Laboratory activities play a very important role in understanding theoretical concepts as well as encouraging students to confront practical challenges. However, the implementation of a laboratory of dynamical systems and process control encounters several challenges, such as space limitations, financial support, difficulties of building a real system, etc. In this context, an electronic analogue can be extremely helpful in developing experimental activities destined to teach dynamical systems and process control, since the dynamical behaviour of a specific system can be easily reproduced through an analogous electronic system.1
Using robust, compact, versatile and inexpensive electronic circuits, it is possible to perform a qualitative study of the nonlinear equations of dynamical behaviour to obtain insights into the nature of their solutions.2 Since the dynamical behaviour of real systems can be expressed by differential equations, this electronic analogy can be easily used to implement educational platforms that reproduce mechanical, electrical, chemical, thermal, hydraulic, economic or biological systems. In this analogy, the original system variables and its derivatives are represented by electric signals, and the system behaviour can be directly observed and recorded on oscilloscopes and/or acquisition boards, avoiding the use of expensive and complex sensor devices.

Although an electronic prototype does not completely reproduce the real system, it can incorporate several aspects of practical implementations such as unpredictable noises, uncertainties, measurement problems and failures that are very difficult to reproduce by means of computational simulation. Another advantage of the electronic analogy is the possibility of on-line adjustments on system parameters.3 However, since electric signals are generally subject to severe limitations, to assure the correct operation and the integrity of the electronic devices, correct reproduction of a dynamical system from a direct electronic implementation can become relatively difficult.

This paper presents a methodology to design analogous electronic circuits that reproduce the dynamical behaviour of physical systems. The implementation of these electronic circuits can be used in practical experiments for the study of dynamical systems and other related subjects. It is always desirable to obtain an electronic version of any system which is as simple as possible to reduce difficulties with its implementation. If necessary, the original model must be modified to restrict the amplitudes and frequencies of electrical signals aiming to respect the limitations imposed by electronic devices and to assure the correct reproduction of a dynamical system. Case studies are presented, in which electronic circuits are designed and experimentally implemented to reproduce the dynamical behaviour of three natural chaotic systems: the forced Duffing system, the Lorenz system and the Rössler system.

Principles of electronic analogy

The first step towards understanding, analysis, design and control of an intricate real system is to obtain its mathematical model,1,4 which usually is expressed as a set of ordinary differential equations and can be considered as the mathematical analogue of a real system.5 The development of an adequate mathematical model requires a deep knowledge of the dynamical system under study in order to describe uncertainties and hypotheses related to its practical operation.3,5

Considering that real implementations of electric and electronic systems are easily realised, electronic circuits can be used to emulate the dynamical behaviour of other physical systems, where the original system variables are represented by electric signals. This concept is known as analogue simulation, and it has practically been forgotten in consequence of the development of fast digital computers and efficient software packages, such as MATLAB/Simulink or VISSIM. Nevertheless, since a physical realisation of several real systems can be very difficult and expensive, this technique remains as an interesting solution to the practical teaching of dynamical systems and process control, allowing an efficient experimental analysis.