scholarly journals Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

2021 ◽  
Vol 31 (9) ◽  
pp. 093124
Author(s):  
Shohei Takata ◽  
Yuzuru Kato ◽  
Hiroya Nakao
Keyword(s):  
Limit Cycle ◽  
Floquet Theory ◽  
Phase Amplitude ◽  
Periodic Inputs

scholarly journals Periodic Forcing of Inhibition-Stabilized Networks: Nonlinear Resonances and Phase-Amplitude Coupling

2015 ◽  
Vol 27 (12) ◽  
pp. 2477-2509 ◽  
Author(s):  
Romain Veltz ◽  
Terrence J. Sejnowski
Keyword(s):  
Pyramidal Neurons ◽  
Phase Coupling ◽  
Periodic Forcing ◽  
Circuit Element ◽  
Damped Oscillations ◽  
Inhibitory Population ◽  
Multiple Frequencies ◽  
Phase Amplitude ◽  
Periodic Inputs ◽  
Phase Phase

Inhibition-stabilized networks (ISNs) are neural architectures with strong positive feedback among pyramidal neurons balanced by strong negative feedback from inhibitory interneurons, a circuit element found in the hippocampus and the primary visual cortex. In their working regime, ISNs produce damped oscillations in the [Formula: see text]-range in response to inputs to the inhibitory population. In order to understand the properties of interconnected ISNs, we investigated periodic forcing of ISNs. We show that ISNs can be excited over a range of frequencies and derive properties of the resonance peaks. In particular, we studied the phase-locked solutions, the torus solutions, and the resonance peaks. Periodically forced ISNs respond with (possibly multistable) phase-locked activity, whereas networks with sustained intrinsic oscillations respond more dynamically to periodic inputs with tori. Hence, the dynamics are surprisingly rich, and phase effects alone do not adequately describe the network response. This strengthens the importance of phase-amplitude coupling as opposed to phase-phase coupling in providing multiple frequencies for multiplexing and routing information.

On the Stability of Forced Radial Oscillations of an Experimental Spring Pendulum

1993 ◽  
Author(s):  
Philip V. Bayly ◽  
Lawrence N. Virgin
Keyword(s):  
Fixed Point ◽  
Limit Cycle ◽  
Internal Resonance ◽  
Floquet Theory ◽  
Degree Of Freedom ◽  
Restoring Force ◽  
Linear Spring ◽  
Basic Ideas ◽  
The Stability ◽  
Elastic Pendulum

Abstract The elastic pendulum is a 2-degree-of-freedom, nonlinear device in which the pendulum bob may slide up and down the pendulum arm subject to the restoring force of a linear spring. In this study, radial motion (motion along the arm) is excited directly. Responses to this excitation include purely radial oscillations as well as swinging motion due to a 2:1 internal resonance. Changes in the behavior of the nonlinear spring pendulum occur when, under the control of a parameter, radial oscillations become unstable and are replaced by radial plus swinging motion. This bifurcation is explored analytically, numerically and experimentally, using the basic ideas of Floquet theory. Poincaré sampling is used to reduce the problem of describing the stability of a limit cycle to the easier task of defining the stability of the fixed point of a Poincaré map.

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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