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Register a new userPhase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force
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Added by
Viktor Novi\\v{c}enko
Publication date
2015
fields
Physics
and research's language is
English
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The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.
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We analyze limit cycle oscillators under perturbation constructed as a product of two signals, namely, an envelope with a period close to natural period of an oscillator and a high-frequency carrier signal. A theory for obtaining an envelope waveform that achieves the maximal frequency interval of entrained oscillators is presented. The optimization problem for fixed power and maximal allowed amplitude is solved by employing the phase reduction method and the Pontryagins maximum principle. We have shown that the optimal envelope waveform is a bang-bang-type solution. Also, we have found inversion symmetry that relates two signals with different powers, but the same interval of entrained frequencies. The theoretical results are confirmed numerically on FitzHugh-Nagumo oscillators.
The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We introduce and describe periodic coherent structures of the CGLE, called Modulated Amplitude Waves (MAWs). MAWs of various period P occur naturally in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period P, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures occur which evolve toward defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.
We present an analytical calculation of the response of a driven Duffing oscillator to low-frequency fluctuations in the resonance frequency and damping. We find that fluctuations in these parameters manifest themselves distinctively, allowing them to be distinguished. In the strongly nonlinear regime, amplitude and phase noise due to resonance frequency fluctuations and amplitude noise due to damping fluctuations are strongly attenuated, while the transduction of damping fluctuations into phase noise remains of order $1$. We show that this can be seen by comparing the relative strengths of the amplitude fluctuations to the fluctuations in the quadrature components, and suggest that this provides a means to determine the source of low-frequency noise in a driven Duffing oscillator.
We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node bifurcations (SN). These MAWs can be characterized by their average phase gradient $ u$ and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary $ u$ and P. MAWs are found to be stable for large enough $ u$ and intermediate values of P. For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient $ u$ > 0 (``wound-up phase chaos) is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.
The paper presents bifurcation behavior of a single phase induction motor. This paper also attempts to discuss the bifurcation behavior of the system based on the evolution of different state variables. The bifurcation diagrams drawn looking at different state variables are different in terms of periodicity and route to chaos. The knowledge of the dynamics of the system obtained from bifurcation diagrams give useful guidelines to control the operation of the induction motor depending on the need of an application for better performance.
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