Limit cycle

Summary

In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912). Definition We consider a two-dimensional dynamical system of the form x'(t)=V(x(t)) where V : \R^2 \to \R^2 is a smooth function. A trajectory of this system is some smooth function x(t) with values in \mathbb{R}^2 which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some t_0>0

Official source

https://en.wikipedia.org/wiki/Limit_cycle

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Mean-field theory revives in self-oscillatory fields with non-local coupling

A simple mean-field idea is applicable to the pattern dynamics of large assemblies of limit-cycle oscillators with non-local coupling. This is demonstrated by developing a mathematical theory for the following two specific examples of pattern dynamics. Firstly we disscuss propagation of phase waves in noisy oscillatory media, with particular concern with the existence of a critical condition for persistent propagation of the waves throughout the medium, and also with the possibility of noise-induced turbulence. Secondly, we discuss the existence of an exotic class of patterns peciliar to non-local coupling called chimera where the system is composed of two distinct domains, one coherent and the other incoherent, separated from each other with sharp boundaries.

2006

Geometrical multiscale model of an idealized left ventricle with fluid-structure interaction effects coupled to a one-dimensional viscoelastic arterial network

Simone Deparis, Toni Mikael Lassila, Adelmo Cristiano Innocenza Malossi

A geometrical multiscale model for blood flow through an ideal left ventricle and the main arteries is presented. The blood flow in the three-dimensional idealized left ventricle is solved through a monolithic fluid-structure interaction solver. To account for the interaction between the heart and the circulatory system the heart flow is coupled through an ideal valve with a network of viscoelastic one-dimensional models representing the arterial network. The geometrical multiscale approach used in this work is based on the exchange of averaged/integrated quantities between the fluid problems. The peripheral circulation is modelled by zero-dimensional windkessel terminals. We demonstrate that the geometrical multiscale model is (i) highly modular in that component models can be easily replaced with higher-fidelity ones whenever the user has a specific interest in modelling a particular part of the system, (ii) passive in that it reaches a stable limit cycle of flow rate and pressure in a few heartbeat cycles when driven by a periodic force acting on the epicardium, and (iii) capable of operating at physiological regimes.

2011

We show that memory can be encoded in a model amorphous solid subjected to athermal oscillatory shear deformations, and in an analogous spin model with disordered interactions, sharing the feature of a deformable energy landscape. When these systems are subjected to oscillatory shear deformation, they retain memory of the deformation amplitude imposed in the training phase, when the amplitude is below a "localization" threshold. Remarkably, multiple persistent memories can be stored using such an athermal, noise-free, protocol. The possibility of such memory is shown to be linked to the presence of plastic deformations and associated limit cycles traversed by the system, which exhibit avalanche statistics also seen in related contexts.

Amer Physical Soc2014