Table 2 Chaos system equations and their corresponding parameters.

VB11 chaotic system⁴⁸

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=y{\text{+}}z \hfill \\ \frac{{dy}}{{dt}}={y^2} - az \hfill \\ \frac{{dz}}{{dt}}=x+by \hfill \\ \end{gathered} \right.\) (4)

where a = 1, b = 2.7.

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=y{\text{+}}z \hfill \\ \frac{{dy}}{{dt}}={y^2} - az \hfill \\ \frac{{dz}}{{dt}}=x+by \hfill \\ \frac{{dw}}{{dt}}=cz - dw+{d_2} \hfill \\ \end{gathered} \right.\) (5)

Lorenz chaotic system⁴⁹

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=\delta (y - x) \hfill \\ \frac{{dy}}{{dt}}=\gamma x - y - xz \hfill \\ \frac{{dz}}{{dt}}=xy - \eta z \hfill \\ \end{gathered} \right.\) (6)

where δ = 10, η = 8/3, γ = 28.

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=\delta (y - x) \hfill \\ \frac{{dy}}{{dt}}=\gamma x - y - xz \hfill \\ \frac{{dz}}{{dt}}=xy - \eta z \hfill \\ \frac{{dw}}{{dt}}=cz - dw+{d_2} \hfill \\ \end{gathered} \right.\) (7)

Chen chaotic system⁵⁰

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=A(y - x) \hfill \\ \frac{{dy}}{{dt}}=(C - A)x - xz+cy \hfill \\ \frac{{dz}}{{dt}}=xy - Bz \hfill \\ \end{gathered} \right.\) (8)

where A = 35, B = 3, C = 28.

\(\left\{ \begin{gathered} \frac{{dx}}{{dt}}=A(y - x) \hfill \\ \frac{{dy}}{{dt}}=(C - A)x - xz+cy \hfill \\ \frac{{dz}}{{dt}}=xy - Bz \hfill \\ \frac{{dw}}{{dt}}=cz - dw+{d_2} \hfill \\ \end{gathered} \right.\) (9)

where c, d and d₂ are constant terms for the system parameters.