ABSTRACT
This chapter covers some relevant methods used for proving boundedness of certain forms of discrete time and continuous time systems. In Sec. 7.2, we present the main method (Lyapunov function) used in proving the boundedness of certain forms of 3-D quadratic continuous-time systems. In Sec. 7.3, we present a method used in proving the bounded jerky dynamics. This method is based on a result about the boundedness of solutions of a certain type of third-order nonlinear differential equations with bounded delay. Another variant of this result is presented in Sec. 7.3.1 and it is used to prove the boundedness of general forms of jerky dynamics. In Sec. 7.3.2, some example of bounded jerky and minimal chaotic attractors are given. Sec. 7.4, deals with some forms of bounded hyperjerky dynamics. In Sec. 7.5, we give some proofs that certain forms of 4-D systems are bounded, i.e., the generalized 4-D hyperchaotic model containing Lorenz-Stenflo and Lorenz-Haken systems. In Sec. 7.5.1, an estimation of the bounds for the Lorenz-Haken system is given with a detailed proof. The same result for the Lorenz–Stenflo system is presented in Sec. 7.5.2. In Sec. 7.6, we present some results regarding the boundedness of certain forms of 2-D Hénon-like mappings with unknown bounded function. In this case, the values of parameters and initial conditions domains, for which the dynamics of this equation is bounded or unbounded, are rigorously derived. Some numerical examples are also given and discussed. In Sec. 7.7, some examples of fully bounded chaotic attractors for all bifurcation parameters are presented and discussed.
