For the systems considered in this paper we introduce the concept of a controlled switch. In its simplest form, this can be a gate or CMOS switch fed by the output of a simple analogue operational amplifier working as a comparator, which senses the size of a voltage or current, or other measurable quantity from a transducer. The actual circuits we had constructed to display intermittent behaviour contain no controlled switches explicitly embedded in their implementation, but we show that they behave isomorphically with systems which do contain such switches. That is, with some increase in complexity, they could be replaced by circuits containing only ideal amplifiers and controlled switches. We emphasise that the controlled switches are introduced in addition to the other linear electronic components; these systems are therefore not analogous to a network consisting solely of Boolean gates or elements. The controlled switches introduce piecewise-linear non-linear properties to the system.
The piecewise-linear transfer function system simulated previously [1] was implemented experimentally using the natural saturation properties of an operational amplifier, such that when the output reaches 12 volts, it saturates, and a further increase in input results in no further increase in output. In other words, the ``differential gain" has fallen to zero. Such an amplifier may be replaced by an ``ideal amplifier" with unrestricted input and output ranges, which is disconnected from the output line when the input reaches the appropriate threshold. The output line is instead connected via another controlled switch to a constant 12 volt source.
Alternatively, the various linear sections of the piecewise linear transfer function may be produced by amplifiers having gains A1,A2,A3 appropriate to the slopes under consideration, switched in and out of circuit (with appropriate offsets O1-O5) by controlled switches driven from the input. To demonstrate the equivalence of the saturating amplifier transfer function (Figure 4) circuit to the VSS explicitly, Figure 1 shows a ``controlled switch" version of the circuit; this version has not been implemented as it is an unnecessary complication from a constructional point of view.
Figure 1: The transfer function circuit drawn explicitly as a VSS
A controlled switch may be used either to alter the system state space trajectory in a discontinuous manner, (for example, by adding an offset to the output) or alternatively to alter the configuration or structure of the system.
If controlled switches are used to alter the circuit configuration or structure, we arrive at the idea of a variable structure system [8] or VSS. In a VSS the structure of the circuit or system is not time-invariant, but fluctuates according to the instantaneous values of the state vector. Since the state vector in turn varies according to the structure of the dynamical system, a loop is established in which the chaos can develop unpredictably; an example is given below showing trapping, in which the chaos stops altogether, and in the network traffic example the global behaviour of the state vector over a complex system consisting of many interlinked simple VSSs is shown to evolve with time.
In digital electronics a device much used to reconfigure a system of gates is the FPGA, or Field Programmable Gate Array. However, we are not concerned only with restructuring the topology of deterministic digital logic, but also with altering the structure of continuous analogue systems which may contain noise which can push the signal which controls the switch across an analogue threshold. Thus there is the possibility of stochastic development as well as deterministic chaos in some of these VSSs.
In the circuits and systems described here, feedback is applied around a structure containing embedded controlled switches or their equivalents, as well as containing other logic or circuit components. These equivalents are shown explicitly by means of circuit block diagrams. For the simple electronic example, the feedback takes the form of a two-stage sample-and-hold analogue shift register, which transfers the output value to the input on a clock pulse and then acquires the resulting new output value for transfer on the next clock pulse. This is an iteration circuit. Such a system is a combination of the discrete and the continuous. Iterations are discrete but the variable being iterated is continuous with its intrinsic noise.
Thus, adaptive behaviour may be engineered in systems containing controlled switches, and in variable structure systems generally. Such behaviour need not be cyclic, repeating exactly, but can be emergent and result in progressive modifications to the system. For those people happier with biological terminology, we have the potential to apply selection to a fluctuating system, resulting in evolution. Here, our use of the term ``evolution" implies no special adaptation to ``fitness for purpose" or other measure of utility, but merely to the fact that the dynamics change as time progresses, and such changes can sometimes happen non-reversibly.