**3.3 Torus-chaotic system with hyperbolic sine nonlinearity**

By introducing a nonlinear feedback controller to system Eq. (5), the following system is obtained:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 - \rho \sinh\left(\rho \mathbf{x}\_3\right) \\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_3 - \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_3 = \mathbf{x}\_4 \\ \dot{\mathbf{x}}\_4 = \mathbf{x}\_5 \\ \dot{\mathbf{x}}\_5 = -c\mathbf{x}\_5 - \rho \sinh\left(\rho \mathbf{x}\_4\right) - 5\mathbf{x}\_3 - 5\mathbf{x}\_2 - 0.1\mathbf{x}\_1 \end{cases} \tag{8}$$

When c = 1, the Lyapunov exponents are ð*λ*1, *λ*2, *λ*3, *λ*4, *λ*5Þ ¼ ð Þ 0*:*47, 0, 0, �1*:*10, �1*:*37 , which suggests Eq. (8) is exhibiting torus-chaos behavior [20].

When c = 1.55 and the initial conditions are set to be ð*x*1, *x*2, *x*3, *x*4, *x*5Þ ¼ ð Þ �0*:*1, �0*:*1, �0*:*1, �0*:*1, �0*:*1 and ð*x*1, *x*2, *x*3, *x*4, *x*5Þ ¼ ð Þ 0*:*1, 0*:*1, 0*:*1, 0*:*1, 0*:*1 , the system has two coexisting attractors as shown in **Figure 9**.

**Figure 10** shows the Lyapunov exponent spectrum, Kaplan–Yorke dimension spectrum and bifurcations of Eq. (8) as the coefficient c is varied over the range c ∈ [0.3, 2]. Those figures suggest there is an interesting route leading to chaos [21].


**Figure 8.** *Coexistence attractors of Eq. (7).*

**Figure 9.** *Coexistence attractors of Eq. (8).*

*Chaotic Systems with Hyperbolic Sine Nonlinearity DOI: http://dx.doi.org/10.5772/intechopen.94518*

#### **Figure 10.**

*LEs spectrum, Kaplan–Yorke dimension spectrum and bifurcations of Eq. (8) as the coefficient c is varied over the range c* ∈ *[0.3, 2].*

