Electronic Design Automation of Multi-scroll Chaos Generators

In the twenty century advances on the realization of chaotic systems by using electronic devices were introduced. This chapter includes an historical overview of electronics and the realization of chaotic oscillators. From the introduction of the computer, electronic design automation (EDA) born several decades ago, and its usefulness in the synthesis of chaotic systems is described herein. Furthermore, some key-points in performing EDA in dynamical systems at different levels of hierarchy are summarized.

The preceding chapter introduced the motivation for further improving in automated synthesis methodologies for chaotic systems. Increasing the efficiency of synthesis, however, can be achieved if one uses higher abstraction levels to capture the behaviour of the analog systems. Behavioural modelling is seen as a promising solution that, combined with the appropriate synthesis tools, can be used to design multi-scrolls chaotic systems. Hence, this chapter first describes the basic concepts about nonlinear dynamical systems and the existing approaches to design chaotic systems as well as the applications of this class of systems. Afterwards, several important aspects in behavioural modelling are introduced. The steps performed during different design stages are represented by identifying the abstraction and description levels. Finally, the approaches for EDA of nonlinear dynamical systems are reviewed, and the interaction of four basic operations is discussed to introduce the definition of a synthesis methodology for analog designs based on behavioural modelling.

The goal of this book consists in the improvement of the EDA process for chaotic systems by means of the definition, development, and demonstration of an automatic synthesis methodology, which helps to reduce the design complexity for this class of systems and allows exploiting the possibilities of chaotic systems in future applications (e.g. chaos-based information systems). For this, it is necessary to properly define the level of abstraction used in the proposed synthesis approach to take full advantage of behavioural modelling as reviewed in the previous chapter. Therefore, the chaotic systems are modelled herein at the highest level of abstraction (ESL), by applying state variables approach and PWL approximations. In particular, the following chaotic systems are considered: Chua’s circuit, Generalized Chua’s circuit and a chaotic oscillator implemented using saturated functions. Further, time-efficient evaluation of chaotic behaviour, via numerical simulation, is presented by using multistep algorithms, from which chaotic dynamical behaviours and basic dynamical properties can be explored. The simulation is executed by automatic determination and control of step-size based on calculating the minimum eigenvalue of the state variables. Regarding to circuit implementation it is necessary to consider the real limitations of the electronic devices, therefore, two procedures (excursion levels scaling and frequency scaling) to modify the behaviour of chaotic systems are also introduced. Furthermore, by applying state variables and PWL approximations one can also simulate 2D and 3D scrolls attractors as shown in this chapter.

The design of a nonlinear function with multi-segments is a basis for generating chaotic attractors with multidirectional orientation and with a large number of scrolls. However, it has been identified that it is quite difficult to synthesize nonlinear functions with multi-segments by using analog electronic circuits. Therefore, to cope with this problem, the electronic design automation (EDA) industry is developing design tools with a high degree of abstraction (behavioural modelling). Henceforth, the synthesis approach presented in this chapter depends on the behavioural modelling of the chaotic systems introduced in Chapter. 3. In particular, this approach is focused on designing multi-scrolls chaotic attractors with multidirectional orientation. In this manner, this chapter first presents a basic cell based on opamps to synthesize saturated nonlinear functions series. Further, a new synthesis approach is introduced to design the saturated functions in current and voltage modes by using a general scheme of connection for basic cells. Finally, at the end of this chapter, a Verilog-A model for considering no-ideal effects of the opamps is defined to execute SPICE simulations.

The behavioural modelling strategy based on state variables and PWL approximations allows us to represent several chaotic systems at the electronic system level (ESL) of abstraction. The design space exploration derived from numerical simulation approaches in the time domain can be adopted to estimate the performance values of the chaotic systems. That is only one part of a design strategy. Indeed, a question remains: which architecture and parameters values must be selected during the refinement operation of the behavioural models? Therefore, the synthesis methodology presented in this chapter completes the opamps based-synthesis of saturated functions introduced in Chap. 4. First, the steps of a new synthesis methodology proposed in this book for the design of chaotic systems based on behavioural modelling are summarized. The new methodology of circuit synthesis is performed by three hierarchical levels. Finally, numerical results are confirmed by H-SPICE simulations to show the usefulness of the proposed synthesis approach.

In this final chapter, the results of the previous chapters are recapitulated and some lines for future research are provided.

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