The expectation of a complete understanding of the universe as a mechanical clock was to be tempered by the realisation that while it was possible to write the equations it was not always possible to solve them. Joseph-Louis Lagrange a mathematician in Turin, in the late 1700, raised the difficulties in solving the equations for the motion of a small planet under the influence of two large celestial bodies. This is the so called ‘three body problem’. In 1887 King Oscar II of Sweden offered a prize of 2,500 crowns for an answer to the fundamental question in astronomy linked to the three body problem: “Is the Solar system stable?”. The French mathematician Henri Poincaré (1854-1911) participated and won the competition with a memoir of 270 pages published in 1890: On the problem of the three bodies and the equation dynamics. He concluded that it was not possible to predict with absolute certainty the future of such a system. Although he did not solve the problem, while attempting it he invented the ‘analysis situs‘ or topological analysis of the dynamics of a system. This method enables to portray graphically, the overall behaviour of the system without having to solve analytically the differential equations.
A system can thus be portrayed graphically in what is called a state space or phase space, a mathematically constructed conceptual space where each dimension corresponds to one variable of the system1. Every point in state in this phase space, called a phase plot, a phase diagram or a phase portrait, represents a full representation of the system in one given state. No matter how a complicated the system under consideration may be, a single point in phase space encapsulates the entire system’s state of motions at a particular instant. The evolution of the system manifests itself as the tracing out of a path, ‘orbit’ or ‘trajectory’ in ‘state space’. A particular set of occurrences can be connected in time, to give a graphic representation of the evolution of a physical system.
As Poincaré was interested in orbits of planets, he focused on those trajectories that repeat themselves at regular intervals. Such periodic orbits, after wandering about the phase-space for a certain amount of time, always precisely returned to their starting point before setting off again on the same path.
To find these loops in phase-space, Poincaré chose to examine not the entire space and the families of geometrical surfaces defining all possible trajectories, but cross sections of the trajectories through that surface2. The result is a trajectory or orbit that traverses an essentially imaginary space quite different from the 3D space through which planets wander.
In effect, he turned the differential equations describing the motions as a continuous flow, into a set of discrete steps specifying what happens to the motion between one given time and the next regular intervals. Hence the name difference equations which are discrete versions of differential equations3.
It is like shining a stroboscopic light on a pendulum to show its position only at discrete intervals corresponding to each flash of light. This enables the transformation of differential equations which portray continuous changes into discrete values.
Mathematically, this is a form of iterative mapping of continuous events into discrete steps. In Poincaré’s words “instead of considering in its entirety the progressive development of a phenomenon, one simply seeks to relate one instant to an instant immediately preceding; one supposes that the actual state of the world depends only on its most recent past. Thanks to this postulate, rather than studying directly a phenomenon’s whole succession, one can limit oneself to writing out its difference equation“.
The state of the system at a particular time is therefore calculated from the state immediately before that time and is an iterative method with a recursive process. The equation that describes the trajectory is in effect a series of equations that relate the parameters at an immediately later time (t+1) to the state of the system at the previous time (t). Non-linear systems with non-linear interacting forces can be graphically modelled following Poincaré’s iterative method to represent dynamic progression of phenomena in a simple way, where the trajectory in time of the system is taken in a step-wise fashion. With this he could depict the tendency of dynamic states towards stability or towards instability.
A regular periodic event generates an orbit cycle which crosses the section of the plot always in the same spot. If every cycle is slightly different, then the points crossing the section will be scattered and depending on their distribution, the periodic or aperiodic nature of the events can be estimated. Solving the equations is equivalent to constructing curves of the directional arrow at every point. This geometrical method applied to the entire system describes its dynamics in a powerful way by turning numbers into a kind of contoured road map of all allowed possibilities.
With interactions of 2 bodies, the dynamical states are determined and predictable as all the future states are delineated by the trajectory of the system. With 3 or more interacting bodies a system becomes ‘complex’. The orbit of a small planet under the influence of two large ones is a good example of the difficulty that we encounter in predicting the dynamics of a ‘chaotic system‘. After a few orbits, a small planet appears to take a different path, every time giving an impression of unpredictability. And yet the system is mathematically completely determined. Poincaré realised this and concluded that there was room for unpredictability in deterministic systems. He had to reach this conclusion for he established that, although the equations representing three gravitational interacting bodies yield a well-defined relationship between time and position, no all-purpose, computational short cut – no magic formula – exists for making accurate predictions of positions far into the future.
Poincaré paved the way for a new kind of model that indicates the range of possibilities the future holds in store but doesn’t predict specifically which one will occur. Poincaré first caught a glimpse of what we call now dynamic chaos. In his revised paper on the three-body system, he realised that his geometrical topological portraits revealed not stability but a bewildering dynamical domain of wondrous complexity.4
Non-linear equations can generate complex phenomena even from relatively simple processes. Conversely, a system that appears to be tremendously complex may be the result of few identifiable variables interacting in a non-linear fashion5.
Models of non-linear dynamics
The physics of a pendulum in a vacuum with no air friction is a good simple model of a dynamic non-linear mechanical system6. An idealised pendulum can be in one of three stable conditions: it can be at rest; it may oscillate back and forth; or may rotate clockwise or anticlockwise. With the use a phase space diagram in which velocity is plotted versus position, the dynamics of the pendulum can be visualised easily by applying ‘topological analysis’, a geometrical method to portray dynamic events as developed by Poincaré. The dynamics of the pendulum portrayed in a phase space show ‘orbits‘ of the ‘trajectory‘ of the system. Stable orbits are called ‘attractors‘.
(from Wikipedia: https://en.wikipedia.org/wiki/Pendulum).
When the pendulum is at its highest position, it is at its most unstable. A gentle push will determine the rotation in one or the other direction. Depending on its initial position (its ‘initial condition’), it will rotate one way or the other. In this situation, it is said to have ‘high sensitivity to the initial conditions, ie, similar changes in initial conditions do not result in proportional changes in the resultant states. Because of small differences in the initial conditions, the orbits (attractors) in the phase space of a real pendulum are not regular. Thus, the system is said to follow non-linear dynamics.
Regular or periodic orbits in the phase space can shift to an unstable aperiodic orbit. When the orbits become irregular and change in an unpredictable way, they are referred to as ‘strange attractors’. Conversely, in ‘limit cycles‘, systems may return to stable modes7.
Plotting the solutions geometrically generates what is known as ‘bifurcation diagram or bifurcation map’, a special case of a logistic map, where periods of order (limit cycle attractors) are interspersed with chaotic states of ‘aperiodic’ cycles generating strange attractors8. As the number of oscillators in a system increases, the system can display periods of chaos interspersed between periods of ordered behaviour, ie, periods of strange attractors interspersed with limit cycles. When graphed as a logistic map, the evolution of this type of chaotic behaviour from relatively simple non-linear equations can be visualised. Such chaotic behaviour has much in common with the generation of fractal geometric dimensions.
The analysis of the properties of non-linear systems has demonstrated that complex systems often show behaviour that has series of periodically repeated ordered states interspersed with periods of irregularities with aperiodic time trajectory. This is modern Chaos Theory, defined as the development of unforeseeable (unpredictable) behaviour within a deterministic system.
Perhaps the world even in its enormous complexity may not as mysterious as feared. But it is necessary to accept that, precisely because of the non-linear nature of most physical phenomena, the unpredictable can arise from systems that follow deterministic rules9.
Graphical ways of showing the dynamics of systems can be applied to any regular or irregular cyclic activity. We will see further down their importance in the study of biological systems including the nervous system.
- https://en.wikipedia.org/wiki/Phase_space ↩︎
- https://en.wikipedia.org/wiki/Poincaré_map ↩︎
- https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients ↩︎
- For more recent analysis of the three-body problem, see C Marchal (1990): The three-body problem. Elsevier Amsterdam, Oxford, New York, Tokyo. Also: http://www.scholarpedia.org/article/Three_body_problem. ↩︎
- For example, Robert M May (1976) Simple mathematical models with very complicated dynamics. Nature 261, 459-467. Click here to download PDF. ↩︎
- For a complete – and complex – analysis of the motion of a pendulum, see https://en.wikipedia.org/wiki/Pendulum_(mechanics) ↩︎
- For more on attractors in biological systems, see Arthur T Winfree (1980). The Geometry of Biological Time. Springer-Verlag. ↩︎
- See Robert May, Footnote 5. ↩︎
- For more on Chaos Theory, see James Gleick (1987): Chaos: Making a New Science Viking Books. See also work by René Thom and his Catastrophe Theory. ↩︎