**1. Introduction**

Since Lorenz discovered chaos in a third-order ordinary differential equations, a new field of science has been launched [1]. The fact that simple equations can exhibit incredible complex behavior continues enthrall engineers to apply chaotic systems to cryptosystem, secure communication, spread spectrum communication, etc. [2].

There is no doubt that nonlinear term is very important to design chaotic systems, which has peculiar complex properties such as ergodicity, highly initial value sensitivity, non-periodicity and long-term unpredictability. According to the literature, the nonlinearities can be piecewise nonlinear function [3], trigonometric function [4], absolute value function [5], or power function [6]. With different nonlinearities, the chaotic system can have various strange attractors as single-scroll [7], double-scroll [8], multi-scroll [9], etc. The majority of such chaotic systems are known for many years, and some chaotic systems with hidden attractors are derived from them [10–12].

In recent years, chaotic systems with hyperbolic sine nonlinearities have gained the interest of many researchers. With two back-to-back diodes to approximate the hyperbolic sine nonlinearity, these chaotic systems can achieve simplicity of the electrical circuit without any multiplier or sub-circuits. Compared to single-scroll chaotic systems, the chaotic system with hyperbolic sine nonlinearity has richer dynamic behavior because it is symmetrical and can exhibit symmetry breaking, and offers the possibility that attractors will split or merge as some bifurcation parameter is changed [13].

In this chapter, we will systematically summarize the design method, the dynamic behavior and typical engineering applications of chaotic systems with hyperbolic sine nonlinearity. The genesis and general method of generating nthorder (n > 3) chaotic systems with hyperbolic sine nonlinearity are introduced in Section II. Some derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity is discussed in Section III. The application such as random number generator algorithm, spread spectrum communication and image encryption schemes are introduced in Section IV. Conclusions are finally drawn in Section V.
