scholarly journals Phase-Coupled Oscillations in the Brain: Nonlinear Phenomena in Cellular Signalling

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vikas Rai ◽  
Sreenivasan Rajamoni Nadar ◽  
Riaz A. Khan
Keyword(s):  
Limit Cycles ◽  
Initial Conditions ◽  
Phase Relationship ◽  
Oscillatory System ◽  
Calcium Currents ◽  
Nonlinear Phenomena ◽  
Inhibitory Interneurons ◽  
Stable Limit ◽  
Principal Cells ◽  
Coupled Oscillations

We report the existence of phase-coupled oscillations in a model neural system. The model consists of a group of excitatory principal cells in interaction with local inhibitory interneurons. The voltages across the membranes of excitatory cells are governed primarily by calcium and potassium ion conductivities. The number of potassium channels open at any given instant changes in accordance with a deterministic law. The time scale of this change is set by a constant which depends on midpoint potentials at which potassium and calcium currents are half-activated. The growth of mean membrane potential of excitatory principal cells is controlled by that of the inhibitory interneurons. Nonlinear oscillatory system associated with these limit cycles starting from two different initial conditions maintain a definite phase relationship. The phase-coupled oscillations in electrical activity of the neuronal cells carry together amplitude, phase, and time information for cellular signaling. This mechanism supports an energy efficient way of information processing in the central nervous system. The information content is encoded as persistent periodic oscillations represented by stable limit cycles in the phase space.

Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

2005 ◽  
Author(s):  
Michael M. Bernitsas ◽  
Joa˜o Paulo J. Matsuura
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Resonance Phenomenon ◽  
Nonlinear Phenomena ◽  
Mooring System ◽  
Stable Limit ◽  
Dynamic Interactions ◽  
Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

scholarly journals STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

2019 ◽  
pp. 71-77
Author(s):  
Е.Р. Новикова ◽  
Р.И. Паровик
Keyword(s):  
Limit Cycle ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Viscous Friction ◽  
Oscillatory System ◽  
Stable Limit ◽  
Van Der Pol ◽  
Nonlinear Oscillatory System ◽  
Stable Limit Cycles

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS). В работе с помощью численного моделирования построены осциллограммы и фазовые траектории с целью исследования предельных циклов нелинейной колебательной системы Ван-дер-Поля Дуффинга со степенной памятью. Результаты моделирования показали, что в случае отсутствия степенной памяти (α = 2, β = 1) или классической динамической системы Ван-дер-Поля Дуффинга, существует единственный устойчивый предельный цикл, т.е. выполняется теорема Льенара. В случае вязкого трения (α = 2, 0 < β < 1), существует семейство устойчивых предельных циклов различной формы. В остальных случаях происходит разрушение предельного цикла по двум сценариям: бифуркация Хопфа (предельный цикл-предельная точка) или (предельный циклапериодический процесс). Дальнейшее продолжение исследований может быть связано с построением спектра максимальных показателей Ляпунова с целью идентификации хаотических колебательных режимов для рассматриваемой эредитарной динамической системы (ЭДС).

Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Michael M. Bernitsas ◽  
João Paulo J. Matsuura
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Resonance Phenomenon ◽  
Nonlinear Phenomena ◽  
Mooring System ◽  
Stable Limit ◽  
Dynamic Interactions ◽  
Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drifts and mooring systems. We were able to reveal new phenomena, based on the design methodology developed at the University of Michigan for autonomous mooring systems, and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed, based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency, estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or nonperiodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes two to three orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

MULTI–INERT OSCILLATORY MECHANISM

2020 ◽  
Vol 2 ◽  
pp. 19-25
Author(s):  
I.P. POPOV
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Oscillation Frequency ◽  
Free Oscillation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Harmonic Oscillations ◽  
Oscillatory Systems ◽  
Oscillatory Mechanism ◽  
Mutual Conversion

A mechanical oscillatory system with homogeneous elements, namely, with n massive loads (multi– inert oscillator), is considered. The possibility of the appearance of free harmonic oscillations of loads in such a system is shown. Unlike the classical spring pendulum, the oscillations of which are due to the mutual conversion of the kinetic energy of the load into the potential energy of the spring, in a multi–inert oscillator, the oscillations are due to the mutual conversion of only the kinetic energies of the goods. In this case, the acceleration of some loads occurs due to the braking of others. A feature of the multi–inert oscillator is that its free oscillation frequency is not fixed and is determined mainly by the initial conditions. This feature can be very useful for technical applications, for example, for self–neutralization of mechanical reactive (inertial) power in oscillatory systems.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

Analysis of chemical kinetic systems over the entire parameter space - I. The Sal’nikov thermokinetic oscillator

1988 ◽  
Vol 416 (1851) ◽  
pp. 391-402 ◽  
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Chemical Kinetic ◽  
Phase Portraits ◽  
Stable Steady State ◽  
Stable Limit ◽  
Unstable Steady State ◽  
Undamped Oscillations

In this series of papers we re-examine, using recently developed techniques, some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit. Each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters. We find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found. Quite often these regions are small and it seems unlikely that they would be found via classical methods. In part I of the series we consider one of the thermally coupled kinetic oscillator models studied by Sal’nikov. He showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle. Our analysis has revealed two further regions in which the phase portraits contain, respectively, two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state. This latter region is extremely small, so much so that it could be reasonably neglected in any predictions made from the model.

A Test-Based Methodology for A Priori Selection of Gain/Phase Relationships in Proportional, Phase-Shifting Control of Combustion Instabilities

2000 ◽  
Author(s):  
Michael A. Vaudrey ◽  
William R. Saunders ◽  
Bryan Eisenhower
Keyword(s):  
Feedback Control ◽  
Frequency Response ◽  
Limit Cycle ◽  
Design Method ◽  
Phase Relationship ◽  
Operating Conditions ◽  
Phase Shifting ◽  
Nonlinear Phenomena ◽  
The Absolute ◽  
Instability Frequency

Feedback control system design, for general single-in-single-out (SISO) applications, requires accurate knowledge of the loop transfer function. Active combustion control design is usually implemented using such SISO architectures, but is quite challenging because the thermoacoustic response results from a relatively unknown, self-excited system and nonlinear processes that must be understood before learning the gain/phase relationship of the system precisely at the instability frequency. However, recent experiments have shown that it is possible to obtain accurate measurements of the relevant loop transfer (frequency response) functions at frequencies adjacent to the instability frequency. Using a simple tube combustor, operating with a premixed, gaseous, burner-stabilized flame, the loop frequency response measurements have been used to develop a methodology that leads to ‘test-based predictions’ of the absolute phase settings and ‘best’ gain settings for a proportional, phase-shifting controller commanding an acoustic actuator in the combustor. The contributions of this methodology are twofold. First, it means that a manual search for the required phase setting of the controller is no longer necessary. In fact, this technique allows the absolute value of controller phase to be determined without running the controller. To the authors’ knowledge, this has not been previously reported in the literature. In addition, the ‘best’ gain setting of the controller, based on this new design approach, can be defined as one that eliminates or reduces the limit cycle amplitude as much as possible within the constraint of avoiding generation of any controller-induced instabilities. (This refers to the generation of ‘new’ peaks in the controlled acoustic pressure spectrum.) It is shown that this tradeoff in limit cycle suppression and avoidance of controller-induced instabilities is a manifestation of the well-known tradeoff in the sensitivity/complementary sensitivity function for feedback control solutions. The focus of this article is limited to the presentation of the design method and does not discuss the detailed nonlinear phenomena that must be understood to determine the optimal gain/phase settings at the limit cycle frequency for a real (versus theoretical) combustor system. A companion paper describes how the proposed design method can be used to generate an AI controller that maintains stabilizing control for a range of changing operating conditions.

Response : Stable Limit Cycles in Prey-Predator Populations

Science ◽  
1973 ◽  
Vol 181 (4104) ◽  
pp. 1074-1074
Author(s):  
Robert M. May
Keyword(s):  
Limit Cycles ◽  
Stable Limit ◽  
Stable Limit Cycles
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