Richard Rand - Academia.edu (original) (raw)
Papers by Richard Rand
Cornell University - arXiv, Sep 23, 2016
Delay or queue length information has the potential to influence the decision of a customer to us... more Delay or queue length information has the potential to influence the decision of a customer to use a service system. Thus, it is imperative for service system managers to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant ∆. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of ∆. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations
Journal of Microelectromechanical Systems
In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mecha... more In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mechanically coupled, doubly clamped, silicon micromechanical oscillators, and numerically investigate the dynamics of the corresponding lumpedparameter model. Coupled limit cycle oscillators exhibit striking nonlinear dynamics and bifurcations in response to variations in system parameters. We show that the input laser power influences the frequency detuning between two non-identical oscillators. As the laser power is varied, different regimes of oscillations such as the synchronized state, the drift state, and the quasi-periodic state are mapped at minimal and high coupling strengths. For nonidentical oscillators, coexistence of two states, the synchronized state and the quasi-periodic state, is demonstrated at high coupling and high laser power. Experimentally, this bistability manifests as irregular oscillations as the system rapidly switches between the two states due to the system's sensitive dependence on initial conditions in the presence of noise. We provide a qualitative comparison of the experimental and numerical results to elucidate the behavior of the system.
International Journal of Bifurcation and Chaos, 2017
Delay or queue length information has the potential to influence the decision of a customer to jo... more Delay or queue length information has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant [Formula: see text]. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of [Formula: see text]. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations and asynchronou...
Springer Proceedings in Mathematics & Statistics, 2014
Nonlinear oscillators with hardening and softening cubic Duffing nonlinearity are considered. Suc... more 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, conservation laws, as well as solutions for motion.
SIAM Journal on Applied Mathematics, 1998
In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu e... more In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu equation given byψ + [δ + (cos t + cos ωt)] ψ = 0 for small and irrational ω. Of interest is the generation of stability diagrams that identify the points or regions in the δ-ω parameter plane (for fixed) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots of Lyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves δ = δ(ω;) are each expanded in powers of , and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that predominant regions of instability are clearly coincident. And, both analytic techniques enable us to gain insight into the structure of the corresponding numerical plots. However, the perturbation method fails in the neighborhood of resonant values of ω due to the problem of small divisors; the results obtained by harmonic balance do not display this undesirable feature.
Physical Review E, 2007
Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated ... more Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated in the presence of vertical quasiperiodic forcing having two incommensurate frequencies 1 and 2. By means of a Galerkin projection truncated to the first order, the governing linear system corresponding to the onset of convection is reduced to a damped quasiperiodic Mathieu equation. The threshold of convection corresponding to quasiperiodic solutions is determined in the cases of heating from below and heating from above. We show that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect depending on the frequencies ratio = 2 / 1. The effect of the Prandtl number in a stabilizing zone is also examined for different frequency ratios.
Journal of Mathematical Biology, 2005
This paper presents a size-structured dynamical model of plant growth. The model takes the form o... more 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 selfshading by leaves. Closed form solutions are presented for the equilibrium size density distribution. Analytic conditions are derived for community persistence, and the self-thinning exponent is obtained as a function of species characteristics and environmental conditions.
Ecology Letters, 2003
We show that explicit mathematical and biological relationships exist among the scaling exponents... more We show that explicit mathematical and biological relationships exist among the scaling exponents and the allometric constants (a and b, respectively) of log-log linear treecommunity size frequency distributions, plant density N T , and minimum, maximum and average stem diameters (D min , D max , and D D, respectively). As individuals grow in size and D max increases, N T is predicted to decrease as reflected by a decrease in the numerical value of a and an increase in the value of b. Our derivations further show that N T decreases as D D increases even if D min or D max remain unchanged. Because D max and the age of the largest individuals in a community are correlated, albeit weakly, we argue that the interdependent relationships among the numerical values of a, b, N T , and D D shed light on the extent to which communities have experienced recent global disturbance. These predicted relationships receive strong statistical support using two large datasets spanning a broad spectrum of tree-dominated communities.
Communications in Nonlinear Science and Numerical Simulation, 2011
We investigate a problem in evolutionary game theory based on replicator equations with periodic ... more 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 coefficients models variations in the environment due to seasonal effects and results in parametric excitation which is studied through the use of perturbation series and numerical integration.
Communications in Nonlinear Science and Numerical Simulation, 2011
A new model of coupled oscillators is proposed and investigated. All phase variables and paramete... more 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 of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.
We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate... more We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate relative to one another about a common shaft S. A motor on one of the bodies provides a small constant internal torque which influences the relative motion of the two bodies, and which may influence the orientation of their common shaft S. Resonant capture refers to the phenomenon that the spacecraft may end up in one of several possible orientations, including a nearly flat spin (transverse to S), in addition to the expected simple rotation about S. The method of averaging is used to treat the original equations of motion, and it is shown that the essential mathematical problem involves separatrix crossing in a problem with slowly moving separatrices. Energy changes represented by Melnikov integrals are used to supplement the averaged equations in the neighborhood of the heteroclinic motions. The method is used to predict which initial conditions lead to capture into each of three distinct capture regions. The asymptotic results are compared to those obtained by direct numerical integration of the equations of motion.
Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise, 2001
We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu e... more We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation,(1)x..+[δ+ϵ(cosω1t+cosω2t)]x+αx3=0. We assume that ϵ ≪ 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions — rather than the usual trigonometric functions — to study subharmonic resonances associated with orbits in 2m : 1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, ϵ, ω1, ω2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov’s overlap criterion to obtain an overview of the O(ϵ) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and ϵ fixed.
IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems, 2019
We investigate the dynamics of two limit cycle MEMS oscillators connected via spring coupling. Ea... more We investigate the dynamics of two limit cycle MEMS oscillators connected via spring coupling. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, has been previously presented (Rand et al in Proceedings of 9th European Nonlinear Dynamics Conference (ENOC 2017), 2017, [3]). For the coupled system, a perturbation method is used to obtain a slow flow which is investigated using AUTO and numerical integration. Various bifurcations which occur as a result of changing the coupling strength are identified. Keywords Coupled oscillators • MEMS • Bifurcations • Perturbations 20.1 Introduction This work is motivated by a type of MEMS device in which a laser is used to determine the motion of the device by interference. The MEMS device is typically a clampedclamped beam fabricated from a thin layer of Si and suspended above a Si substrate. Laser light is focused onto the beam surface and is partially reflected, absorbed and
arXiv: Dynamical Systems, 2017
Understanding how delayed information impacts queueing systems is an important area of research. ... more Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time varying arrivals. In the first model, customers receive queue length information that is delayed by a constant Delta. In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is Delta. We analyze the impact of the time varying arrival rate and show using asymptotic analysis that the time varying arrival rate does not impact the critical delay unless the frequency of the...
Mathematics of Operations Research, 2020
Many service systems provide queue length information to customers, thereby allowing customers to... more Many service systems provide queue length information to customers, thereby allowing customers to choose among many options of service. However, queue length information is often delayed, and it is often not provided in real time. Recent work by Dong et al. [Dong J, Yom-Tov E, Yom-Tov GB (2018) The impact of delay announcements on hospital network coordination and waiting times. Management Sci. 65(5):1969–1994.] explores the impact of these delays in an empirical study in U.S. hospitals. Work by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] uses a two-dimensional fluid model to study the impact of delayed information and determine the exact threshold under which delayed information can cause oscillations in the dynamics of the queue length. In this work, we confirm that the fluid model analyzed by Pender et al. [Pender J, Rand RH, Wesson E (2017) Que...
Volume 8: 26th Conference on Mechanical Vibration and Noise, Aug 17, 2014
Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscilla... more 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, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m:n. These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.
Journal of Applied Nonlinear Dynamics, 2014
Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscilla... more 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, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m : n. These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.
Nonlinear Dynamics, 2017
We investigate the dynamics of the nonlinear DDE (delay-differential equation) d 2 x dt 2 (t) + x... more We investigate the dynamics of the nonlinear DDE (delay-differential equation) d 2 x dt 2 (t) + x(t − T) + x(t) 3 = 0, where T is the delay. For T = 0, this system is conservative and exhibits no limit cycles. For T > 0, no matter how small T is, an infinite number of limit cycles exist, their amplitudes going to infinity in the limit as T approaches zero. We investigate this situation in three ways: (1) harmonic balance, (2) Melnikov's integral, and (3) adding damping to regularize the singularity.
Nano letters, Jan 12, 2017
We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in th... more We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in the presence of incident laser light and a reflecting Si mirror. Situated in a standing wave of optical intensity and subject to photothermal forces, the nanowire undergoes self-induced oscillations at low incident light thresholds of <1 μW due to engineered strong temperature-position (T-z) coupling. Along with inducing self-oscillation, laser light causes large changes to the mechanical resonant frequency ω0 and equilibrium position z0 that cannot be neglected. We present experimental results and a theoretical model for the motion under laser illumination. In the model, we solve the governing nonlinear differential equations by perturbative means to show that self-oscillation amplitude is set by the competing effects of direct T-z coupling and 2ω0 parametric excitation due to T-ω0 coupling. We then study the linearized equations of motion to show that the optimal thermal time constant...
Cornell University - arXiv, Sep 23, 2016
Delay or queue length information has the potential to influence the decision of a customer to us... more Delay or queue length information has the potential to influence the decision of a customer to use a service system. Thus, it is imperative for service system managers to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant ∆. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of ∆. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations
Journal of Microelectromechanical Systems
In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mecha... more In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mechanically coupled, doubly clamped, silicon micromechanical oscillators, and numerically investigate the dynamics of the corresponding lumpedparameter model. Coupled limit cycle oscillators exhibit striking nonlinear dynamics and bifurcations in response to variations in system parameters. We show that the input laser power influences the frequency detuning between two non-identical oscillators. As the laser power is varied, different regimes of oscillations such as the synchronized state, the drift state, and the quasi-periodic state are mapped at minimal and high coupling strengths. For nonidentical oscillators, coexistence of two states, the synchronized state and the quasi-periodic state, is demonstrated at high coupling and high laser power. Experimentally, this bistability manifests as irregular oscillations as the system rapidly switches between the two states due to the system's sensitive dependence on initial conditions in the presence of noise. We provide a qualitative comparison of the experimental and numerical results to elucidate the behavior of the system.
International Journal of Bifurcation and Chaos, 2017
Delay or queue length information has the potential to influence the decision of a customer to jo... more Delay or queue length information has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant [Formula: see text]. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of [Formula: see text]. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations and asynchronou...
Springer Proceedings in Mathematics & Statistics, 2014
Nonlinear oscillators with hardening and softening cubic Duffing nonlinearity are considered. Suc... more 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, conservation laws, as well as solutions for motion.
SIAM Journal on Applied Mathematics, 1998
In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu e... more In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu equation given byψ + [δ + (cos t + cos ωt)] ψ = 0 for small and irrational ω. Of interest is the generation of stability diagrams that identify the points or regions in the δ-ω parameter plane (for fixed) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots of Lyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves δ = δ(ω;) are each expanded in powers of , and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that predominant regions of instability are clearly coincident. And, both analytic techniques enable us to gain insight into the structure of the corresponding numerical plots. However, the perturbation method fails in the neighborhood of resonant values of ω due to the problem of small divisors; the results obtained by harmonic balance do not display this undesirable feature.
Physical Review E, 2007
Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated ... more Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated in the presence of vertical quasiperiodic forcing having two incommensurate frequencies 1 and 2. By means of a Galerkin projection truncated to the first order, the governing linear system corresponding to the onset of convection is reduced to a damped quasiperiodic Mathieu equation. The threshold of convection corresponding to quasiperiodic solutions is determined in the cases of heating from below and heating from above. We show that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect depending on the frequencies ratio = 2 / 1. The effect of the Prandtl number in a stabilizing zone is also examined for different frequency ratios.
Journal of Mathematical Biology, 2005
This paper presents a size-structured dynamical model of plant growth. The model takes the form o... more 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 selfshading by leaves. Closed form solutions are presented for the equilibrium size density distribution. Analytic conditions are derived for community persistence, and the self-thinning exponent is obtained as a function of species characteristics and environmental conditions.
Ecology Letters, 2003
We show that explicit mathematical and biological relationships exist among the scaling exponents... more We show that explicit mathematical and biological relationships exist among the scaling exponents and the allometric constants (a and b, respectively) of log-log linear treecommunity size frequency distributions, plant density N T , and minimum, maximum and average stem diameters (D min , D max , and D D, respectively). As individuals grow in size and D max increases, N T is predicted to decrease as reflected by a decrease in the numerical value of a and an increase in the value of b. Our derivations further show that N T decreases as D D increases even if D min or D max remain unchanged. Because D max and the age of the largest individuals in a community are correlated, albeit weakly, we argue that the interdependent relationships among the numerical values of a, b, N T , and D D shed light on the extent to which communities have experienced recent global disturbance. These predicted relationships receive strong statistical support using two large datasets spanning a broad spectrum of tree-dominated communities.
Communications in Nonlinear Science and Numerical Simulation, 2011
We investigate a problem in evolutionary game theory based on replicator equations with periodic ... more 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 coefficients models variations in the environment due to seasonal effects and results in parametric excitation which is studied through the use of perturbation series and numerical integration.
Communications in Nonlinear Science and Numerical Simulation, 2011
A new model of coupled oscillators is proposed and investigated. All phase variables and paramete... more 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 of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.
We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate... more We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate relative to one another about a common shaft S. A motor on one of the bodies provides a small constant internal torque which influences the relative motion of the two bodies, and which may influence the orientation of their common shaft S. Resonant capture refers to the phenomenon that the spacecraft may end up in one of several possible orientations, including a nearly flat spin (transverse to S), in addition to the expected simple rotation about S. The method of averaging is used to treat the original equations of motion, and it is shown that the essential mathematical problem involves separatrix crossing in a problem with slowly moving separatrices. Energy changes represented by Melnikov integrals are used to supplement the averaged equations in the neighborhood of the heteroclinic motions. The method is used to predict which initial conditions lead to capture into each of three distinct capture regions. The asymptotic results are compared to those obtained by direct numerical integration of the equations of motion.
Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise, 2001
We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu e... more We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation,(1)x..+[δ+ϵ(cosω1t+cosω2t)]x+αx3=0. We assume that ϵ ≪ 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions — rather than the usual trigonometric functions — to study subharmonic resonances associated with orbits in 2m : 1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, ϵ, ω1, ω2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov’s overlap criterion to obtain an overview of the O(ϵ) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and ϵ fixed.
IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems, 2019
We investigate the dynamics of two limit cycle MEMS oscillators connected via spring coupling. Ea... more We investigate the dynamics of two limit cycle MEMS oscillators connected via spring coupling. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, has been previously presented (Rand et al in Proceedings of 9th European Nonlinear Dynamics Conference (ENOC 2017), 2017, [3]). For the coupled system, a perturbation method is used to obtain a slow flow which is investigated using AUTO and numerical integration. Various bifurcations which occur as a result of changing the coupling strength are identified. Keywords Coupled oscillators • MEMS • Bifurcations • Perturbations 20.1 Introduction This work is motivated by a type of MEMS device in which a laser is used to determine the motion of the device by interference. The MEMS device is typically a clampedclamped beam fabricated from a thin layer of Si and suspended above a Si substrate. Laser light is focused onto the beam surface and is partially reflected, absorbed and
arXiv: Dynamical Systems, 2017
Understanding how delayed information impacts queueing systems is an important area of research. ... more Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time varying arrivals. In the first model, customers receive queue length information that is delayed by a constant Delta. In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is Delta. We analyze the impact of the time varying arrival rate and show using asymptotic analysis that the time varying arrival rate does not impact the critical delay unless the frequency of the...
Mathematics of Operations Research, 2020
Many service systems provide queue length information to customers, thereby allowing customers to... more Many service systems provide queue length information to customers, thereby allowing customers to choose among many options of service. However, queue length information is often delayed, and it is often not provided in real time. Recent work by Dong et al. [Dong J, Yom-Tov E, Yom-Tov GB (2018) The impact of delay announcements on hospital network coordination and waiting times. Management Sci. 65(5):1969–1994.] explores the impact of these delays in an empirical study in U.S. hospitals. Work by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] uses a two-dimensional fluid model to study the impact of delayed information and determine the exact threshold under which delayed information can cause oscillations in the dynamics of the queue length. In this work, we confirm that the fluid model analyzed by Pender et al. [Pender J, Rand RH, Wesson E (2017) Que...
Volume 8: 26th Conference on Mechanical Vibration and Noise, Aug 17, 2014
Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscilla... more 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, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m:n. These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.
Journal of Applied Nonlinear Dynamics, 2014
Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscilla... more 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, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m : n. These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.
Nonlinear Dynamics, 2017
We investigate the dynamics of the nonlinear DDE (delay-differential equation) d 2 x dt 2 (t) + x... more We investigate the dynamics of the nonlinear DDE (delay-differential equation) d 2 x dt 2 (t) + x(t − T) + x(t) 3 = 0, where T is the delay. For T = 0, this system is conservative and exhibits no limit cycles. For T > 0, no matter how small T is, an infinite number of limit cycles exist, their amplitudes going to infinity in the limit as T approaches zero. We investigate this situation in three ways: (1) harmonic balance, (2) Melnikov's integral, and (3) adding damping to regularize the singularity.
Nano letters, Jan 12, 2017
We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in th... more We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in the presence of incident laser light and a reflecting Si mirror. Situated in a standing wave of optical intensity and subject to photothermal forces, the nanowire undergoes self-induced oscillations at low incident light thresholds of <1 μW due to engineered strong temperature-position (T-z) coupling. Along with inducing self-oscillation, laser light causes large changes to the mechanical resonant frequency ω0 and equilibrium position z0 that cannot be neglected. We present experimental results and a theoretical model for the motion under laser illumination. In the model, we solve the governing nonlinear differential equations by perturbative means to show that self-oscillation amplitude is set by the competing effects of direct T-z coupling and 2ω0 parametric excitation due to T-ω0 coupling. We then study the linearized equations of motion to show that the optimal thermal time constant...