Françoise J-P, Piquet C
Géométrie Différentielle, Systèmes Différentiels, Imagerie Géométrique et Biomédicale, Université de Paris VI, 175 Rue du Chevaleret, Paris 75013, France.
Acta Biotheor. 2005;53(4):381-92. doi: 10.1007/s10441-005-4892-1.
This article describes new aspects of hysteresis dynamics which have been uncovered through computer experiments. There are several motivations to be interested in fast-slow dynamics. For instance, many physiological or biological systems display different time scales. The bursting oscillations which can be observed in neurons, beta-cells of the pancreas and population dynamics are essentially studied via bifurcation theory and analysis of fast-slow systems (Keener and Sneyd, 1998; Rinzel, 1987). Hysteresis is a possible mechanism to generate bursting oscillations. A first part of this article presents the computer techniques (the dotted-phase portrait, the bifurcation of the fast dynamics and the wave form) we have used to represent several patterns specific to hysteresis dynamics. This framework yields a natural generalization to the notion of bursting oscillations where, for instance, the active phase is chaotic and alternates with a quiescent phase. In a second part of the article, we emphasize the evolution to chaos which is often associated with bursting oscillations on the specific example of the Hindmarsh-Rose system. This evolution to chaos has already been studied with classical tools of dynamical systems but we give here numerical evidence on hysteresis dynamics and on some aspects of the wave form. The analytical proofs will be given elsewhere.
本文描述了通过计算机实验揭示的滞后动力学的新方面。对快慢动力学感兴趣有几个动机。例如，许多生理或生物系统表现出不同的时间尺度。在神经元、胰腺β细胞和种群动态中观察到的爆发性振荡主要通过分岔理论和快慢系统分析进行研究（基纳和斯内德，1998年；林泽尔，1987年）。滞后是产生爆发性振荡的一种可能机制。本文的第一部分介绍了我们用来表示滞后动力学特定的几种模式的计算机技术（点相图、快动力学的分岔和波形）。这个框架自然地推广了爆发性振荡的概念，例如，活跃相是混沌的，并与静止相交替。在本文的第二部分，我们以欣德马什 - 罗斯系统为例，强调了通常与爆发性振荡相关的向混沌的演化。这种向混沌的演化已经用动力系统的经典工具进行了研究，但我们在此给出关于滞后动力学和波形某些方面的数值证据。解析证明将在其他地方给出。
