Coupled Oscillators and Synchronization

2019 ◽  
pp. 177-204
Author(s):  
David D. Nolte
Keyword(s):  
Phase Locking ◽  
Beat Frequency ◽  
Normal Modes ◽  
Coupled Systems ◽  
Torus Action ◽  
Slow Phase ◽  
Nonlinear Coupling ◽  
Frequency Locking ◽  
Frequency Entrainment ◽  
Central Paradigm

Coupled linear oscillators provide a central paradigm for the combined behavior of coupled systems and the emergence of normal modes. Nonlinear coupling of two autonomous oscillators provides an equally important paradigm for the emergence of collective behavior through synchronization. Simple asymmetric coupling of integrate and fire oscillators captures the essence of frequency locking. Quasiperiodicity on the torus (action-angle oscillators) with nonlinear coupling demonstrates phase locking, while the sine-circle map is a discrete map that displays multiple Arnold tongues at frequency-locking resonances. External synchronization of a phase oscillator is analyzed in terms of the “slow” phase difference, resulting in a beat frequency and frequency entrainment that are functions of the coupling strength.

Bispectral analysis of complex patterns of sympathetic nerve discharge

1996 ◽  
Vol 271 (5) ◽  
pp. R1173-R1185 ◽  
Author(s):  
G. L. Gebber ◽  
S. Zhong ◽  
Y. Paitel
Keyword(s):  
Sympathetic Nerve ◽  
Phase Locking ◽  
Nonlinear Coupling ◽  
Cardiac Sympathetic Nerve ◽  
Bispectral Analysis ◽  
Frequency Locking ◽  
Linear Superposition ◽  
Frequency Components ◽  
Complex Patterns ◽  
Nerve Discharge

Bispectral analysis was used to demonstrate quadratic nonlinear coupling (i.e., phase locking) of different frequency components in inferior cardiac sympathetic nerve discharge (SND) of urethan-anesthetized rats. The complex patterns of SND analyzed included mixtures of 1) the cardiac-related and 10-Hz rhythms, 2) the 10-Hz rhythm and irregular 2-to 6-Hz oscillations, and 3) the 10-Hz rhythm and a lower frequency non-cardiac-related rhythm near 4 Hz. In some cases, the bicoherence function (normalized bispectrum) showed no phase locking of these frequency components. Cases of nil bicoherence are equated with linear superposition of frequency components, which implies the existence of multiple and noninteractive central circuits. Increased complexity of SND was observed in other cases, as evidenced by significant phase locking of different frequency components with or without frequency locking. Frequency locking (higher frequency rhythm is a multiple of lower) was confirmed by constructing Lissajous orbital plots showing covariation of voltages in selectively filtered bands of SND. We equate frequency locking with nonlinear coupling of the central generators of different sympathetic nerve rhythms and phase locking without frequency locking possibly with nonlinearities arising at levels below noncoupled central rhythm generators.

GENERALIZED SYNCHRONIZATION, FREQUENCY-LOCKING AND PHASE-LOCKING OF COUPLED SINE CIRCLE MAPS

2001 ◽  
Vol 11 (08) ◽  
pp. 2245-2253
Author(s):  
WEN-XIN QIN
Keyword(s):  
Dynamical Systems ◽  
Invariant Manifold ◽  
Coupling Strength ◽  
Phase Locking ◽  
Coupled System ◽  
Generalized Synchronization ◽  
Frequency Locking ◽  
Local Systems ◽  
Circle Maps ◽  
The Individual

Applying invariant manifold theorem, we study the existence of generalized synchronization of a coupled system, with local systems being different sine circle maps. We specify a range of parameters for which the coupled system achieves generalized synchronization. We also investigate the relation between generalized synchronization, predictability and equivalence of dynamical systems. If the parameters are restricted in the specified range, then all the subsystems are topologically equivalent, and each subsystem is predictable from any other subsystem. Moreover, these subsystems are frequency locked even if the frequencies are greatly different in the absence of coupling. If the local systems are identical without coupling, then the widths of the phase-locked intervals of the coupled system are the same as those of the individual map and are independent of the coupling strength.

PHASE SYNCHRONIZATION OF LINEARLY AND NONLINEARLY COUPLED OSCILLATORS WITH INTERNAL RESONANCE

2009 ◽  
Vol 23 (30) ◽  
pp. 5715-5726
Author(s):  
YONG LIU
Keyword(s):  
Coupling Strength ◽  
Coupled Oscillators ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Phase Synchronization ◽  
Coupled Systems ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
Phase Dynamics ◽  
Diffuse Clouds

Phase synchronization between linearly and nonlinearly coupled systems with internal resonance is investigated in this paper. By introducing the conception of phase for a chaotic motion, it demonstrates that the detuning parameter σ between the two natural frequencies ω1and ω2affects phase dynamics, and with the increase in the linear coupling strength, the effect of phase synchronization between two sub-systems was enhanced, while increased firstly, and then decayed as nonlinear coupling strength increases. Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the Lyapunov exponents, which can also be explained by the diffuse clouds.

Resonance Assisted Synchronization of Coupled Oscillators: Frequency Locking without Phase Locking

2011 ◽  
Vol 107 (10) ◽  
Author(s):  
J. Thévenin ◽  
M. Romanelli ◽  
M. Vallet ◽  
M. Brunel ◽  
T. Erneux
Keyword(s):  
Coupled Oscillators ◽  
Phase Locking ◽  
Frequency Locking

Quasi-normal modes in coupled systems

1993 ◽  
Vol 10 (10) ◽  
pp. 2067-2076 ◽  
Author(s):  
T W Baumgarte ◽  
B G Schmidt
Keyword(s):  
Normal Modes ◽  
Coupled Systems

TRACKING PHASE SYNCHRONIZATION OF TWO COUPLED RÖSSLER OSCILLATORS WITH INTERNAL RESONANCE

2009 ◽  
Vol 23 (23) ◽  
pp. 4809-4816 ◽  
Author(s):  
YONG LIU
Keyword(s):  
Coupling Strength ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Phase Synchronization ◽  
Coupling Parameter ◽  
Primary Resonance ◽  
Coupled Systems ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
Phase Dynamics

Phase synchronization between linearly and nonlinearly coupled systems with internal resonance is investigated in this paper. By introducing the conception of phase for a chaotic motion, we tune the linear coupling parameter to obtain the two Rössler oscillators in a synchronized regime and analyze the effect of a nonlinear coupling on the phase synchronized state. It demonstrates that the detuning parameter σ between the two natural frequencies ω1and ω2affects phase dynamics, and with the increase of the nonlinear coupling strength, for the primary resonance, the effect of phase synchronization between two sub-systems was decayed, while increasing with frequency ratio 1:2. Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the nonlinear coupling strength.

Stabilization of gaze during circular locomotion in darkness. II. Contribution of velocity storage to compensatory eye and head nystagmus in the running monkey

1992 ◽  
Vol 67 (5) ◽  
pp. 1158-1170 ◽  
Author(s):  
D. Solomon ◽  
B. Cohen
Keyword(s):  
Time Constant ◽  
Beat Frequency ◽  
Maximum Velocity ◽  
The Body ◽  
Head Movements ◽  
Velocity Storage ◽  
Gait Velocity ◽  
Slow Phase ◽  
Eccentric Rotation ◽  
Body Velocity

1. Yaw eye in head (Eh) and head on body velocities (Hb) were measured in two monkeys that ran around the perimeter of a circular platform in darkness. The platform was stationary or could be counterrotated to reduce body velocity in space (Bs) while increasing gait velocity on the platform (Bp). The animals were also rotated while seated in a primate chair at eccentric locations to provide linear and angular accelerations similar to those experienced while running. 2. Both animals had head and eye nystagmus while running in darkness during which slow phase gaze velocity on the body (Gb) partially compensated for body velocity in space (Bs). The eyes, driven by the vestibuloocular reflex (VOR), supplied high-frequency characteristics, bringing Gb up to compensatory levels at the beginning and end of the slow phases. The head provided substantial gaze compensation during the slow phases, probably through the vestibulocollic reflex (VCR). Synchronous eye and head quick phases moved gaze in the direction of running. Head movements occurred consistently only when animals were running. This indicates that active body and limb motion may be essential for inducing the head-eye gaze synergy. 3. Gaze compensation was good when running in both directions in one animal and in one direction in the other animal. The animals had long VOR time constants in these directions. The VOR time constant was short to one side in one animal, and it had poor gaze compensation in this direction. Postlocomotory nystagmus was weaker after running in directions with a long VOR time constant than when the animals were passively rotated in darkness. We infer that velocity storage in the vestibular system had been activated to produce continuous Eh and Hb during running and to counteract postrotatory afterresponses. 4. Continuous compensatory gaze nystagmus was not produced by passive eccentric rotation with the head stabilized or free. This indicates that an aspect of active locomotion, most likely somatosensory feedback, was responsible for activating velocity storage. 5. Nystagmus was compared when an animal ran in darkness and in light. the beat frequency of eye and head nystagmus was lower, and the quick phases were larger in darkness. The duration of head and eye quick phases covaried. Eye quick phases were larger when animals ran in darkness than when they were passively rotated. The maximum velocity and duration of eye quick phases were the same in both conditions. 6. The platform was counterrotated under one monkey in darkness while it ran in the direction of its long vestibular time constant.(ABSTRACT TRUNCATED AT 400 WORDS)

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