Abstract
From the viewpoint of engineering applications, time delay is useful for constructing a chaotic signal generator, which is the major part of diverse potential applications. Although different mathematical models of time-delay systems have been known, few models can exhibit chaotic behaviors. Motivated by attractive features and potential applications of time-delay models, a new chaotic system with a single scalar time delay and a nonlinearity described by a saturation function is proposed in this chapter. Nonlinear behavior of the system is discovered through bifurcation diagrams and the maximum Lyapunov exponent with the variance of system parameters. Interestingly, the system shows double-scroll chaotic attractors for some suitable chosen system parameters. In order to confirm the correction and feasibility of the theoretical model, the system is also implemented with analog electronic circuit. Finally, a practical application of such circuit is discussed at the end of this chapter.
Top1. Introduction
After the discovery of the first classical chaotic attractor in 1963 (Lorenz, 1963), chaos theory has received a great deal of attentions (Anishchenko, 1993; Grebogi, 1997; Davies, 2004; Banerjee, 2011). Although the fact that there is not an universal definition of chaos, three remarkable characteristics of a chaotic system are: dynamical instability, topological mixing and dense periodic orbits (Hasselblatt, 2003). Dynamical instability is known as the “butterfly effect”, which means that a small change in initial conditions of system can create significant differences. In other word, this vital characteristic makes the system highly sensitive to initial conditions (Lorenz, 1963; Strogatz, 1994). Topologically mixing is refers as stretching and folding of the phase space, which means that the chaotic trajectory at the phase space will evolve in time so that each given area of this trajectory will eventually cover part of any particular region. Dense periodic orbits means that the trajectory can come arbitrarily close every possible asymptotic state. Hence, the future behaviors of a chaotic system seem to be arbitrary and unpredictable. In fact, chaos appears naturally in weather and climate, biology, sociology, and physics etc. (Strogatz, 1994; Holland, 1998; Hilborn, 2000).
From the view point of engineering, chaotic systems with their complex characteristics have also been successfully utilized in diverse applications, ranging from secure chaotic communications (Cuomo, 1993), information encryption (Volos, 2013), economics (Medio, 2009), robotics (Volos, 2012), to liquid shakers (Zhang, 2007) and so on. One of the most interesting applications is chaotic cryptography (Kocarev, 2001), where chaotic systems provide alternative techniques capable of enhancing cryptographic features (Koc, 2009). The structural relationship between cryptography and chaos seem quite natural (Alvarez, 2006). It has been known that main properties of cryptography like confusion, diffusion, deterministic pseudo randomness and algorithm complexity connect directly with their analogous properties of chaos like ergodicity, sensitivity to initial conditions or system parameters, deterministic dynamics and structural complexity, respectively (Shannon, 1949; Alvarez, 2006; Kocarev, 2011). By now, a great majority of chaos-based cryptographic systems has been developed (Zhang, 2005; Tong, 2009; Wang, 2012; Seyedzadeh, 2012). Recent obtained results of such cryptographic systems can be found in (Kocarev, 2011).
Key Terms in this Chapter
Phase Portrait: A geometric presentation of the orbits of a dynamical nonlinear system in the phase plane.
Lyapunov Exponent: A quantity that characterizes the rate of separation of infinitesimally close trajectories. Chaotic motion is indicated by at least one positive Lyapunov exponent.
Attractor: In dynamical systems, it is a region of phase space, towards which neighboring trajectories asymptotically approach.
Piecewise Linear Function: A function which is composed of piecewise linear segments.
Chaos: A type of behavior of a deterministic nonlinear system, where tiny changes in initial conditions make huge changes over time.
Delay Differential Equation (DDE): A differential equation in which the evolution of dependent variables at a certain time depend on their values at previous times.
Bifurcation: A sudden change that accompanies the onset of chaos at a critical value of a varied control parameter.
Bifurcation Diagram: A diagram which displays the possible long-term changes of system behavior when the value of the control parameter is varied.
Multisim: An electronic schematic capture and simulation program of National Instruments (NI). Multisim provides an advanced, industry-standard SPICE simulation environment and is used widely in the world.
Limit Cycle: A closed trajectory in the phase space associated with a nonlinear system.