Limit Cycle Solutions Research Articles

Modelling the coupling between intracellular calcium release and the cell cycle during cortical brain development

Most neocortical neurons formed during embryonic brain development arise from radial glial cells which communicate, in part, via ATP mediated calcium signals. Although the intercellular signalling mechanisms that regulate radial glia proliferation are not well understood, it has recently been demonstrated that ATP dependent intracellular calcium release leads to an increase of nearly 100% in overall cellular proliferation. It has been hypothesised that cytoplasmic calcium accelerates entry into S phase of the cell cycle and/or acts to recruit otherwise quiescent cells onto the cell cycle. In this paper we study this cell cycle acceleration and recruitment by forming a differential equation model for ATP mediated calcium-cell cycle coupling via Cyclin D in a single radial glial cell.Bifurcation analysis and numerical simulations suggest that the cell cycle period depends only weakly on cytoplasmic calcium. Therefore, the accelerative impact of calcium on the cell cycle can only account for a small fraction of the large increase in proliferation observed experimentally. Crucially however, our bifurcation analysis reveals that stable fixed point and stable limit cycle solutions can coexist, and that calcium dependent Cyclin D dynamics extend the oscillatory region to lower Cyclin D synthesis rates, thus rendering cells more susceptible to cycling. This supports the hypothesis that cycling glial cells recruit quiescent cells (in G0 phase) onto the cell cycle, via a calcium signalling mechanism, and that this may be the primary means by which calcium augments proliferation rates at the population scale. Numerical simulations of two coupled cells demonstrate that such a scenario is indeed feasible.

Limit cycle analysis of nonsmooth aeroelastic system of an airfoil by extended variational iteration method

The variational iteration method (VIM) was extended to solve the aeroelastic system of an airfoil with both cubic and nonsmooth stiffnesses (a piecewise freeplay), expressed by integral-differential equations. Limit cycle (LC) frequency and amplitude are unknown to be updated at every iteration stage. If the VIM is routinely applied, the LC solution needs to be embedded into both the integrals and the freeplay, which can result in two intractable problems: (1) the integrals lead to coupled terms between frequency and amplitude and (2) the Fourier expansion of the freeplay cannot be realized. In order to tackle these problems, the solution obtained at the previous iteration rather than the current unknown one is applied to the freeplay and the integrals. By this means, the frequency and amplitude are decoupled, and the freeplay can be expanded as Fourier series. Numerical examples show that very accurate LC solutions can be obtained by the presented method. This method is suitable for solving LCs of nonlinear dynamical systems with nonsmooth nonlinearities.

Collective phase description of oscillatory convection

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.

Chemical oscillator as a generalized Rayleigh oscillator

We derive the conditions under which a set of arbitrary two dimensional autonomous kinetic equations can be reduced to the form of a generalized Rayleigh oscillator which admits of limit cycle solution. This is based on a linear transformation of field variables which can be found by inspection of the kinetic equations. We illustrate the scheme with the help of several chemical and bio-chemical oscillator models to show how they can be cast as a generalized Rayleigh oscillator.

Dynamic Modeling for the Design and Cyclic Operation of an Atomic Layer Deposition (ALD) Reactor

A laboratory-scale atomic layer deposition (ALD) reactor system model is derived for alumina deposition using trimethylaluminum and water as precursors. Model components describing the precursor thermophysical properties, reactor-scale gas-phase dynamics and surface reaction kinetics derived from absolute reaction rate theory are integrated to simulate the complete reactor system. Limit-cycle solutions defining continuous cyclic ALD reactor operation are computed with a fixed point algorithm based on collocation discretization in time, resulting in an unambiguous definition of film growth-per-cycle (gpc). A key finding of this study is that unintended chemical vapor deposition conditions can mask regions of operation that would otherwise correspond to ideal saturating ALD operation. The use of the simulator for assisting in process design decisions is presented.

Quantum-classical transition of correlations of two coupled cavities

We study the difference between quantum and classical behavior in a pair of nonidentical cavities with second-harmonic generation. In the classical limit, each cavity has a limit-cycle solution, in which the photon number oscillates periodically in time. Coupling between the cavities leads to synchronization of the oscillations and classical correlations between the cavities. In the quantum limit, there are quantum correlations due to entanglement. The quantum correlations persist even when the cavities are far off resonance with each other, in stark contrast with the classical case. We also find that the quantum and classical limits are connected by an intermediate regime of almost no correlations. Our results can be extended to a wide variety of quantum models.

Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers

In this contribution we provide an in depth theoretical analysis of the bifurcations leading to nonlinear polarization dynamics in a free-running vertical-cavity surface-emitting laser (VCSEL). We detail the sequence of bifurcations that occurs when increasing the injection current, and which brings the laser from linear to elliptical polarization emission and then self-pulsating or even more complex chaotic dynamics of the light intensity. Continuation techniques allow us to follow the stable and unstable limit cycle solutions emerging from Hopf bifurcations, and therefore to interpret the frequency of the self-pulsating polarization dynamics. The fundamental frequency of the pulsating dynamics is either close to the laser relaxation oscillation frequency or close to the linear-birefringence-induced polarization mode frequency splitting, depending on the laser parameters. A systematic analysis of the parameter space allows us to identify two scenarios that are in excellent qualitative agreement with those reported in recent experiments. Our results provide, moreover, evidence for an interesting polarization mode hopping mechanism, i.e., a so-called deterministic mode hopping where the laser exhibits a chaotic and therefore random-like hopping between two states that are elliptically polarized and nonorthogonal.

A limit cycle oscillator model for cycling mood variations of bipolar disorder patients derived from cellular biochemical reaction equations

We derive a nonlinear limit cycle model for oscillatory mood variations as observed in patients with cycling bipolar disorder. To this end, we consider two signaling pathways leading to the activation of two enzymes that play a key role for cellular and neural processes. We model pathway cross-talk in terms of an inhibitory impact of the first pathway on the second and an excitatory impact of the second on the first. The model also involves a negative feedback loop (inhibitory self-regulation) for the first pathway and a positive feedback loop (excitatory self-regulation) for the second pathway. We demonstrate that due to the cross-talk the biochemical dynamics is described by an oscillator equation. Under disease-free conditions the oscillatory system exhibits a stable fixed point. The breakdown of the self-inhibition of the first pathway at higher concentration levels is studied by means of a scalar control parameter ξ, where ξ equal to zero refers to intact self-inhibition at all concentration levels. Under certain conditions, stable limit cycle solutions emerge at critical parameter values of ξ larger than zero. These oscillations mimic pathological cycling mood variations that emerge due to a disease-induced bifurcation. Consequently, our modeling analysis supports the notion of bipolar disorder as a dynamical disease. In addition, our study establishes a connection between mechanistic biochemical modeling of bipolar disorder and phenomenological nonlinear oscillator approaches to bipolar disorder suggested in the literature.

An experimental setup for autonomous operation of surface vessels in rough seas

SUMMARYA small-scale experimental setup for autonomous target tracking of a surface vessel in the presence of obstacles is presented. The experiments are performed in simulated rough seas through wave, current, and wind generation in a small indoor pool. Absolute position of the agent and the target as well as the obstacle size and position are provided through an overhead camera by detecting color light emitting diodes installed on all objects. Ordinary differential equations with stable limit-cycle solutions are used to define transitional trajectories around obstacles based on the camera data. A sliding mode control law is implemented for real-time tracking control which is capable of rejecting large disturbances from the generated waves and wind. The sliding mode control signals are sent to wireless receivers on the autonomous vessel where a proportional integral speed controller maintains the commanded speed. A special scaling method is presented to show that the environmental forces are similar to those of moderate through high sea states. Several experiments are presented where the autonomous vessel catches and follows a target boat moving in arbitrary trajectories in both the presence and absence of obstacles.

Time-Dependent Ferrofluid Dynamics in Symmetry Breaking Transverse

We investigate the Taylor-Couette flow of a rotating ferrofluid under the influence of symmetry breaking transverse magnetic field in counter-rotating small-aspect-ratio setup. We find only changing the magnetic field strength can drive the dynamics from time-periodic limit-cycle solution to time-independent steady fixed-point solution and vice versa. Thereby both solutions exist in symmetry related offering mode-two symmetry with left-or right-winding characteristics due to finite transverse magnetic field. Furthermore the time-periodic limit-cycle solutions offer alternately stroboscoping both helical left-and right-winding contributions of mode-two symmetry. The Navier-Stokes equations are solved with a second order time splitting method combined with spatial discretization of hybrid finite difference and Galerkin method.

Nonlinear Analysis of Electrically Actuated Carbon Nanotube Resonator Using a Novel Discretization Technique

Static as well as dynamic analyses have been performed on clamped-clamped carbon nanotube (CNT) resonator. The nonlinear CNT model is investigated with a novel discretization technique: a differential quadrature method (DQM) to discretize the spatial variables and a finite difference method (FDM) for limit-cycle solutions. Parametric study is performed by varying the electric load, as well as the initial curvature (due to fabrication). It is found that the pull-in voltage decreases nonlinearly with initial curvature and linearly with residual stresses. The eigenvalue problem is also solved to obtain the bending natural frequencies of the system as function of the DC voltage as well as the initial curvature of the CNT. Frequency-response curves near some selected resonant frequencies are plotted to better understand the nanotubes' dynamic behavior. Different linear and nonlinear phenomena are depicted such as dynamic pull-in, hardening, and softening behavior and veering of the odd modes. We have found that even when exciting the CNT near its first natural frequency, the vibration mode located at the veering process significantly alters the CNT's motion and hence may decrease its overall quality factor.

Dynamic mode decomposition analysis of detonation waves

Dynamic mode decomposition is applied to study the self-excited fluctuations supported by transversely unstable detonations. The focus of this study is on the stability of the limit cycle solutions and their response to forcing. Floquet analysis of the unforced conditions reveals that the least stable perturbations are almost subharmonic with ratio between global mode and fundamental frequency λi/ωf = 0.47. This suggests the emergence of period doubling modes as the route to chaos observed in larger systems. The response to forcing is analyzed in terms of the coherency of the four fundamental energy modes: acoustic, entropic, kinetic, and chemical. Results of the modal decomposition suggest that the self-excited oscillations are quite insensitive to vortical forcing, and maintain their coherency up to a forcing turbulent Mach number of 0.3.

On the thermo-acoustic Fant equation

A ‘reduced complexity’ equation is derived to investigate combustion instabilities of a Rijke burner. The equation is nonlinear and furnishes limit cycle solutions for finite amplitude burner modes. It is a generalisation to combustion flows of the Fant equation used to investigate the production of voiced speech by unsteady throttling of flow by the vocal folds [G. Fant, Acoustic Theory of Speech Production. Mouton, The Hague, 1960]. In the thermo-acoustic problem the throttling occurs at the flame holder. The Fant equation governs the unsteady volume flow past the flame holder which, in turn, determines the acoustics of the entire system. The equation includes a fully determinate part that depends on the geometry of the flame holder and the thermo-acoustic system, and terms defined by integrals involving thermo-aerodynamic sources, such as a flame and vortex sound sources. These integrals provide a clear indication of what must be known about the flow to obtain a proper understanding of the dynamics of the thermo-acoustic system. Illustrative numerical results are presented for the linearised equation. This governs the growth rates of the natural acoustic modes, determined by system geometry, boundary conditions and mean temperature distribution, which are excited into instability by unsteady heat release from the flame and damped by large scale vorticity production and radiation losses into the environment. In addition, the equation supplies information about the ‘combustion modes’ excited by the local time-delay feedback dynamics of the flame.

Diffusion-Induced Oscillations of Extended Defects

From a simple model for the driven motion of a planar interface under the influence of a diffusion field we derive a damped nonlinear oscillator equation for the interface position. Inside an unstable regime, where the damping term is negative, we find limit-cycle solutions, describing an oscillatory propagation of the interface. In the case of a growing solidification front this offers a transparent scenario for the formation of solute bands in binary alloys and, taking into account the Mullins-Sekerka instability, of banded structures.

Phase Description of Stable Limit-cycle Solutions in Reaction-diffusion Systems

Phase reduction theory for stable limit-cycle solutions of one-dimensional reaction-diffusion systems is developed. By locally approximating the isochrons of the limit-cycle orbit, we derive the phase sensitivity function, which is a key quantity in the phase description of limit cycles. As an example, synchronization of traveling pulses in a pair of mutually interacting reaction-diffusion systems is analyzed. It is shown that the traveling pulses can exhibit multi–modal phase locking.

Collective phase description of globally coupled excitable elements

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type I for the saddle-node bifurcation and type II for the Hopf bifurcation.

Yarkovsky-O’Keefe-Radzievskii-Paddack effect on tumbling objects

A semi-analytical model of the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect on an asteroid spin in a non-principal axis rotation state is developed. The model describes the spin-state evolution in Deprit–Elipe variables, first-order averaged with respect to rotation and Keplerian orbital motion. Assuming zero conductivity, the YORP torque is represented by spherical harmonic series with vectorial coefficients, allowing us to use any degree and order of approximation. Within the quadrupole approximation of the illumination function we find the same first integrals involving rotational momentum, obliquity and dynamical inertia that were obtained by Cicalo & Scheeres. The integrals do not exist when higher degree terms of the illumination function are included, and then the asymptotic states known from Vokrouhlický et al. appear. This resolves an apparent contradiction between earlier results. Averaged equations of motion admit stable and unstable limit cycle solutions that were not previously detected. Non-averaged numerical integration by the Taylor series method for an exemplary shape of 3103 Eger is in good agreement with the semi-analytical theory.

A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage

In this paper, we propose the design of an ohmic contact RF microswitch with low voltage actuation, where the upper and lower microplates are displaceable. We develop a mathematical model for the RF microswitch made up of two electrostatically actuated microplates; each microplate is attached to the end of a microcantilever. We assume that the microbeams are flexible and that the microplates are rigid. The electrostatic force applied between the two microplates is a nonlinear function of the displacements and applied voltage. We formulate and solve the static and eigenvalue problems associated with the proposed microsystem. We also examine the dynamic behavior of the microswitch by calculating the limit-cycle solutions. We discretize the equations of motion by considering the first few dominant modes in the microsystem dynamics. We show that only two modes are sufficient to accurately simulate the response of the microsystem under DC and harmonic AC voltages. We demonstrate that the resulting static pull-in voltage and switching time are reduced by 30 and 45%, respectively, as compared to those of a single microbeam-microplate RF-microswitch. Finally, we investigate the global stability of the microswitch using different excitation conditions.

Theoretical and experimental study of Chen chaotic system with notch filter feedback control

Since the past two decades, the time delay feedback control method has attracted more and more attention in chaos control studies because of its simplicity and efficiency compared with other chaos control schemes. Recently, it has been proposed to suppress low-dimensional chaos with the notch filter feedback control method, which can be implemented in a laser system. In this work, we have analytically determined the controllable conditions for notch filter feedback controlling of Chen chaotic system in terms of the Hopf bifurcation theory. The conditions for notch filter feedback controlled Chen chaoitc system having a stable limit cycle solution are given. Meanwhile, we also analysed the Hopf bifurcation direction, which is very important for parameter settings in notch filter feedback control applications. Finally, we apply the notch filter feedback control methods to the electronic circuit experiments and numerical simulations based on the theoretical analysis. The controlling results of notch filter feedback control method well prove the feasibility and reliability of the theoretical analysis.

Strongly Nonlinear Oscillators Subject to Delay

A large class of strongly nonlinear conservative oscillators subject to a delayed feedback are modeled mathematically by second-order delay differential equations. Recent applications include the control of crane oscillations and lasers subject to optoelectronic feedback. We apply the method of averaging in the case of weak damping and weak feedback and determine the bifurcation diagram of the limit-cycle solutions. We find that the coexistence of a stable equilibrium with one or several stable periodic solutions is unavoidable if the delay is sufficiently large.