# Analysis, Synchronization and Circuit Design of a 4D Hyperchaotic Hyperjerk System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}− 1), which exhibits chaos for a = 0.6 and b = 0.58.

## 2. Description of the 4D Hyperchaotic Hyperjerk System

#### 2.1. Model of the 4D Hyperjerk System

_{1}= 0.132, L

_{2}= 0.035, L

_{3}= 0 and L

_{4}= −1.250. For these values of Lyapunov exponents, the Kaplan–Yorke dimension [54,55,56] of the hyperjerk system (5) is defined as:

_{1}≥ ... ≥ L

_{j}are the Lyapunov exponents of the chaotic system and j is the largest integer for which L

_{1}+ L

_{2}+ ... + L

_{j}≥ 0. Thus, the Kaplan–Yorke dimension of the hyperjerk system (5) is easily calculated as D

_{KY}= 3.13.

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1), has Lyapunov exponents as: L

_{1}= 0.1555, L

_{2}= 0.0330, L

_{3}= 0 and L

_{4}= −1.6100, while the Kaplan–Yorke dimension of system (7) is D

_{KY}= 3.1171. Thus, system (7) is also a hyperchaotic 4D hyperjerk system.

_{1}= 0.1909, L

_{2}= 0.06462, L

_{3}= 0 and L

_{4}= −1.81846. It is easily seen that the maximal Lyapunov exponent (MLE) of our novel hyperchaotic hyperjerk system (8) is L

_{1}= 0.1809, which is greater than the respective MLE of systems (5) and (7). In addition, the Kaplan–Yorke dimension of the novel hyperjerk system (8), has been calculated as: D

_{KY}= 3.1405. Thus, the Kaplan–Yorke is also greater than the respective D

_{KY}of hyperjerk systems (5) and (7). This shows that the proposed hyperchaotic 4D hyperjerk system (8) exhibits more complex behavior than systems (5) and (7). Furthermore, the proposed in this work system has better results concerning others reported in literature hyperchaotic hyperjerk systems concerning their MLE and only one (see Ref. [59]) has greater Kaplan–Yorke dimensions (see Table 1). Therefore, the high-dimensional phase space of the proposed hyperjerk system combined with its high Kaplan–Yorke dimension, especially in regard to the other reported systems, guarantees the system’s complex dynamical behavior, which contributes to the required robustness.

_{1}–x

_{4}, show the system’s strange attractor for the selected set of parameters and initial conditions. Furthermore, in Figure 2, the Poincaré maps, which are produced by selecting two different planes, display the system (8)’s strange attractor. The study of continuous dynamical systems, like the proposed system (8), through a Poincaré map is one of the most popular topics in nonlinear dynamical analysis. This is done by taking intersections of the system’s orbit into the plane x

_{2}= 0 with dx

_{3}/dt > 0. Naturally, for a n dimensional attractor, the Poincaré map gives rise to (n − 1) points, which can describe the dynamics of the attractor properly. Thus, this is an artificial way of reducing the map by dropping its dimension. In this direction, the organized set of points in the Poincaré maps of Figure 2 is an indication of system’s chaotic behavior, while the discrete number of n points is an indication of periodic state of period-n. Finally, a closed curve in the Poincaré map is an indication of the system’s quasiperiodic behavior.

#### 2.2. Equilibrium Point Analysis

_{1}= 0.16385 + 1.89874i, λ

_{2}= 0.16385 − 1.89874i, λ

_{3}= −0.16385 + 0.498477i and λ

_{4}= −0.16385 − 0.498477i. Based on the theory, according to the aforementioned eigenvalues, equilibrium point E is unstable.

#### 2.3. Dissipativity and Invariance

_{1}, x

_{2}, x

_{3}, x

_{4}) → (−x

_{1}, −x

_{2}, −x

_{3}, x

_{4}). Therefore, (−x

_{1}, −x

_{2}, −x

_{3}, x

_{4}) is also a solution for the same values of parameters a, b, c and the system can display symmetric attractors.

## 3. Analysis of the 4D Hyperjerk Dynamics

_{2}= 0 with dx

_{3}/dt > 0, while the parameter a increases with very small step. In addition, the initial conditions at each iteration are (x

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1).

## 4. Circuit Realization of the Proposed System

_{C}is the voltage accross the capacitor. Other three operational amplifiers have been used as inverting amplifiers (U5–U7), which use negative feedback to amplify the input voltage. The last ones (U8, U9) with the two diodes (1N4007) are used for implementing the absolute nonlinearity ($|{x}_{2}|$). The selected configuration for the absolute nonlinearity gives the best results according to literature. Furthermore, the three multipliers (U10–U12) are used for implementing the quintic term c ${x}_{1}^{4}{x}_{4}$. The low-cost analog multipliers AD633, which are used for this purpose, is the most common approach for realizing polynomial terms or products, like the quintic term cx

^{4}x

_{1}.

_{1}, x

_{2}, x

_{3}and x

_{4}correspond to the voltages on the integrators (U1–U4), respectively, while the power supply is ±15 V. System (12) is normalized by using τ = t/RC. It can thus be suggested that system (12) is equivalent to system (8) with a = R/R

_{a}, b = R/R

_{b}and c = R/R

_{C}. The values of circuit components are R = 100 kΩ, R

_{1}= 10 kΩ, R

_{2}= 90 kΩ, R

_{b}= 1 MΩ, R

_{c}= 66.666 kΩ, and C = 1 nF. In order to change the parameter a, a variable resistor R

_{a}can be used.

## 5. Synchronization Scheme

_{i}, (i = 1, ..., 4) are the states and u

_{i}, (i = 1, ..., 4) are the adaptive controls to be determined. As it is mentioned, the parameters a, b, c, in systems (8) and (13) are unknown and the design goal is to find adaptive feedback controls u

_{i}that uses estimates for the parameters a, b, c, respectively, so as to render the states of the systems (8) and (13) fully synchronized asymptotically.

_{i}, (i = 1, ..., 4) are positive gain constants and $\widehat{a},\widehat{b},\widehat{c}$ are the parameter update laws.

**Theorem**

**1.**

_{i}, (i = 1, ..., 4) are positive constants.

**Proof.**

^{7}, is used. Therefore, the time derivative of V, by substituting the parameter update laws (23) into (22) is obtained as:

^{7}. In addition, it is concluded that the synchronization error vector e(t) = (e

_{1}(t), e

_{2}(t), e

_{3}(t), e

_{4}(t)) and the parameter estimation error (e

_{a}(t), e

_{b}(t), e

_{c}(t)) are globally bounded.

_{1}, k

_{2}, k

_{3}, k

_{4}), then it follows from Equation (24) that

_{i}= 10, for i = 1, 2, 3, 4.

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1), while the set of initial conditions of the slave system (13), is (y

_{1}(0), y

_{2}(0), y

_{3}(0), x

_{4}(0)) = (–0.1, 0.2, –0.2, 0). In addition, we use the values of $\widehat{a}(0)=3.5,\widehat{b}(0)=0.15,\widehat{c}(0)=1.2$, as initial conditions of the parameter estimates. The synchronization of the states of the master system (8) and slave system (13) are depicted in Figure 8, while the time-history of the synchronization errors e

_{1}(t), e

_{2}(t), e

_{3}(t), e

_{4}(t) is depicted in Figure 9. In more details, Figure 8 shows the fast convergence of the respective signals of the two coupled systems by using the proposed synchronization scheme. Furthermore, Figure 9 confirms the feasibility of the synchronization method in the coupled system by confirming that the synchronization errors converge to zero, as it is expected. Therefore, a chaotic complete synchronization has been achieved.

## 6. Discussion

_{1}, x

_{2}, x

_{3}, x

_{4}) → (−x

_{1}, −x

_{2}, −x

_{3}, x

_{4}), which is a very important feature that could drive the system to coexisting attractors.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Hyperchaotic attractors of system (8) in (

**a**) x

_{1}–x

_{2}plane; (

**b**) x

_{1}–x

_{3}plane; (

**c**) x

_{1}–x

_{4}; (

**d**) x

_{2}–x

_{3}plane; (

**e**) x

_{2}–x

_{4}plane and (

**f**) x

_{3}–x

_{4}plane, for a = 3.8, b = 0.1 and c = 1.5 and initial conditions (x

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1).

**Figure 2.**Poincaré maps of system (8) in (

**a**) x

_{1}–x

_{3}plane and (

**b**) x

_{1}–x

_{4}plane, for a = 3.8, b = 0.1 and c = 1.5 and initial conditions (x

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1).

**Figure 3.**Bifurcation diagram of system (8)’s variable x

_{4}versus the parameter a, for b = 0.1 and c = 1.5.

**Figure 4.**Spectrum of three largest Lyapunov exponents of system (8) versus the parameter a, for b = 0.1 and c = 1.5.

**Figure 5.**Phase portraits in x

_{1}–x

_{2}plane and the respective Poincaré maps in x

_{1}–x

_{3}plane, for (

**a**,

**b**) a = 5 (period-1 state); (

**c**,

**d**) a = 4.5 (period-2 state); and (

**e**,

**f**) a = 4.209 (quasiperiodic state). The rest of the parameters and initial conditions are: b = 0.1 and c = 1.5 and (x

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1).

**Figure 7.**PSpice simulation results of system (8) in different phase portraits (

**a**) x

_{1}–x

_{2}plane; (

**b**) x

_{1}–x

_{3}plane; (

**c**) x

_{1}–x

_{4}; (

**d**) x

_{2}–x

_{3}plane; (

**e**) x

_{2}–x

_{4}plane and (

**f**) x

_{3}–x

_{4}plane, for a = 3.8, b = 0.1 and c = 1.5 and initial conditions (x

_{1}(0), x

_{2}(0), x

_{3}(0), x

_{4}(0)) = (0.1, 0.1, 0.1, 0.1).

**Figure 8.**Synchronization of the states (

**a**) x

_{1}(t) and y

_{1}(t); (

**b**) x

_{2}(t) and y

_{2}(t); (

**c**) x

_{3}(t) and y

_{3}(t) and (

**d**) x

_{4}(t) and y

_{4}(t).

**Table 1.**Comparison of system (8) with other reported hyperchaotic hyperjerk systems regarding the maximal Lyapunov exponent and the Kaplan–Yorke dimension.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Daltzis, P.A.; Volos, C.K.; Nistazakis, H.E.; Tsigopoulos, A.D.; Tombras, G.S.
Analysis, Synchronization and Circuit Design of a 4D Hyperchaotic Hyperjerk System. *Computation* **2018**, *6*, 14.
https://doi.org/10.3390/computation6010014

**AMA Style**

Daltzis PA, Volos CK, Nistazakis HE, Tsigopoulos AD, Tombras GS.
Analysis, Synchronization and Circuit Design of a 4D Hyperchaotic Hyperjerk System. *Computation*. 2018; 6(1):14.
https://doi.org/10.3390/computation6010014

**Chicago/Turabian Style**

Daltzis, Petros A., Christos K. Volos, Hector E. Nistazakis, Andreas D. Tsigopoulos, and George S. Tombras.
2018. "Analysis, Synchronization and Circuit Design of a 4D Hyperchaotic Hyperjerk System" *Computation* 6, no. 1: 14.
https://doi.org/10.3390/computation6010014