Open access peer-reviewed chapter - ONLINE FIRST

Chaotic Systems with Hyperbolic Sine Nonlinearity

By Jizhao Liu, Yide Ma, Jing Lian and Xinguo Zhang

Submitted: July 29th 2020Reviewed: October 16th 2020Published: November 18th 2020

DOI: 10.5772/intechopen.94518

Downloaded: 14

Abstract

In recent years, exploring and investigating chaotic systems with hyperbolic sine nonlinearity has gained the interest of many researchers. With two back-to-back diodes to approximate the hyperbolic sine nonlinearity, these chaotic systems can achieve simplicity of the electrical circuit without any multiplier or sub-circuits. In this chapter, the genesis of chaotic systems with hyperbolic sine nonlinearity is introduced, followed by the general method of generating nth-order (n > 3) chaotic systems. Then some derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity is discussed. Finally, the applications such as random number generator algorithm, spread spectrum communication and image encryption schemes are introduced. The contribution of this chapter is that it systematically summarizes the design methods, the dynamic behavior and typical engineering applications of chaotic systems with hyperbolic sine nonlinearity, which may widen the current knowledge of chaos theory and engineering applications based on chaotic systems.

Keywords

  • chaotic systems
  • torus chaos
  • hyperbolic sine nonlinearity
  • spread spectrum communication
  • image encryption

1. Introduction

Since Lorenz discovered chaos in a third-order ordinary differential equations, a new field of science has been launched [1]. The fact that simple equations can exhibit incredible complex behavior continues enthrall engineers to apply chaotic systems to cryptosystem, secure communication, spread spectrum communication, etc. [2].

There is no doubt that nonlinear term is very important to design chaotic systems, which has peculiar complex properties such as ergodicity, highly initial value sensitivity, non-periodicity and long-term unpredictability. According to the literature, the nonlinearities can be piecewise nonlinear function [3], trigonometric function [4], absolute value function [5], or power function [6]. With different nonlinearities, the chaotic system can have various strange attractors as single-scroll [7], double-scroll [8], multi-scroll [9], etc. The majority of such chaotic systems are known for many years, and some chaotic systems with hidden attractors are derived from them [10, 11, 12].

In recent years, chaotic systems with hyperbolic sine nonlinearities have gained the interest of many researchers. With two back-to-back diodes to approximate the hyperbolic sine nonlinearity, these chaotic systems can achieve simplicity of the electrical circuit without any multiplier or sub-circuits. Compared to single-scroll chaotic systems, the chaotic system with hyperbolic sine nonlinearity has richer dynamic behavior because it is symmetrical and can exhibit symmetry breaking, and offers the possibility that attractors will split or merge as some bifurcation parameter is changed [13].

In this chapter, we will systematically summarize the design method, the dynamic behavior and typical engineering applications of chaotic systems with hyperbolic sine nonlinearity. The genesis and general method of generating nth-order (n > 3) chaotic systems with hyperbolic sine nonlinearity are introduced in Section II. Some derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity is discussed in Section III. The application such as random number generator algorithm, spread spectrum communication and image encryption schemes are introduced in Section IV. Conclusions are finally drawn in Section V.

2. General chaotic systems with hyperbolic sine nonlinearity

2.1 The genesis of chaotic systems with hyperbolic sine nonlinearity

In 2011, Sprott and Munmuangsaen proposed an exponential chaotic system [14], which happens to be an example of the simplest chaotic system [15]. In the same year, Sprott used common resistors, capacitors, operational amplifiers, and a diode to successfully implement this system in a circuit [16]. Few years later, the simplest hyperbolic sine chaotic system is proposed [17]. Compared to the exponential chaotic system, the hyperbolic sine chaotic system changed the nonlinearity from exponential function (asymmetric function) to hyperbolic sine function (symmetric function), which can exhibit symmetry breaking, and offers the possibility that attractors will split or merge as some bifurcation parameter is changed [18].

The simplest chaotic system with a hyperbolic sine is described as follows:

x+cx¨+x+ρsinhφẋ=0E1

Where c is considered as the bifurcation parameter, sinhφẋ=eφẋeφẋ2, ρ=1.2106and φ=10.026, which have been chosen to facilitate circuit implementation using diodes. The corresponding circuit schematic diagram of Eq. (1) is shown as Figure 1.

When c=0.75, the Eq. (1) can exhibit chaotic behavior, which is shown as Figure 2.

Figure 1.

The corresponding circuit schematic diagram of Eq. (1).

Figure 2.

Numerical and actual circuit state space plot in x − x ¨ plane.

2.2 The general equations of generating chaotic systems with hyperbolic sine nonlinearity

It is obvious that Eq. (1) can be written in the form with jerk equations:

ẋ1=x2ẋ2=x3ẋ3=cx3fx2x1E2

where fx2=ρsinhφx2. Therefore, the higher order chaotic systems with hyperbolic sine nonlinearity can be generated by adding jerk cabins, which is described by:

ẋ1=x2x1ẋ2=x3x2ẋn3=xn2xn3ẋn2=xn1ẋn1=xnẋn=cxnfxn1nxn2nxn312nx1E3

where ẋk1=xkxk1is the jerk cabin. With Eq. (3), we can construct nth-order (n > 3) chaotic systems with hyperbolic sine nonlinearity.

When n = 4, the equations of fourth-order chaotic systems will be:

ẋ1=x2x1ẋ2=x3ẋ3=x4ẋ4=x4fx35x20.125x1E4

The corresponding circuit schematic diagram of Eq. (4) is shown as Figure 3.

Figure 3.

The corresponding circuit schematic diagram of Eq. (4).

Its numerical and actual circuit state space plot is shown as Figure 4.

Figure 4.

Numerical and actual circuit state space plot in x 2 − x 3 plane and x 3 − x 4 plane.

When n = 5, the equations of fifth-order chaotic systems will be:

ẋ1=x2x1ẋ2=x3x2ẋ3=x4ẋ4=x5ẋ5=x5fx45x35x20.1x1E5

The corresponding circuit schematic diagram of Eq. (5) is shown as Figure 5.

Figure 5.

The corresponding circuit schematic diagram of Eq. (5).

Its numerical and actual circuit state space plot is shown as Figure 6.

Figure 6.

Numerical and actual circuit state space plot in x 1 − x 5 plane and x 2 − x 3 plane.

3. Derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity

3.1 Multi-nonlinearities hyperbolic sine chaotic system

One way to construct the derived chaotic systems is to add more nonlinear terms of the equations. For example, the new chaotic system can be constructed by Eq. (4), which is described as follows:

ẋ1=x2ρsinhφx1ẋ2=x30.3x2ρsinhφx2ẋ3=x4ẋ4=0.25x4ρsinhφx30.5x24x1E6

Where ρ=1.2106, φ=10.026. These equations can exhibit chaotic behavior as shown in Figure 7.

Figure 7.

Numerical phase space plot of Eq. (6).

3.2 Simple chaotic system with hyperbolic sine nonlinearity

The other way to construct the derived chaotic systems is to simplify the known chaotic systems. For example, if we remove the parameter ρand φ, search the parameter space, we will have the following chaotic system:

ẋ1=6x2x1ẋ2=x3ẋ3=x4ẋ4=x4sinhx3x1E7

When initial conditions are set to be x1x2x3x4=0.7,0.9,1.0,1.3, or x1x2x3x4=0.70.91.01.3, the system exhibits period behavior. When the initial conditions are set to be x1x2x3x4=791013and x1x2x3x4=791013, the system exhibits chaotic behavior. Therefore, this system has four coexistence attractors [19], as shown in Figure 8.

Figure 8.

Coexistence attractors of Eq. (7).

3.3 Torus-chaotic system with hyperbolic sine nonlinearity

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

ẋ1=x2ρsinhφx3ẋ2=x3x2ẋ3=x4ẋ4=x5ẋ5=cx5ρsinhφx45x35x20.1x1E8

When c = 1, the Lyapunov exponents are λ1λ2λ3λ4λ5=0.47001.101.37, which suggests Eq. (8) is exhibiting torus-chaos behavior [20].

When c = 1.55 and the initial conditions are set to be x1x2x3x4x5=0.10.10.10.10.1and x1x2x3x4x5=0.10.1,0.1,0.1,0.1, the system has two coexisting attractors as shown in Figure 9.

Figure 9.

Coexistence attractors of Eq. (8).

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].

  1. When c ∈ [0.3, 0.4639], there exists a period-doubling behavior along with ẋ2and ẋ3subspace. However, the system shows torus behavior along with ẋ2and ẋ3subspace. It is like saddle point: the system is stable in one direction but unstable in the other direction.

  2. When c ∈ [0.4640, 0.5574], the system exhibits two-torus-chaos behavior except for some 2-torus windows. When the parameter passed c = 0.4639 to c = 0.4640, two-torus-chaos is born by replacing the 2-torus behavior. The Lyapunov exponents at these two critical values are λ1λ2λ3λ4λ5=000.010.570.88for c = 0.4639 and λ1λ2λ3λ4λ5=0.02,000.600.88for c = 0.4640. This may cause by the period-doubling route along with ẋ2and ẋ3subspace.

  3. When c ∈ [0.5575, 0.5901], the system exhibits 2-torus behavior.

  4. When c ∈ [0.5902, 1.5575], the system exhibits 2-torus-chaos behavior except for 2-torus windows. The route leading to chaos is same to point 3.

  5. When c ∈ [1.5575, 2] the system exhibits 2-torus behavior, except for some 3-torus windows like c = 1.6157.

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].

4. Engineering applications with chaotic systems with hyperbolic sine nonlinearity

4.1 Random number generator

Sensitivity to initial conditions is one of the most important property of chaotic systems. Therefore, chaotic systems are very suitable for the cryptography purpose. But before that, it should be noticed that the probability density distributions (PDD) of chaotic systems are not uniform distribution. Figure 11(a) and 11(b) are the waveform and PDD of x4of Eq. (4). It shows that PDD of the output sequences has physical characteristic. The cryptosystem with these sequences cannot resist side channel attack.

Figure 11.

Waveform and PDD before and after de-correlation operation of x 4 of Eq. (4): (a) is the waveform of x 4 before de-correlation operation; (b) is the PDD of x 4 before de-correlation operation; (c) is the waveform of x 4 after de-correlation operation; (b) is the PDD of x 4 after de-correlation operation.

To remove physical characteristic, one can use the following de-correlation operation:

Sout=Sin106floorSin106E9

In fact, Eq. (9) can be applied in all chaotic/torus-chaotic/hyperchaotic systems. The output sequences can pass fifteen random tests of NIST 800-22, as shown as in Table 1, which indicated the proposed method can provide high security Level. This proposed method can be used as a part of some cyber security systems such as the verification code, secure QR code and some secure communication protocols.

TestP-valueResult
Frequency0.841481Success
Block frequency0.900704Success
Runs0.744455Success
Longest run0.172897Success
Rank0.368065Success
FFT0.762020Success
Non-overlapping template0.813121Success
Overlapping template0.532736Success
Universal0.856573Success
Linear complexity0.408679Success
Serial0.967366Success
Approximate entropy0.433157Success
Cumulative sums0.688582Success
Random excursions0.075229Success
Random excursions variant0.102049Success

Table 1.

Pseudo-random properties of x3of Eq. (8) after de-correlation operation.

4.2 Image encryption

Image encryption is another widely used engineering application of chaotic system. In this section, we will use Eq. (7) for image encryption purpose.

A flowchart of the encryption scheme is shown in Figure 12.

Figure 12.

A flowchart of the encryption scheme.

The detailed encryption process includes the following steps.

Input: Plain image; Initial conditions for the chaotic system; Control parameters of the chaotic system.

Output: Ciphered image.

Step 1: Calculate the average pixel value of the plain image and generate the pseudorandom sequence.

Step 2: Transform the pseudorandom sequence and change pixel value of the image via XOR.

Step 3: Sort the pseudorandom sequence for permutation.

Step 4: Shift the pixel positions by column using the sorted elements.

Step 5: Shift the pixel positions by row using the sorted elements.

To provide a better understanding of this scheme, the pseudocode is provided in Table 2.

Input: Plain image Org_Img, Initial conditions for the chaotic system, Control parameter for the chaotic system,
Output: Ciphered Image En_Img
[m,n] ← size(Org_Img);

Avg_pixel_value ← mean2(Org_Img)*10^(-5) % mean2 is a function that
returns the

          % average value of a matrix

x(1) ← x(1) + Avg_pixel_value
y(1) ← y(1)
z(1) ← z(1)
u(1) ← u(1)
s(1) ← u(1)*10^4 – floor(u(1)*10^4)

For i=1:1:m*n     % Generate pseudorandom sequence that will
                % be used for diffusion and permutation
  [dx, dy, dz, du] ← Runge-Kutta (x(i), y(i), z(i), u(i))
  x(i+1) ← x(i) +dx
  y(i+1) ← y(i) +dy
  z(i+1) ← z(i) +dz
  u(i+1) ← u(i) +du
s(i+1) ← u(i+1)*10^4 – floor(u(i+1)*10^4)
End

Count=1   % Count flag
For i=1:m   % Diffusion Operation
 For j=i:n
  diff(Count) ← mod (s(Count)*10^14, 256) % transform s, which could be used for XOR
 En_Dif(i,j)=bitxor(Org_Img(i,j), diff (Count)); % Bitwise exclusive OR
  Count= Count+1;
 End
End

S_index ← Sort(s)
For i=1:n    % Column-wise permutation
 For j=1:m
 En_per_col (i,j) ← Sort (En_Dif, S_index)
 End
End
For i=1:m    % Row-wise permutation
 For j=1:n
 En_Img (i,j) ← Sort (En_per_col, S_index)
 End
End

Table 2.

Image encryption scheme.

The decryption process of the proposed algorithm is the reverse process of the encryption algorithm. A flowchart of the decryption process is shown in Figure 13.

Figure 13.

A flowchart of the decryption scheme.

The detailed decryption process includes the following steps.

Input: Plain image; Initial conditions for the chaotic system; Control parameter of the chaotic system; Average pixel value of the plain image

Output: Decrypted image

Step 1: Generate the pseudorandom sequence via the initial conditions and the average pixel values of the plain image

Step 2: Sort the pseudorandom sequence for row and column recovery.

Step 3: Shift the pixel positions by row

Step 4: Shift the pixel positions by column

Step 5: Transform the pseudorandom sequence and recover the pixel values of the image via XOR

To provide a better understanding of this scheme, the pseudo-code is provided in Table 3

 Input: Ciphered image En_Img, Initial conditions for the chaotic system, control parameter for the chaotic system, Avg_pixel_value of Org_Img
 Output: Plain Image Org_Img
 [m,n] ← size(En_Img);
 x(1) ← x(1) + Avg_pixel_value
 y(1)← y(1)
 z(1) ← z(1)
 u(1) ← u(1)
 s(1) ← u(1)*10^4 – floor(u(1)*10^4)

 For i=1:1:m*n     % Generate a pseudorandom sequence that will
 % be used for decryption
 [dx, dy, dz, du] ← Runge-Kutta (x(i), y(i), z(i), u(i))
 x(i+1) ← x(i) +dx
 y(i+1) ← y(i) +dy
 z(i+1) ← z(i) +dz
 u(i+1) ← u(i) +du
  s(i+1) ← u(i+1)*10^4 – floor(u(i+1)*10^4)
 End

 S_index ← Sort(s)
 For i=1:m      % Row-wise permutation recovery
  For j=1:n
  De_per_row (i,j) ← Sort (En_Img, S_index)
  End
 End

 For i=1:n        % Column-wise permutation recovery
 For j=1:m
 De_per_col (i,j) ← Sort (De_per_row, S_index)
 End
 End

Count=1       % Count flag
For i=1:m      % Diffusion recovery
 For j=i:n
 diff(Count) ← mod (s(Count)*10^14, 256) % transform s, which could be used for XOR
 Org_Img (i,j)=bitxor(De_per_col (i,j), diff (Count)); % Bitwise exclusive OR
  Count= Count+1;
 End
 End

Table 3.

Image decryption scheme.

The testing results of encryption and decryption are shown in Figure 14.

Figure 14.

The testing results of encryption and decryption: (a) is the plain image of cameraman; (b) is the encrypted image of cameraman; (c) is the decrypted image of cameraman; (d) is the plain image of breast CT image; (e) is the encrypted image of breast CT image; (f) is the decrypted image of breast CT image; (g) is the plain image of thorax CT image; (h) is the encrypted image of thorax CT image; (i) is the decrypted image of thorax CT image.

In this system, all the initial conditions and control parameters can be considered as secret keys. Because the basin of attraction of each initial condition is greater than 1, it could have more than 1015∗4 =1060 choices via a resolution of 10−15, in terms of a numeric calculation. Moreover, if a range of control parameters are considered for the key space, the key space of this system would far exceed 1090. Such a large key space provides sufficient security against brute-force attacks.

Correlation coefficients of adjacent pixels in the plain and encrypted image are shown in Table 4.

Figure nameDirectionPlain-imageCiphered image
Cameraman ImageHorizontal0.9831460.001731
Cameraman ImageVertical0.9900250.004141
Cameraman ImageDiagonal0.9732490.000324
Breast CT imageHorizontal0.9782920.002500
Breast CT imageVertical0.9554810.006207
Breast CT imageDiagonal0.9407370.003071
Thorax CT imageHorizontal0.9945850.001267
Thorax CT imageVertical0.9947610.001267
Thorax CT imageDiagonal0.9919730.001558

Table 4.

Correlation coefficients of adjacent pixels in the plain and encrypted image.

The NPCR and UACI score of CT image are 99.5804% and 33.3227%.

From the above security analysis, the proposed scheme can provide high security for cryptographic applications.

4.3. Spread spectrum communication

Chaotic systems can also use for spread spectrum communication propose. Different chaos shift keying (DCSK) technology employs nonperiodic and wideband chaotic signals as carriers so as to achieve the effect of spectrum spreading in the process of digital modulation. Figure 15 shows the scheme of modulation for DCSK.

Figure 15.

Scheme of DCSK modulation.

In this scheme, every bit has two time slots. The first time slot is used for transmission of a chaotic sequence for the reference signal. The second time slot is used for transmission of another chaotic sequence for the reference signal which has the same length as the first time slot. If the information bit is +1, then the information signal is exactly the same as the reference signal. If the information signal bit is −1, then the information signal is the negative of the reference signal. For bits bk, the signal at time k is:

si=xi2<i2k+1βbkxiβ2k+1β<i2k+1βE10

Where βis the number of sampling points. The spreading factor (SF) in the DCSK system is SF=2β.

For demodulation as shown in Figure 16, the receiver calculates the correlation between the received signal riand the signal riβ, which is ridelayed by β. After a time k, the output of the correlator is:

Figure 16.

Scheme of the DCSK demodulation.

Zk=2k+1βi=2k+1β+1ririβE11

Thus, the information bit bkcan be restored by the sign of the decision variable:

b̂k=sgnZkE12

The obtained BER performance under additive white Gaussian noise (AWGN) channels for spreading factor 2β=200is shown in Figure 17. From the comparison results, DCSK can have a lower BER when using this system as a carrier signal in the presence of noise.

Figure 17.

Comparison of the bit error rate for a Chebyshev sequence and the hyperbolic sine system with DCSK.

5. Conclusions

In this chapter, we first described a third order chaotic system with hyperbolic sine nonlinearity, then we introduced the method to expend this chaotic system to high order chaotic systems. After that, we introduced the method to construct the derived chaotic torus-chaotic systems. Finally, we introduced some applications such as random number generator algorithm, spread spectrum communication and image encryption schemes. The contribution of this chapter is that it systematically summarizes the design method, the dynamic behavior and typical engineering application of chaotic systems with hyperbolic sine nonlinearity, which may widen the current knowledge of chaos theory and engineering applications based on chaotic systems.

Acknowledgments

Jizhao Liu has received research grants from Sun Yat-sen University.

This study was supported by the Fundamental Research Funds for the Central Universities. No. 19lgpy230.

Conflict of interest

The authors declare that they have no conflict of interest.

Notes/thanks/other declarations

The authors would like to thank professor Julien Clinton Sprott for helpful discussion.

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Jizhao Liu, Yide Ma, Jing Lian and Xinguo Zhang (November 18th 2020). Chaotic Systems with Hyperbolic Sine Nonlinearity [Online First], IntechOpen, DOI: 10.5772/intechopen.94518. Available from:

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