A simple multi-stable chaotic jerk system with two saddle-foci equilibrium points: analysis, synchronization via backstepping technique and MultiSim circuit design

This paper announces a new three-dimensional chaotic jerk system with two saddle-focus equilibrium points and gives a dynamic analysis of the properties of the jerk system such as Lyapunov exponents, phase portraits, Kaplan-Yorke dimension and equilibrium points. By modifying the Genesio-Tesi jerk dynamics (1992), a new jerk system is derived in this research study. The new jerk model is equipped with multistability and dissipative chaos with two saddle-foci equilibrium points. By invoking backstepping technique, new results for synchronizing chaos between the proposed jerk models are successfully yielded. MultiSim software is used to implement a circuit model for the new jerk dynamics. A good qualitative agreement has been shown between the MATLAB simulations of the theoretical chaotic jerk model and the MultiSIM result

Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

This paper deals with the problem of pseudo-state sliding mode control of fractional SISO nonlinear systems with model inaccuracies. Firstly, a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding mode control law is designed using the theory of Mittag-Leffler stability. Further, we utilize the control methodology to synchronize two fractional chaotic systems, which serves as an example of verifying the viability and effectiveness of the proposed technique

Nonlinear Unknown‐Input Observer‐Based Systems for Secure Communication

Secure communication employing chaotic systems is considered in this chapter. Chaos‐based communication uses chaotic systems as its backbone for information transmission and extraction, and is a field of active research and development and rapid advances in the literature. The theory and methods of synchronizing chaotic systems employing unknown input observers (UIOs) are investigated. New and novel results are presented. The techniques developed can be applied to a wide class of chaotic systems. Applications to the estimation of a variety of information signals, such as speech signal, electrocardiogram, stock price data hidden in chaotic system dynamics, are presented

Transcritical and zero-Hopf bifurcations in the Genesio system

Agraïments: The first author is supported by FAPESP Grant No. 2013/24541-0. Both authors are supported by CAPES Grant 88881.030454/2013-01 Program CSF-PVE.In this paper we study the existence of transcritical and zero--Hopf bifurcations of the third--order ordinary differential equation a b c x - x^2 = 0, called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first order differential system in \R^

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure