A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems

Jay Prakash Singh; Binoy Krishna Roy

doi:10.1007/s11071-018-4249-3

Nonlinear Dynamics

pp 1–28 | Cite as

A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems

Authors
Authors and affiliations

Jay Prakash SinghEmail author
Binoy Krishna Roy

Original Paper

First Online: 12 April 2018

Received: 14 August 2017

Accepted: 28 March 2018

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Abstract

This paper attempts to construct a new 3-D chaotic system which is easily hardware realisable and fulfil the requirement of a real-life application. The proposed system is relatively more chaotic (based on the first Lyapunov exponent) and has larger bandwidth than 50 available chaotic systems. Lyapunov spectrum and bifurcation diagram of the system reveal that it has chaotic behaviour for a wider range of its parameters. Such characteristic is helpful for an easy hardware realisation of the system. It is to be noted that the reported systems with hidden attractors are not considered here for the comparison. The proposed system has more complexity and disorder due to several unique properties like asymmetry to principle coordinates, dissimilar and asymmetrical equilibria, and non-uniform contraction and expansion of volume in phase space. The proposed system also exhibits asymmetric pairs of coexisting attractors during its operation in two modes. The new system has different routes to chaos including crisis, an inverse crisis, period-doubling and reverse period-doubling routes to chaos with the variation of parameters. MATLAB simulation results confirm the claims, and the results of hardware circuit realisation validate the simulation results. An application of the new system is shown by masking and retrieving an information signal. It is also shown that the proposed system is better than a well-known Lorenz chaotic system for this application. A system with the above unique properties is rare in the literature.

Keywords

New chaotic system: more chaotic Wider spectrum Hardware implementation Crisis route to chaos Inverse crisis route to chaos Communication breaking

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Appendix A: Comparison with other chaotic systems

Lyapunov exponents (LEs), Lyapunov dimension (LD), bandwidth (BW) of proposed system (1) are compared with 50 reported systems and are presented in Table 4, 5 and 6.

Table 4

Different chaotic systems with Lyapunov exponents, Lyapunov dimension, bandwidth and other properties

Sl. no.	System	Dynamics	LEs	LD	BW
1.	Lorenz [23]	\({\dot{x}}_1=10\left( x_2-x_1\right) \),	0.9022,	2.062	4
		\({\dot{x}}_2=28x_1-x_2-x_1x_3\)	0.003,
		\({\dot{x}}_3=-\,2.667x_3+x_1x_2\)	\(-\,14.569\)
2.	Rossler [24]	\({\dot{x}}_1=-\,x_2-x_3\),	0.069,	2.013	1
		\({\dot{x}}_2=x_1+0.2x_2\)	\(-\,0.0002,\)
		\({\dot{x}}_3=0.2+x_3{(x}_1-5.7)\)	\(-\,5.392\)
3.	Chen [16]	\({\dot{x}}_1=35\left( x_2-x_1\right) \),	0.6971,	2.1752	6
		\({\dot{x}}_2=-\,7x_1+28x_2-x_1x_3\),	\(-\,0.039,\)
		\({\dot{x}}_3=-\,3x_3+x_1x_2\)	\(-\,3.7549\)
4.	Lu [14]	\({\dot{x}}_1=36\left( x_2-x_1\right) \),	1.5046,	2.066	7
		\({\dot{x}}_2=20x_2-x_1x_3\)	0,
		\({\dot{x}}_3=-\,3x_3+x_1x_2\)	\(-\,22.504\)
5.	Lorenz [31]	\({\dot{x}}_1=16\left( x_2-x_1\right) \),	2.160,	2.066	6
		\({\dot{x}}_2=45.92x_1-x_2-x_1x_3\)	0,
		\({\dot{x}}_3=-\,4x_3+x_1x_2\)	\(-\,32.4\)
6.	Lorenz [31]	\({\dot{x}}_1=10\left( x_2-x_1\right) \),	1.253,	2.0895	7
		\({\dot{x}}_2=142x_1-x_2-x_1x_3\)	0,
		\({\dot{x}}_3=-\,2.667x_3+x_1x_2\)	\( -14\)
7.	Liu et al. [17]	\({\dot{x}}_1=10\left( x_2-x_1\right) \),	1.64328,	2.1161	1
		\({\dot{x}}_2=40x_1-x_1x_3\),	0,
		\({\dot{x}}_3=-\,2.5x_3+4x^2_2\)	\(-\,14.142\)
8.	Modified Lu [47]	\({\dot{x}}_1=35\left( x_2-x_1+x_2x_3\right) \),	0.967,	2.034	7
		\({\dot{x}}_2=14x_2-x_1x_3\),	0,
		\({\dot{x}}_3=-\,5x_3+x_1x_2\)	\(-\,27.9671\)
9.	Tigan et al. [18]	\({\dot{x}}_1=2.1\left( x_2-x_1\right) \),	0.37,	2.1205	1
		\({\dot{x}}_2=30x_1-x_2-x_1x_3\),	0,
		\({\dot{x}}_3=x_1x_2-0.6x_3\)	\(-\,3.07\)
10.	Cai et al. [41]	\({\dot{x}}_1=20\left( x_2-x_1\right) \),	2.3947,	2.1640	6
		\({\dot{x}}_2=14x_1+10.6x_2-x_1x_3\),	0,
		\({\dot{x}}_3=-\,2.8x_3+x^2_1\)	\(-\,14.5943\)
11.	Zhou et al. [38]	\({\dot{x}}_1=10\left( x_2-x_1\right) \),	0.7822,	2.0970	3
		\({\dot{x}}_2=16x_1-x_1x_3\),	0.01507,
		\({\dot{x}}_3=x_1x_2-x_3\)	\(-\,9.5257\)
12.	Li [48]	\({\dot{x}}_1=40\,(x_2-x_1+0.16x_1x_3,\)	0.231,	2.11	5
		\({\dot{x}}_2=55x_1+20x_2-x_1x_3\),	0,
		\({\dot{x}}_3=-\,0.65x^2_1+x_1x_2+1.83x_3\)	\(-\,1.9872\)
13.	Chen et al. [49]	\({\dot{x}}_1=4.6x_1+x_2-x_2x_3\),	1.4,	2.010	8.5
		\({\dot{x}}_2=-\,12x_2-x_3+x_1x_3\),	0,
		\({\dot{x}}_3=-\,x_1-5x_3+x_1x_2\ \)	\(-\,13.81\)
14.	Li et al. [40]	\({\dot{x}}_1=8.17(x_2-x_1)\),	3.06,	2.291	8
		\({\dot{x}}_2=-\,x_2+x_1x_3\),	0,
		\({\dot{x}}_3=-\,x_1x_2+290.5-cx_3\)	\(-\,10.54\)
15.	Li et al. [50]	\({\dot{x}}_1=5(x_2-x_1)\),	0.77, 0,	2.10	8
		\({\dot{x}}_2=55x_1-5x_1x_3\),	\(-\,7.77\)
		\({\dot{x}}_3=x_1x_2-2x_3\)
16.	Munmuangsaen et al. [51]	\({\dot{x}}_1=-\,5x_1+5x_2\),	1.4913, 0,	2.2300	7
		\({\dot{x}}_2=-\,x_1x_3\),	\( -\,6.4939\)
		\({\dot{x}}_3=-\,90+x_1x_2\)
17.	Liu et al. [52]	\({\dot{x}}_1=-\,x_1-x^2_2\),	0.4412, 0,	2.1192	2
		\({\dot{x}}_2=2.5x_2-4x_1x_3\),	\(-\,3.9418\)
		\({\dot{x}}_3=-\,5x_3+4x_1x_2\)
18.	Dadras et al. [53]	\({\dot{x}}_1=x_2-3x_1+2.7x_2x_3\)	\(Max LE<2\)		6
		\({\dot{x}}_2=4.7x_2-x_1x_3+x_3\)
		\({\dot{x}}_3=-\,9x_3+2x_1x_2\)

Table 5

Different chaotic systems with Lyapunov exponents, Lyapunov dimension and bandwidth

Sl. no.	System	Dynamics	LEs	LD	BW
19.	Zhu et al. [37]	\({\dot{x}}_1=-\,x_1-1.5x_2+x_2x_3\),	0.6747, 0,	2.166	4
		\({\dot{x}}_2=2.5x_2-x_1x_3\),	\(-\,4.073\)
		\({\dot{x}}_3=-\,4.9x_3+x_1x_2\)
20.	Liu et al. [52]	\({\dot{x}}_1=10\left( x_2-x_1\right) \),	1.0742, 0,	2.0274	4
		\({\dot{x}}_2=16x_1-x_1x_3\),	\(-\,39.074\)
		\({\dot{x}}_3=-2.66x_3+x_1x_2 \)
21.	Pan [19]	\({\dot{x}}_1=60\left( x_2-x_1\right) +0.4x_1x_3\),	1.07, 0,	2.06	5
		\({\dot{x}}_2=-\,35x_1+25x_2-x_1x_3\)	\(-\,15.572\)
		\({\dot{x}}_3=-\,0.65x^2_1+0.83x_3+x_1x_2\)
22.	Pehlivan [20]	\({\dot{x}}_1=\left( x_2-x_1\right) \),	0.198, 0,	2.276	1
		\({\dot{x}}_2=0.5x_2-x_1x_3\),	\( -\,0.68\)
		\({\dot{x}}_3=x_1x_2-0.5\)
23.	Wei et al. [54]	\({\dot{x}}_1=-\,x_2,\)	0.0793, 0,	2.0528	1
		\({\dot{x}}_2=x_1+x_3\),	\(-\,1.5034\)
		\({\dot{x}}_3=2x^2_2+x_1x_3-0.35\)
24.	Li et al. [55]	\({\dot{x}}_1=10\left( x_2-x_1\right) ,\)	0.4265, 0,	2.0574	2
		\({\dot{x}}_2=6x_2-x_1x_3\)	\( -7.42645\)
		\({\dot{x}}_3=-\,3x_3+x^2_2\)
25.	San-um et al. [56]	\({\dot{x}}_1=x_2-x_1\)	0.4299, 0,	2.3004	7
		\({\dot{x}}_2=x_3\mathrm{tanh}(x_1)\)	\(-\,1.4294\)
		\({\dot{x}}_3=-\,60+x_2x_3+\left\| x_2\right\| \)
26.	Li et al. [39]	\({\dot{x}}_1=-\,16x_1+0.5x_2x_3\),	1.868, 0,	2.1051	1
		\({\dot{x}}_2=10x_2-6x_1x_3\),	\(-\,17.736\)
		\({\dot{x}}_3=-\,5x_3+18x_2\)
27.	Kim et al. [36]	\({\dot{x}}_1=30\left( x_2-x_1\right) \),	0.9727, 0,	2.0335	6
		\({\dot{x}}_2=15x_2+x_1x_2\),	\(-\,26.9007\)
		\({\dot{x}}_3=-\,11x_3-x^2_1\)
28.	Liu et al. [57]	\({\dot{x}}_1=-\,3x_1+x_2x_3-x^2_2\),	\(\mathrm{Max\,LE}<2.5\),		2
		\({\dot{x}}_2=5x_2-x_1x_3+16x_3\)
		\({\dot{x}}_3=x_1x_2-10x_3\)
29.	Abooee [58]	\({\dot{x}}_1=10\left( x_2-x_1\right) +x_2\) \(x^2_{3}\)	0.58, 0,	2.037	6
		\({\dot{x}}_2=5x_1-x_1x^2_3\)	\(-\,15.59\)
		\({\dot{x}}_3=-\,6x^2_1-5x_3\)
30.	Qiao et al. [59]	\({\dot{x}}_1=10\left( x_2-x_1\right) \)	0.660, 0,	2.0374	4
		\({\dot{x}}_2=-\,7x_2-x_1x_3\)	\(-\,17.632\)
		\({\dot{x}}_3=-\,50+x_1x_2-0.2x^2_1\)
31.	Xu et al. [35]	\({\dot{x}}_1=\mathrm{ln}(0.1+e^{y-x})\),	0.1139, 0,	2.5003	1
		\({\dot{x}}_2=x_1x_3\),	\( -\,0.2970\)
		\({\dot{x}}_3=0.25-x_1x_2\)
32.	Deng et al. [60]	\({\dot{x}}_1=x_2\),	0.1928, 0,	2.117	1
		\({\dot{x}}_2=x_3-0.5x_2\),	\(-\,1.635\)
		\({\dot{x}}_3=-\,6x_1-2.85x_2\)
		\(0.5x_3+3x^2_1\)
33.	Wang et al. [61]	\({\dot{x}}_1=-\,x^2_2-x^2_3-0.25x_1+2\)	\(\mathrm{Max\,LE}<0.5\)		2
		\({\dot{x}}_2=x_1x_2-{4x}_1x_3-\,x_2+1\)
		\({\dot{x}}_3=4x_1x_2+x_1x_3-x_3\)
34.	Zhou et al. [62]	\({\dot{x}}_1=14x_2\)	\(\mathrm{Max\,LE}<1\)		7
		\({\dot{x}}_2=-\,x_3sgn(x_1)-cx_2\),	\(c=0.5{-}4.5\)
		\({\dot{x}}_3=3x^2_1-1\)
35.	Liu et al. [63]	\({\dot{x}}_1=-\,x_1+2x_2\)	0.4135, 0,	2.2922	1.5
		\({\dot{x}}_2=9x_1-x_1x_3\),	\(-\,1.4129\)
		\({\dot{x}}_3=x_1x_2-x_3\)

Table 6

Different chaotic systems with Lyapunov exponents, Lyapunov dimension and bandwidth

Sl. no.	System	Dynamics	LEs	LD	BW
36.	Gholizadeh [64]	\({\dot{x}}_1=-\,16x_1+0.5x_2x_3\)	1.8652, 0,	2.105	4
		\({\dot{x}}_2=10x_2-{6x}_1x_3\),	\(-\,17.736\)
		\({\dot{x}}_3=18x^2_2-5x_3\)
37.	Wu et al. [65]	\({\dot{x}}_1=6\left( x_2-x_1\right) \),	0.3564, 0,	2.0145	7
		\({\dot{x}}_2=9x_1-x_1x_3\),	\(-\,7.330\)
		\({\dot{x}}_3=x_1x_2-x_3\)
38.	Bhalekar [66]	\({\dot{x}}_1=-\,2.667{x_1-x}^2_2,\)	0.9635, 0,	2.06	3
		\({\dot{x}}_2=10\left( x_3-x_2\right) \)	\(-\,14.634\)
		\({\dot{x}}_3=x_1x_2-x_3+27.3x_2\)
39.	LR [29]	\({\dot{x}}_1=19x_2-20x_1-x_3, \)	0.0181, 0,	2.011	6
		\({\dot{x}}_2=20x_1+8x_2-20x_1x_3\)	\( -19.52\)
		\({\dot{x}}_3=5x_1x_2-8.5x_3+ x_1(x_3-8)\)
40.	Su [67]	\({\dot{x}}_1=-18x_1+5x_2, \)	0.6, 0,	2.044	5
		\({\dot{x}}_2=10x_2-2x_1x_3\)	\( -13.6\)
		\({\dot{x}}_3=-\,x_3+5x_2^2\)
41.	Cicek [68]	\({\dot{x}}_1=x_2-0.6x_1+3x_1x_3, \)	0.4379, 0	2.35	6
		\({\dot{x}}_2=-\,10x_1x_3-0.3x_1+x_2x_3+1\)	\(-\,1.228\)
		\({\dot{x}}_3=1+x_1x_2\)
42.	Akgul [69]	\({\dot{x}}_1=1.8x_2-1.8x_1, \)	0.231, 0	2.024	5
		\({\dot{x}}_2=-\,7.2x_2+x_1x_3+0.02x_3\)	\(-\,9.3662\)
		\({\dot{x}}_3=2.7x_3+x_1^3x_2-0.07x_3^2\)
43.	Zhang [70]	\({\dot{x}}_1=-\,2x_1+10x_2x_3, \)	1.059, 0	2.024	5
		\({\dot{x}}_2=-\,6x_2^3+3x_1x_3\)	\(-\,18.293\)
		\({\dot{x}}_3=3x_3-x_1x_2\)
44.	Golamin et al. [71]	\({\dot{x}}_1=-\,ax_{1}+bx_{2}x_{3}\),	0.2404, 0	2.055	6
		\({\dot{x}}_2=x_2-7x_1x_3\),	\(-\,4.223\)
		\({\dot{x}}_3=x_3+2x_1x_2\)
45.	Lai et al. [72]	\({\dot{x}}_1=5(x_2-x_1)\),	1.2199, 0,	2.196	6
		\({\dot{x}}_2=2x_1x_3(x_{3}^2-9)\),	\(-\,6.2199\)
		\({\dot{x}}_3=1-x_1x_2\)
46.	Tuna et al. [73]	\({\dot{x}}_1=x_2(x_3-1.3),\)	0.8, 0	2.19	4
		\({\dot{x}}_2=-\,x_1(x_3+1.3),\)	\(-\,4.2\)
		\({\dot{x}}_3=-\,x_2(1.3x_1-x_2)-4(x_3-1.3)\)
47.	Vaidyanathan et al. [74]	\({\dot{x}}_1=40\,(x_2-x_1)+0.16x_1x_3,\)	1.501, 0	2.511	6
		\({\dot{x}}_2=55x_1+12x_2-x_1x_3\),	\(-\,2.936\)
		\({\dot{x}}_3=24-0.65x_1^2+x_1x_2+2x_3\)
48.	Vaidyanathan et al. [75]	\({\dot{x}}_1=30(x_2-x_1)+14x_2x_3,\)	3.2085, 0,	2.1357	4
		\({\dot{x}}_2=14x_2-x_1x_3^2\),	\(-\,23.635\)
		\({\dot{x}}_3=-\,4.5x_3+x_1x_2\)
49.	Volos et al. [76]	\({\dot{x}}_1=-x_2,\)	0.2124, 0,	2.1733	2.5
		\({\dot{x}}_2=-\,x_3,\)	\(-\,1.2254\)
		\({\dot{x}}_3=-\,x_1-0.7x_3+0.00038 \sinh (x_2)\)
50.	Kengne et al. [77]	\({\dot{x}}_1=-x_3^2+7,\)	\(\mathrm{Max\,LE}<0.20\)		3
		\({\dot{x}}_2=x_1^2-x_3^2+1\),
		\({\dot{x}}_3=x_1^2-x_2^2+x_3^2\)
51.	System (1)	\({\dot{x}}_1=32x_2-{33x}_1-x_3,\)	5.5897, 0,	2.16	13
		\({\dot{x}}_2=46.6x_1+12x_2\pm 10x_1x_3,\)	\( -\,34.486\)
		\({\dot{x}}_3=\pm 10x_1x_2-6x_3+ x_1(\pm x_3-11)\)

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Jay Prakash Singh
- 1
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Binoy Krishna Roy
- 1

1.Department of Electrical EngineeringNational Institute of Technology SilcharSilcharIndia

A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems

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