Cxdxdt2+Ddxdt3=0We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the r...

We investigate the nonlinear dynamics of the classical Mathieu equation to which is added a nonlinearity which is a general cubic in x, ẋ. We use a perturbation method (averaging) which is valid in the neighborhood of 2:1 resonance, and in the limit of small forcing and small nonlinearity. By comparing the predictions of first-order averaging with...

In this paper, we present an perturbation method that utilizes Lie transform perturbation theory and elliptic functions to investigate subharmonic resonances in the non-linear Mathieu equationIt is assumed that the parametric perturbation, , is small and that the coefficient of the non-linear term, α, is positive but not necessarily small. We deriv...

In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [ + (cos 1 t + cos 2 t)] x + x3 = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the u...

Experimental results for vortex-induced vibrations of both pivoted and non-pivoted cylinders in a crossflow are presented in this report. Simplified models involving two ordinary differential equations are reviewed and shown to exhibit many of the features of the experimental results. Results of perturbation analysis of the simplified model are pre...

We investigate the dynamics of a system involving the planar motionof a rigid body which is restrained by linear springs and whichpossesses a skate-like nonholonomic constraint known as \frac12\frac{1}{2} degrees of freedom. The resulting phase flow is shownto involve a curve of nonisolated equilibria. Using second-orderaveraging, the system is s...

Non-linear modal interactions in the dynamics of a vibrating drop are examined. The partial di!erential equations governing the drop vibrations are formulated assuming potential #ow and incompressibility. The solution is expressed in terms of the eigenfunctions of the (linearized) Laplace operator in spherical coordinates. A small parameter is intr...

The boundaries of the basin of attraction are usually assumed to be rather elementary for Hamiltonian systems with autonomous perturbations. In the case of one saddle point, the sequences of orbits before capture are unique for each basin. However, we show that for two saddle points each with double heteroclinic orbits, there is an infinite number...

We obtain power series solutions to the “abc equation” dydx=a+bcosy+ccosx, valid for small c, and for small b. This equation is shown to determine the stability of the quasiperiodic Mathieu equation, z¨+(δ+ϵA1cost+ϵA2cosωt)z=0, in the small ϵ limit. Perturbation results of the abc equation are shown to compare favorably to numerical integration of...