scholarly journals Amplitude and Frequency Control: Stability of Limit Cycles in Phase-Shift and Twin-T Oscillators

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
J. P. Dada ◽  
J. C. Chedjou ◽  
S. Domngang
Keyword(s):  
Phase Shift ◽  
Limit Cycles ◽  
Frequency Control ◽  
Equilibrium Points ◽  
Control Voltage ◽  
Stable Limit ◽  
Control Stability ◽  
Oscillatory Network ◽  
Total Impedance ◽  
Oscillator Output

We show a technique for external direct current (DC) control of the amplitudes of limit cycles both in the Phase-shift and Twin-T oscillators. We have found that amplitudes of the oscillator output voltage depend on the DC control voltage. By varying the total impedance of each oscillator oscillatory network, frequencies of oscillations are controlled using potentiometers. The main advantage of the proposed circuits is that both the amplitude and frequency of the waveforms generated can be independently controlled. Analytical, numerical, and experimental methods are used to determine the boundaries of the states of the oscillators. Equilibrium points, stable limit cycles, and divergent states are found. Analytical results are compared with the numerical and experimental solutions, and a good agreement is obtained.

Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5271-5293
Author(s):  
A.K. Pal ◽  
P. Dolai ◽  
G.P. Samanta
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Toxic Substances ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Competitive System ◽  
Interval Numbers ◽  
Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

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.

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.

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

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

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