A Colpitts-Like Chaotic System With Constant Lyapunov Exponents And Its Synchronization

Chaos, a well known complex nonlinear behavior, has applications in various fields such as physics, engineering, information, biology and chemistry. The research about applications of chaos has attracted increasing attention and has rapidly become one of the frontier directions. Because chaotic systems possess certain features, such as high randomicity, board spectra of its Fourier transform, and hyper sensitivity to initial conditions, the application of chaos can be found in secure communications, signal processing, image processing etc. Chaotic system construction and implementation as well as chaos synchronization have become the key processes in applying chaos.A Colpitts-like Chaotic System with Constant Lyapunov Exponents (CCSCLE) is proposed in this dissertation based on the chaos phenomenon found in Colpitts circuit which possesses special structure. The chaos model is constructed; its dynamical property is analyzed. Based on the equation of CCSCLE, and so a chaotic family with constant Lyapunov exponent spectrum is proposed. The synchronization of CCSCLE is studied by using various synchronization methods and the particularity is then expounded. Theory proving and numerical simulation on the dynamical behaviour and synchronization particularity of the chaotic system are achieved, and after experimental simulation, corresponding implemental chaotic circuits and synchronization circuit is designed. The main innovations and contributions are listed as follows:1. A Colpitts-like chaotic system with constant Lyapunov exponent spectrum is proposed, its special signal variation law is revealed and it is implemented in physics after the experimental circuit is designed.Based on the nonlinear function of chaos system, a novel chaotic attractor is found by using simple absolute term to substitute for exponent term in normalized chaotic Colpitts system. By differentiating the parameters of constant term and coefficient term, it is verified that the Lyapunov exponent spectrum remains constant when the amplitude of the system variables are modulated by the constant term and the phase of a certain variable is inversed by the coefficient term. The system is implemented in physics after the analog circuit is designed.2. Followed with the extension of CCSCLE, a chaotic family with invariable Lyapunov exponent spectrum is proposed and which is also implemented successfully by an analog switchable circuit. By introducing linear and constant terms in system equations of CCSCLE, and based on that, after further adjusting the absolute term and introducing a new absolute term in the dynamical equation, the extension system of CCSCLE is obtained. A class of subsystems with the same properties but different phase trajectories is achieved through different combinations of linear terms, and another novel chaotic attractor is found, all state variables can also be modified linearly by constant term while the Lyapunov exponent spectrum remains stable. An analog switchable circuit is proposed, by the choosing of a jump line or switch and all kinds of novel chaotic attractors generated in various systems are shown on an oscillograph.3. Synchronization study on CCSCLE is conducted, the adjustable flexibility on state variables of synchronization system is pointed out, the synchronization system is constructed and based on modularization theory a synchronization circuit is designed.Synchronization study using feedback control, generalized synchronization and generalized projective synchronization methods for CCSCLE are conducted. The proper controller gain region to realize synchronization is obtained, the proper driving and response systems are constructed and the appropriate nonlinear feedback controllers are designed, so then the synchronization systems are constructed, the synchronization circuits are also designed. There exists adjustable flexibility on amplitude and phase of state variables owing to the special parameters, i.e. constant term and coefficient of CCSCLE. Two generalized projective synchronization methods of CCSCLE with the same and different structure are studied, inner and outer amplitude adjusters are obtained, i.e., any scale signal of the driving system state variable can be obtained by adjusting scaling factor, and when the global linear amplitude adjuster is changed, the state variables of the synchronization system and the response system are modulated to increase and decrease in their own evolvement region synchronously in linearity.