An official website of the United States government
Phase reduction is a well-established method to study weakly driven and weakly perturbed oscillators. Traditional phase-reduction approaches characterize the perturbed system dynamics solely in terms of the timing of the oscillations. In the case of large perturbations, the introduction of amplitude (isostable) coordinates improves the accuracy of the phase description by providing a sense of distance from the underlying limit cycle. Importantly, phase-amplitude coordinates allow for the study of both the timing and shape of system oscillations. A parallel tool is the infinitesimal shape response curve (iSRC), a variational method that characterizes the shape change of a limit-cycle oscillator under sustained perturbation. Despite the importance of oscillation amplitude in a wide range of physical systems, systematic studies on the shape change of oscillations remain scarce. Both phase-amplitude coordinates and the iSRC represent methods to analyze oscillation shape change, yet a relationship between the two has not been previously explored. In this work, we establish the iSRC and phase-amplitude coordinates as complementary tools to study oscillation amplitude. We extend existing iSRC theory and specify conditions under which a general class of systems can be analyzed by the joint iSRC phase-amplitude approach. We show that the iSRC takes on a dramatically simple form in phase-amplitude coordinates, and directly relate the phase and isostable response curves to the iSRC. We apply our theory to weakly perturbed single oscillators, and to study the synchronization and entrainment of coupled oscillators.
more »
« less
A universal description of stochastic oscillators
Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function Q 1 * ( x ) that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function Q 1 * ( x ) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ 1 = μ 1 + iω 1 . The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω 1 and half-width μ 1 ; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω 1 ; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.
more »
« less
- Award ID(s):
- 2052109
- PAR ID:
- 10433835
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 120
- Issue:
- 29
- ISSN:
- 0027-8424
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract For a Brownian directed polymer in a Gaussian random environment, with q ( t , ⋅) denoting the quenched endpoint density and Q n ( t , x 1 , … , x n ) = E [ q ( t , x 1 ) … q ( t , x n ) ] , we derive a hierarchical PDE system satisfied by { Q n } n ⩾ 1 . We present two applications of the system: (i) we compute the generator of { μ t ( d x ) = q ( t , x ) d x } t ⩾ 0 for some special functionals, where { μ t ( d x ) } t ⩾ 0 is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O ( t 2/3 ) scaling.more » « less
-
Symmetry-breaking in coupled, identical, fast–slow systems produces a rich, dramatic variety of dynamical behavior—such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast–slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel–Epstein model of chemical oscillators.more » « less
-
The dynamics of mechanical systems, such as turbomachinery with multiple blades, are often modeled by arrays of periodically driven coupled nonlinear oscillators. It is known that such systems may have multiple stable vibrational modes, and transitions between them may occur under the influence of random factors. A methodology for finding most probable escape paths and estimating the transition rates in the small noise limit is developed and applied to a collection of arrays of coupled monostable oscillators with cubic nonlinearity, small damping, and harmonic external forcing. The methodology is built upon the action plot method [Beri et al., Phys. Rev. E 72, 036131 (2005)] and relies on the large deviation theory, the optimal control theory, and the Floquet theory. The action plot method is promoted to non-autonomous high-dimensional systems, and a method for solving the arising optimization problem with a discontinuous objective function restricted to a certain manifold is proposed. The most probable escape paths between stable vibrational modes in arrays of up to five oscillators and the corresponding quasipotential barriers are computed and visualized. The dependence of the quasipotential barrier on the parameters of the system is discussed.more » « less
-
Abstract The striking similarity between biological locomotion gaits and the evolution of phase patterns in coupled oscillatory network can be traced to the role of central pattern generator located in the spinal cord. Bio-inspired robotics aim at harnessing this control approach for generation of rhythmic patterns for synchronized limb movement. Here, we utilize the phenomenon of synchronization and emergent spatiotemporal pattern from the interaction among coupled oscillators to generate a range of locomotion gait patterns. We experimentally demonstrate a central pattern generator network using capacitively coupled Vanadium Dioxide nano-oscillators. The coupled oscillators exhibit stable limit-cycle oscillations and tunable natural frequencies for real-time programmability of phase-pattern. The ultra-compact 1 Transistor-1 Resistor implementation of oscillator and bidirectional capacitive coupling allow small footprint area and low operating power. Compared to biomimetic CMOS based neuron and synapse models, our design simplifies on-chip implementation and real-time tunability by reducing the number of control parameters.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1073/pnas.2303222120
