Abrupt changes in the behaviour of hybrid systems in discontinuity induced multiple attractors bifurcations

我们要解决的基本问题是：混合动力系统的研究如何帮助我们理解人类的平衡？要回答这个问题，我们必须知道什么是混合系统，以及如何在人类平衡的背景下使用它们。简单地说，以连续和离散进化相互作用为特征的系统被称为混合系统。举个日常生活中的例子：考虑一架飞机，它在飞行过程中的位置随着时间的推移不断演变。飞机是由微处理器控制的，这些微处理器在离散的输入上运行，因此整个系统是一个混合系统。工程学领域以外的另一个例子是生物细胞的生长和分裂。增长是一个连续的时间过程，但分裂是一个离散的过渡。事实上，几乎不可能想到任何复杂的系统，也就是由许多相互作用的子系统建立的系统，不具有这里给出的启发式定义意义上的混合特征。用于动力系统研究的一个非常有价值且非常成功的工具是分叉分析。简而言之，分叉给出了关于稳态(平衡点或周期运动)稳定性边界的信息，作为可能变化的系统参数的函数；这些参数可以是温度、压力或其他物理量。结果表明，由于开关的存在，混杂系统可能会出现完全由这些开关引起的分叉(失稳)。由开关的存在引起的分叉(转变)的一个重要特征是它们可能导致系统行为的突变。例如，从稳定的振荡运动到混沌运动的突然转变。它还表明，在混合系统中，由于开关的存在，许多稳定状态，例如振荡状态，可能起源于一个单一状态。任何系统都是在不断变化的环境条件下运行的，如果有可能从单个系统产生不同的稳定运动，那么在某些参数值下，系统很容易通过在其稳定状态之间跳跃来改变其演化。如果这些稳定状态中的一个是不希望的，例如从系统性能的角度来看，这可能会导致系统的灾难性故障。显然，理解这种类型的行为，即产生多个吸引子，对系统设计者来说至关重要。混合系统和人类平衡之间的联系是什么？对混合系统的研究将如何被用来理解人类平衡？近年来，数学模型被用来洞察人类在安静站立时保持平衡的问题。通常假设，作为第一近似值，人体可以被建模为单连杆倒立摆，其中不同的控制反馈律对姿势变化的神经肌肉反应进行建模，从而确保直立姿势。最近，有人指出，控制直立姿势的是像肌肉运动一样的冲动行为，因此，开关类行为似乎在平衡控制中起着关键作用。通过了解具有开关的系统的动力学，以及它们行为中可能出现的故障的路径，我们可以使用这些知识，例如，理解人类摔倒背后的机制。

The fundamental question we address is: How can the study of hybrid dynamical systems inform our understanding of human balance? To answer this question we have to know what are hybrid systems, and how they can be used in the context of human balance. In simple terms, systems characterised by an interaction of continuous and discrete evolution are termed as hybrid systems. To give an example from everyday life: consider an aircraft whose position during the flight evolves continuously in time. The aircraft is controlled by microprocessors which operate on discrete inputs, and hence the whole system is a hybrid system. Another example, from outside of the field of engineering, is growth and division of biological cells. Growth is a continuous time process but division is a discrete transition. Actually, it is virtually impossible to think of any complex system, that is, a system built from a number of interacting subsystems, that does not have a hybrid characteristic in the sense of a heuristic definition given here. An invaluable and a highly successful tool used for dynamical systems investigation is bifurcation analysis. In simple terms, bifurcations give information on stability boundaries of steady states (equilibrium points or periodic motions) as functions of system parameters that may vary; these parameters could be a temperature, pressure or other physical quantity. It turns out that hybrid systems, due to the presence of switches, may exhibit bifurcations (loss of stability) which are solely caused by these switches. An important feature of bifurcations (transitions) which are induced by the presence of switches is that they may lead to an abrupt change of system's behaviour. For instance, an abrupt transition from a stable oscillatory motion to a chaotic motion. It has also been shown that in hybrid systems many stable states, say oscillatory states, may originate from a single one, again due to the presence of switches. Any system operates in continuously changing environmental conditions, and if there is a possibility of different stable motions originating from a single one there are certain parameter values at which the system is highly susceptible to changing its evolution by jumping between its stable states. And if one of these stable states is undesirable, for instance from the point of view of system's performance, this may lead to a catastrophic failure of a system. Clearly, understanding this type of behaviour, that is birth of multiple attractors, is of critical importance for system designers. What is the link between hybrid systems and human balance and how the research on hybrid systems will be used to understand human balance? In recent years, mathematical models have been used to gain insight into the problem of maintaining balance in humans during quiet standing. It is usually assumed that, as a first approximation, a human body can be modelled as a single link inverted pendulum where different control feedback laws model neuromuscular response to change in posture which then ensures the upright stance. Recently, it has been pointed out that it is impulsive like muscle movements that control upright stance, and hence it is switch like behaviour that seems to play a crucial role in balance control. By understanding the dynamics of systems with switches, routes to possible failures in their behaviour, we may then use this knowledge, for instance, to understand the mechanisms behind falling in humans.