# Design of Initial Value Control for Modified Lorenz-Stenflo System

Authors: Yuan-Long Wu; Cheng-Hsiung Yang; Chang-Hsi Wu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

For the sake of complexity, unpredictability, and exceeding sensitivity to initial conditions in the chaotic systems, there were many studies for information encryption of chaotic systems in recent years. Enhancing the security in information encryption of chaotic systems, an initial value control circuit for chaotic systems is proposed in this paper. By way of changing the initial value, we can change the behavior of chaotic systems and also change the key of information encryption. An analog circuit is implemented to verify the initial value control circuit design.


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## Body

## 1. Introduction

Chaotic systems provide a rich mechanism for signal design and generation, with potential applications to communications and signal processing. Because chaotic signals are typically broadband, noise-like, and difficult to predict, they can be used in various contexts for masking information bearing waveforms. They can also be used as modulating waveforms in spread spectrum systems. This can be useful in many practical circumstances like securing communication channels [1–3], masking signals [4–6], and spreading data sequence or for generating random signals [7]. Four-dimensional Lorenz-Stenflo system is a famous four-dimensional chaotic system; there were many recent works on it [8–11].The cryptographic system is composed of plaintext, ciphertext, encryption algorithm, key, and decryption algorithm. Dynamic update of the key can reduce its probability of being correctly guessed and improve the security of cryptographic system. In this paper, we introduce an initial value control circuit design for the modified four-dimensional Lorenz-Stenflo system. Furthermore, an analog circuit of the chaotic system is implemented to verify the initial value control circuit design. By way of changing the initial value, we can change the behavior of chaotic systems and also change the key of information encryption.This paper is organized as follows. In Section2, nonlinear dynamic characteristics of the modified four-dimensional Lorenz-Stenflo system are analyzed. In Section 3, an analog circuit of the modified chaotic system is implemented to verify the initial value control circuit design. Finally, some concluding remarks are given in Section 4.


## 2. Dynamic Analysis of a New Chaotic System

### 2.1. Chaotic Equations and Phase Portraits

The Lorenz-Stenflo system is described as [12, 13](1)x˙1=ax2-ax1+bx4,x˙2=cx1-x1x3-x2,x˙3=x1x2-dx3,x˙4=-x1-ax4,where x=x1,x2,x3,x4T, a is Prandtl number, b is rotation number, c is Rayleigh number, and d is geometric parameter. Because four-dimensional Lorenz-Stenflo system is a famous four-dimensional chaotic system, we modify four-dimensional Lorenz-Stenflo system as a study example. We modify system equations (1) where new system equations (2) still keep chaotic characteristics [14].(2)x˙1=ax2-ax1,x˙2=cx1-x1x3,x˙3=x1x2-dx3+bx4,x˙4=-x1-ax4.The two-dimensional phase portraits of chaotic system equations (2) are shown in Figure 1, in which symbol “∗” denotes equilibrium points, with EP1, EP2, and EP3, with the parameters a=3.7, b=1.5, c=25, d=0.7, and x1(0)=x2(0)=x3(0)=x4(0)=1.Figure 1
Two-dimensional phase portraits of system equations (2).
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### 2.2. Symmetry and Invariance

New chaotic system equations (2) are invariance under the coordinate transform (x1,x2,x3,x4)→(-x1,-x2,x3,-x4); chaotic system equations (2) are rotation symmetrical about the x3 axis.


### 2.3. Dissipativity and Existence of Attractor

The state space of chaotic system equations (2) is four-dimensional. The vector field on the right-hand sides of chaotic system equations (2) is defined by(3)Φx=Φ1xΦ2xΦ3xΦ4x=ax2-ax1cx1-x1x3x1x2-dx3+bx4-x1-ax4.The divergence of the vector field Φ is calculated as(4)∇Φx=∂Φ1x∂x1+∂Φ2x∂x2+∂Φ3x∂x3+∂Φ4x∂x4=-2a-d.A necessary and sufficient condition for system equations (2) to be dissipative is that the divergence of the vector field Φ is negative. In (4), it is immediate that system equations (2) are dissipative if and only if a,d∈R+ with an exponential rate:(5)dΦdt=e-2a+d.Thus, in dynamical system equations (2), a volume element Φ0 is apparently contracted by the flow into a volume element Φ0e-(2a+d+1) in time domain. This means that each volume containing the trajectories of this dynamical system shrinks to zero as time approached infinity at an exponential rate ∇Φ. So, all the orbits of chaotic system equations (2) will be eventually confined to a special subset that has zero volume, and the asymptotic motion of system equations (2) will settle onto an attractor of the system. Then, the existence of an attractor is proved.


### 2.4. Equilibrium Points Analysis

The equilibrium points of system equations (2) can be found by solving the following algebraic equations simultaneously:(6)ax2-ax1=0,cx1-x1x3=0,x1x2-dx3+bx4=0,-x1-ax4=0.Chaotic system equations (2) have three equilibrium points, respectively, represented as(7)EP10,0,0,0,EP2b+4a2cd+b22a,b+4a2cd+b22a,c,-b+4a2cd+b22a2,EP3b-4a2cd+b22a,b-4a2cd+b22a,c,-b-4a2cd+b22a2.When the parameters are a=3.7, b=1.5, c=25, and d=0.7, chaotic system equations (2) have three equilibrium points, given by EP1(0,0,0), EP2(4.39091,4.39091,25,-1.18673), and EP3(-3.98551,-3.98551,25,1.07716).By linearizing chaotic system equations (2), the Jacobian matrix is obtained as(8)J=-aa00c-x30-x10x2x1-db-100-a.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP1(0,0,0,0) is obtained as(9)JEP1=-aa00c-x30-x10x2x1-db-100-ax=EP1=-aa00c00000-db-100-awhich has the eigenvalues(10)λ1EP1=-0.7,λ2EP1=-3.7,λ3EP1=-11.6440,λ4EP1=7.9440.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP2(4.39091,4.39091,25,-1.18673) is obtained as(11)JEP2=-aa00c-x30-x10x2x1-db-100-ax=EP2=-aa00c-250-4.3909104.390914.39091-db-100-awhich has the eigenvalues(12)λ1EP2=0.4161+5.1585i,λ2EP2=0.4161-5.1585i,λ3EP2=-5.5368,λ4EP2=-3.3954.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP3(-3.98551,-3.98551,25,1.07716) is obtained as(13)JEP3=-aa00c-x30-x10x2x1-db-100-ax=EP3=-aa00c-2503.985510-3.98551-3.98551-db-100-awhich has the eigenvalues(14)λ1EP3=0.4488+4.7403i,λ2EP3=0.4488-4.7403i,λ3EP3=-4.7828,λ4EP3=-4.2148.Since the linearization matricesJ(EP1), J(EP2), and J(EP3) have at least one eigenvalue with positive real parts, according to Lyapunov stability method [15], then the equilibrium points EP1, EP2, and EP3 are unstable, and this implies chaos in the dissipative chaotic system equation (2). So, the trajectories of chaotic system equation (2) diverge from the three equilibrium points and orbit into the strange attractor of chaotic system Eq. (2).


## 2.1. Chaotic Equations and Phase Portraits

The Lorenz-Stenflo system is described as [12, 13](1)x˙1=ax2-ax1+bx4,x˙2=cx1-x1x3-x2,x˙3=x1x2-dx3,x˙4=-x1-ax4,where x=x1,x2,x3,x4T, a is Prandtl number, b is rotation number, c is Rayleigh number, and d is geometric parameter. Because four-dimensional Lorenz-Stenflo system is a famous four-dimensional chaotic system, we modify four-dimensional Lorenz-Stenflo system as a study example. We modify system equations (1) where new system equations (2) still keep chaotic characteristics [14].(2)x˙1=ax2-ax1,x˙2=cx1-x1x3,x˙3=x1x2-dx3+bx4,x˙4=-x1-ax4.The two-dimensional phase portraits of chaotic system equations (2) are shown in Figure 1, in which symbol “∗” denotes equilibrium points, with EP1, EP2, and EP3, with the parameters a=3.7, b=1.5, c=25, d=0.7, and x1(0)=x2(0)=x3(0)=x4(0)=1.Figure 1
Two-dimensional phase portraits of system equations (2).
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## 2.2. Symmetry and Invariance

New chaotic system equations (2) are invariance under the coordinate transform (x1,x2,x3,x4)→(-x1,-x2,x3,-x4); chaotic system equations (2) are rotation symmetrical about the x3 axis.


## 2.3. Dissipativity and Existence of Attractor

The state space of chaotic system equations (2) is four-dimensional. The vector field on the right-hand sides of chaotic system equations (2) is defined by(3)Φx=Φ1xΦ2xΦ3xΦ4x=ax2-ax1cx1-x1x3x1x2-dx3+bx4-x1-ax4.The divergence of the vector field Φ is calculated as(4)∇Φx=∂Φ1x∂x1+∂Φ2x∂x2+∂Φ3x∂x3+∂Φ4x∂x4=-2a-d.A necessary and sufficient condition for system equations (2) to be dissipative is that the divergence of the vector field Φ is negative. In (4), it is immediate that system equations (2) are dissipative if and only if a,d∈R+ with an exponential rate:(5)dΦdt=e-2a+d.Thus, in dynamical system equations (2), a volume element Φ0 is apparently contracted by the flow into a volume element Φ0e-(2a+d+1) in time domain. This means that each volume containing the trajectories of this dynamical system shrinks to zero as time approached infinity at an exponential rate ∇Φ. So, all the orbits of chaotic system equations (2) will be eventually confined to a special subset that has zero volume, and the asymptotic motion of system equations (2) will settle onto an attractor of the system. Then, the existence of an attractor is proved.


## 2.4. Equilibrium Points Analysis

The equilibrium points of system equations (2) can be found by solving the following algebraic equations simultaneously:(6)ax2-ax1=0,cx1-x1x3=0,x1x2-dx3+bx4=0,-x1-ax4=0.Chaotic system equations (2) have three equilibrium points, respectively, represented as(7)EP10,0,0,0,EP2b+4a2cd+b22a,b+4a2cd+b22a,c,-b+4a2cd+b22a2,EP3b-4a2cd+b22a,b-4a2cd+b22a,c,-b-4a2cd+b22a2.When the parameters are a=3.7, b=1.5, c=25, and d=0.7, chaotic system equations (2) have three equilibrium points, given by EP1(0,0,0), EP2(4.39091,4.39091,25,-1.18673), and EP3(-3.98551,-3.98551,25,1.07716).By linearizing chaotic system equations (2), the Jacobian matrix is obtained as(8)J=-aa00c-x30-x10x2x1-db-100-a.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP1(0,0,0,0) is obtained as(9)JEP1=-aa00c-x30-x10x2x1-db-100-ax=EP1=-aa00c00000-db-100-awhich has the eigenvalues(10)λ1EP1=-0.7,λ2EP1=-3.7,λ3EP1=-11.6440,λ4EP1=7.9440.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP2(4.39091,4.39091,25,-1.18673) is obtained as(11)JEP2=-aa00c-x30-x10x2x1-db-100-ax=EP2=-aa00c-250-4.3909104.390914.39091-db-100-awhich has the eigenvalues(12)λ1EP2=0.4161+5.1585i,λ2EP2=0.4161-5.1585i,λ3EP2=-5.5368,λ4EP2=-3.3954.The Jacobian matrix for chaotic system equations (2) at equilibrium point EP3(-3.98551,-3.98551,25,1.07716) is obtained as(13)JEP3=-aa00c-x30-x10x2x1-db-100-ax=EP3=-aa00c-2503.985510-3.98551-3.98551-db-100-awhich has the eigenvalues(14)λ1EP3=0.4488+4.7403i,λ2EP3=0.4488-4.7403i,λ3EP3=-4.7828,λ4EP3=-4.2148.Since the linearization matricesJ(EP1), J(EP2), and J(EP3) have at least one eigenvalue with positive real parts, according to Lyapunov stability method [15], then the equilibrium points EP1, EP2, and EP3 are unstable, and this implies chaos in the dissipative chaotic system equation (2). So, the trajectories of chaotic system equation (2) diverge from the three equilibrium points and orbit into the strange attractor of chaotic system Eq. (2).


## 3. Initial Value Control Circuit for Chaotic Systems

The four-dimensional modified Lorenz-Stenflo system can be modeled by an electronic circuit [16–19]. To implement system equations (2), a circuit is constructed in Figure 2; the governing integral equation of the circuit can be written as (15) and some simulation results are showed in Figure 3. A initial value control circuit design for system equations (2) is designed in Figure 4; the two-dimensional phase portraits of chaotic system (2) with initial value control circuit are shown in Figures 5 and 6. From Figures 5 and 6, we can observe the results, when the control switch turn-on time is different, the capacitor is charged to different voltage level. For example, if the control switch turn-on time = 3 sec, the capacitor is charged to Vc1=0.5 V ∗ 3 sec = 1.5 V. By way of changing the charged voltage level in capacitor, we can change the initial value in system equations (2) and the behavior of system equations (2) also is modified. The real circuit of the modified four-dimensional Lorenz-Stenflo system is constructed on a breadboard shown in Figure 7. The experimental setup is shown in Figure 8 and the experimental results are shown in Figure 9.(15)x˙1=1R2C1x2-1R1C1x1,x˙2=1R3C2x1-1R4C2x1x3,x˙3=1R7C3x1x2-1R8C3x3+1R6C3x4,x˙4=-1R9C4x1-1R10C4x4.Figure 2
Circuit of chaotic system (2).Figure 3
Two-dimensional phase portraits of circuit in Figure2 (default initial value voltage x1(0) = Vc1 = 2 V, without initial value control).
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Initial value control circuit of chaotic system equations (2).Figure 5
Two-dimensional phase portrait of Figure4 with control switch turn-on time 3 sec (initial value voltage x1(0)=Vc1=0.5 V ∗ 3 sec = 1.5 V, with initial value control).
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Two-dimensional phase portrait of Figure4 with control switch turn-on time 4 sec (initial value voltage x1(0)=Vc1=0.5 V ∗ 4 sec = 2 V, with initial value control).
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Realization circuit of the modified four-dimensional Lorenz-Stenflo system.Figure 8
Experimental setup for realization of the modified four-dimensional Lorenz-Stenflo system.Figure 9
Realization circuit results for voltage signals outputs with different control switch turn-on time.
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## 4. Conclusions

In summary, this paper introduces an initial value control circuit design and nonlinear dynamic analysis of the modified four-dimensional Lorenz-Stenflo system is presented. Furthermore, an analog circuit of the chaotic system is implemented to verify the initial value control circuit design. We design a charge circuit embedded in modified four-dimensional Lorenz-Stenflo analog circuit and use analog circuit to implement the concept of changing initial values for chaotic systems in mathematics. Because there are operational amplifier (OPA) integrators in the modified four-dimensional Lorenz-Stenflo analog circuit, we only need Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) to implement the charge circuit; it is easy to realize. By way of changing the initial value, we can change the behavior of chaotic system and also change the key of information encryption. Next step we will apply the initial value control circuit of chaotic systems for information encryption application.



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*Source: 8424139-2017-07-05.xml*