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7.1.1 Classical Dynamical Systems Theory
Dynamical systems theory describes the “qualitative” behaviour of processes evolving according to some given law. As a mathematical discipline, it dates back to
Henri Poincaré [29] and Aleksandr Lyapunov [25]; a particular goal of this theory
was to describe properties—particularly stability properties—of celestial orbits.
The fundamental assumption underlying the modelling of celestial mechanics
is that each celestial body is a point particle, whose instantaneous acceleration is
determined from the instantaneous configuration of the positions of all the celestial
bodies, by Newton’s universal law of gravitation. This law stipulates the existence
of a universal non-time-varying constant of nature in terms of which the (similarly time-independent) mathematical relation between forces and distances is then
formulated. This model of celestial mechanics means, in particular, that the future
motion of celestial bodies starting from any given initial configuration of positions
and velocities is in no way dependent on the time at which the bodies start in this initial configuration. In other words, the Newtonian formulation of celestial mechanics
is, in a way, “timeless”: although increments of time are of physical meaning and
significance, absolute times are not. Or in the language of dynamical systems theory: the law specifying the time-evolution of the position-velocity configuration of
a system of interacting celestial bodies is an autonomous dynamical system.
As described in the preceding chapter [24], a dynamical system is called
autonomous if the future evolution that it specifies from any given current state has
no dependence on the current time. Autonomous dynamical systems serve as mathematical models for the time-evolution of the state of an isolated physical system. A
physical system is called isolated if it exchanges neither matter nor energy with its
environment. Since the pioneering work of Poincaré and Lyapunov, the mathematical theory of autonomous dynamical systems has become extremely well-developed
over the last century, and new and important advances continue to be made. This
theory includes mathematical formalisms for some of the most fundamental questions that one can ask about a physical system’s behaviour in time, such as stability or
chaos, as well as how this behaviour depends on the system’s parameters. Such mathematical formalisms include asymptotic stability, stability in the sense of Lyapunov
(formalising the physical concept of “neutral stability”), various mathematical definitions of chaos, and also asymptotic Lyapunov exponents (ALEs) which are often
used to quantify stability or chaos [24]. Two important features of these various
mathematical formalisms for the questions one can ask about a system’s qualitative
or quantitative dynamics are as follows.
• They give the same answer after a change of coordinates of the state space of the
system.
• For any finite time-interval [0, T ] (however large T may be), the answer is independent of the behaviour of the system on the time-interval [0, T ].
Properties of dynamical systems that fulfil the latter of these are referred to as longtime-asymptotic properties. These only make sense if the dynamical system is itself
J. M. I. Newman et al.
7.1.1 Classical Dynamical Systems Theory
Dynamical systems theory describes the “qualitative” behaviour of processes evolving according to some given law. As a mathematical discipline, it dates back to
Henri Poincaré [29] and Aleksandr Lyapunov [25]; a particular goal of this theory
was to describe properties—particularly stability properties—of celestial orbits.
The fundamental assumption underlying the modelling of celestial mechanics
is that each celestial body is a point particle, whose instantaneous acceleration is
determined from the instantaneous configuration of the positions of all the celestial
bodies, by Newton’s universal law of gravitation. This law stipulates the existence
of a universal non-time-varying constant of nature in terms of which the (similarly time-independent) mathematical relation between forces and distances is then
formulated. This model of celestial mechanics means, in particular, that the future
motion of celestial bodies starting from any given initial configuration of positions
and velocities is in no way dependent on the time at which the bodies start in this initial configuration. In other words, the Newtonian formulation of celestial mechanics
is, in a way, “timeless”: although increments of time are of physical meaning and
significance, absolute times are not. Or in the language of dynamical systems theory: the law specifying the time-evolution of the position-velocity configuration of
a system of interacting celestial bodies is an autonomous dynamical system.
As described in the preceding chapter [24], a dynamical system is called
autonomous if the future evolution that it specifies from any given current state has
no dependence on the current time. Autonomous dynamical systems serve as mathematical models for the time-evolution of the state of an isolated physical system. A
physical system is called isolated if it exchanges neither matter nor energy with its
environment. Since the pioneering work of Poincaré and Lyapunov, the mathematical theory of autonomous dynamical systems has become extremely well-developed
over the last century, and new and important advances continue to be made. This
theory includes mathematical formalisms for some of the most fundamental questions that one can ask about a physical system’s behaviour in time, such as stability or
chaos, as well as how this behaviour depends on the system’s parameters. Such mathematical formalisms include asymptotic stability, stability in the sense of Lyapunov
(formalising the physical concept of “neutral stability”), various mathematical definitions of chaos, and also asymptotic Lyapunov exponents (ALEs) which are often
used to quantify stability or chaos [24]. Two important features of these various
mathematical formalisms for the questions one can ask about a system’s qualitative
or quantitative dynamics are as follows.
• They give the same answer after a change of coordinates of the state space of the
system.
• For any finite time-interval [0, T ] (however large T may be), the answer is independent of the behaviour of the system on the time-interval [0, T ].
Properties of dynamical systems that fulfil the latter of these are referred to as longtime-asymptotic properties. These only make sense if the dynamical system is itself