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Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Dynamical System

#### 2.1. System Description

- The system has only five terms. Till now only few chaotic 3D dynamical systems with five terms have been reported in literature. Most of them belong to the class of jerk systems and they have been introduced by Professor Sprott and his colleagues [47,48]. However, there are few others, which are written in the general 3D dynamical system’s form [49,50,51,52,53]. In 1997, Fu and Heidel rigorously proved that there can be no simpler system with a quadratic nonlinearity [54].
- By solving$$\left\{\begin{array}{c}0={x}_{2}\left(t\right){x}_{3}\left(t\right)\hfill \\ 0={x}_{1}\left(t\right)|{x}_{1}\left(t\right)|-{x}_{2}\left(t\right)\left|{x}_{2}\left(t\right)\right|\hfill \\ 0=|{x}_{1}\left(t\right)|-a{x}_{1}\left(t\right){x}_{2}\left(t\right)\hfill \end{array}\right.$$It is easy to verify that (1) has a line of equilibrium points $\mathcal{E}(0,0,{x}_{3}\left(t\right))$. Therefore, system’s attractor is hidden.
- The amplitude ${x}_{3}\left(t\right)$ is easily controllable.
- The system is symmetric through the transformation $({x}_{1},{x}_{2},{x}_{3})\to (-{x}_{1},-{x}_{2},{x}_{3})$.This symmetry is observed in Figure 3, where two symmetric system’s periodic attractors, in the three different planes, for $a=3$ and initial conditions $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(1,0.5,1)$, with red color, and $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(-1,-0.5,1)$, with blue color, has been captured.

#### 2.2. Dynamical Analysis

## 3. Realization of the System

## 4. Application to Secure Communications

- (A.1)
- $rank{({E}^{T},{C}^{T})}^{T}=n$, where n is the number of columns of E. So, by considering the form of E, one can see that this is equivalent to D having full column rank.
- (A.2)
- The nonlinear part satisfies the Lipschitz property, that is, there exists a positive scalar $\gamma >0$ such that$$\left|\right|f\left(x\right)-f\left(y\right)\left|\right|\le \gamma \left|\right|x-y\left|\right|,\phantom{\rule{0.166667em}{0ex}}\forall x,y\in {\mathbb{R}}^{n}$$

**Remark**

**1.**

- Compute the matrix R, following the procedure given in ([67] Appendix A.)
- Compute the matrices $\tilde{E}=RE$, $\tilde{A}=RA$ that correspond to the system$$\tilde{E}\dot{x}=\tilde{A}+Rf\left(x\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}y=Cx$$$$\left(\begin{array}{cc}{\tilde{A}}^{T}P+P\tilde{A}-{C}^{T}{\tilde{K}}^{T}-\tilde{K}C+{\gamma}^{2}I& PR\\ {R}^{T}P& -I\end{array}\right)\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<0$$
- If (9) is solvable, the observer matrices can then be computed by$$\begin{array}{c}\hfill K={P}^{-1}\tilde{K},\phantom{\rule{1.em}{0ex}}N=\tilde{A}-KC\end{array}$$$$\begin{array}{c}\hfill \tilde{E}=I-MC,\phantom{\rule{1.em}{0ex}}L=K+NM\end{array}$$

**Remark**

**2.**

## 5. Simulation Results

## 6. Conclusions and Future Aspects

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Symmetric system’s attractors in (

**a**) ${x}_{1}-{x}_{2}$ plane, (

**b**) ${x}_{1}-{x}_{3}$ plane, and (

**c**) ${x}_{2}-{x}_{3}$ plane, for $a=3$ and initial conditions $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(1,0.5,1)$, with red color, and $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(-1,-0.5,1)$, with blue color.

**Figure 4.**(

**a**) Bifurcation diagram (blue) (with same initial conditions in each iteration) and continuation diagram (red) (with different initial conditions in each iteration), and (

**b**) Lyapunov Exponents (LE) of system (1) versus parameter a.

**Figure 5.**Coexisting chaotic attractors in ${x}_{1}-{x}_{3}$ plane, for $a=0.726$ and initial conditions $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(1,0.5,1)$, with red color, and $({x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right))=(-1,1.27,-2.05)$, with blue color.

**Figure 6.**Phase portraits in ${x}_{1}-{x}_{3}$ plane, for (

**a**) $a=0.5$ (period-1), (

**b**) $a=0.7$ (period-2), (

**c**) $a=0.708$ (period-4), (

**d**) $a=1.5$ (chaos), (

**e**) $a=2.09$ (period-5), (

**f**) $a=2.59$ (period-4), (

**g**) $a=2.7$ (period-2), and (

**h**) $a=3.0$ (period-1).

**Figure 8.**Schematic of the circuit emulating system (1).

**Figure 9.**Chaotic attractors produced from the circuit of Figure 8, for $a=1.5$, in (

**a**) $x-y$ plane, (

**b**) $x-z$ plane and (

**c**) $y-z$ plane.

**Figure 10.**Time response of states ${x}_{1}\left(t\right),{x}_{2}\left(t\right),{x}_{3}\left(t\right)$ and signal $s\left(t\right)$ and their estimations ${\widehat{x}}_{1}\left(t\right),{\widehat{x}}_{2}\left(t\right),{\widehat{x}}_{3}\left(t\right),\widehat{s}\left(t\right)$.

**Figure 11.**Time response of states ${x}_{1}\left(t\right),{x}_{2}\left(t\right),{x}_{3}\left(t\right)$ and signal $s\left(t\right)$ and their estimations ${\widehat{x}}_{1}\left(t\right),{\widehat{x}}_{2}\left(t\right),{\widehat{x}}_{3}\left(t\right),\widehat{s}\left(t\right)$ for a binary information signal.

**Figure 12.**Estimation of signals ${s}_{1}\left(t\right)=0.5cos\left(2\pi t\right)$ and ${s}_{2}\left(t\right)=cos\left(0.5\pi t\right)+0.5sin\left(0.8\pi t\right)+1$.

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**MDPI and ACS Style**

Moysis, L.; Volos, C.; Pham, V.-T.; Goudos, S.; Stouboulos, I.; Gupta, M.K.; Mishra, V.K.
Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer. *Technologies* **2019**, *7*, 76.
https://doi.org/10.3390/technologies7040076

**AMA Style**

Moysis L, Volos C, Pham V-T, Goudos S, Stouboulos I, Gupta MK, Mishra VK.
Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer. *Technologies*. 2019; 7(4):76.
https://doi.org/10.3390/technologies7040076

**Chicago/Turabian Style**

Moysis, Lazaros, Christos Volos, Viet-Thanh Pham, Sotirios Goudos, Ioannis Stouboulos, Mahendra Kumar Gupta, and Vikas Kumar Mishra.
2019. "Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer" *Technologies* 7, no. 4: 76.
https://doi.org/10.3390/technologies7040076