Chapter 2 Yin Chaos Yang Chaos and Tai Ji Chaos of Chen Systems
2.5 Summary
In this thesis, Chen system is divided into Yang, Yin and “Tai Ji” systems and the
“Tai Ji” Chen system is introduced firstly. Chaos of Yang system is studied by time histories, phase portraits, bifurcation diagrams and Lyapunov exponents. Next, the past of the system, Yin system, is studied. Finally, “Tai Ji” system is studied for past and future chaos and is compared with Yang and Yin systems via numerical simulations.
It is firstly discovered that the exact scientific chaos phenomena marvelously match with the ancient Chinese philosophy, “The Yin Classic” with English translation “The book of Change” which states that the original chaotic substance, Tai Ji, begets two fundamental opposites, Yin and Yang. Correspondingly, Tai Ji chaos consists of two opposites, Yin chaos and Yang chaos.
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Fig. 2.1 Time histories of Yang Chen chaos for Yang Chen system with a=35, b=3,c=27.2 and t in seconds.
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Fig. 2.2 Projections of phase portraits and Poincaré maps of Yang Chen chaos with a=35, b=3 and c=27.2.
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Fig. 2.3 3D phase portrait of Yang Chen chaos with a=35, b=3 and c=27.2.
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Fig. 2.4 Time histories of Yin Chen chaos for Yin Chen system with a=-35, b=-3, c=-27.2 and t in seconds.
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Fig. 2.5 Projections of Phase portraits and Poincaré maps of Yin Chen chaos with a=-35, b=-3 and c=-27.2.
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Fig. 2.6 3D phase portrait of Yin Chen system with a=-35, b=-3 and c=-27.2.
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Fig. 2.7 Bifurcation diagram of chaotic Yang Chen system with b=3 and c=27.2.
Fig. 2.8 Lyapunov exponents of chaotic Yang Chen system with b=3 and c=27.2.
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Fig. 2.9 Bifurcation diagram of chaotic Yin Chen system with b=-3 and c=-27.2.
Fig. 2.10 Lyapunov exponents of chaotic Yin Chen system with b=-3 and
c=-27.2..
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Fig. 2.11 Bifurcation diagram of chaotic Yang Chen system with a=35 and
c=27.2.
Fig. 2.12 Lyapunov exponents of chaotic Yang Chen system with a=35 and c=27.2.
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Fig. 2.13 Bifurcation diagram of chaotic Yin Chen system with a=-35 and c=-27.2.
Fig. 2.14 Lyapunov exponents of chaotic Yin Chen system with a=-35 and c=-27.2.
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Fig. 2.15 Bifurcation diagram of chaotic Yang Chen system with a=35 and b=3.
Fig. 2.16 Lyapunov exponents of chaotic Yang Chen system with a=35 and b=3.
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Fig. 2.17 Bifurcation diagram of chaotic Yin Chen system with a=-35 and b=-3.
Fig. 2.18 Lyapunov exponents of chaotic Yin Chen system with a=-35 and b=-3.
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Fig. 2.19 Time histories of Tai Ji Chen system.
Fig. 2.20 Time histories of Tai Ji Chen system for -10sec < t < 10sec.
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Fig. 2.21 Projections of phase portraits of Tai Ji Chen system.
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Fig. 2.22 3D phase portrait of Tai Ji Chen system.
Fig. 2.23 Time histories of errors for Chen system.
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Fig.2.24 Bifurcation diagram of chaotic Yang Chen system for varied parameter a.
Fig. 2.25 Bifurcation diagram of chaotic Yin Chen system for varied parameter a.
Fig.2.26 Bifurcation diagram of chaotic Tai Ji Chen system for varied parameter a.
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Fig.2.27 Bifurcation diagram of chaotic Yang Chen system for varied parameter b.
Fig. 2.28 Bifurcation diagram of chaotic Yin Chen system for varied parameter b.
Fig.2.29 Bifurcation diagram of chaotic Tai Ji Chen system for varied parameter b.
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Fig.2.30 Bifurcation diagram of chaotic Yang Chen system for varied parameter c.
Fig. 2.31 Bifurcation diagram of chaotic Yin Chen system for varied parameter c.
Fig.2.32 Bifurcation diagram of chaotic Tai Ji Chen system for varied parameter c.
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There are two chaotic systems, partner system A and partner system B.
The partner A is given by
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3.2 Simulation Results of Six Different Chaotic Systems
To learn more about the process of synchronization, six different chaotic systems are selected to obtain this synchronization.
Add 2 in the second equation of Eq.(3.6), a new Ge-Ku-Mathieu-Chen system obtained: portrait and time histories of states are shown in Figs 3.1~3.2.
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Similarly, Sprott 19-F system and Sprott G-H [23] system can be generated.
Sprott 19-F system chaotic phase portrait and time histories of states are shown in Figs 3.3~3.6.
3.3 Multiple Symplectic Derivative Synchronization Result
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to keep the error dynamics always in first quadrant. Our goal islim𝑡→∞𝑒𝑖 lim𝑡→∞ 𝐺𝑖 − 𝐹𝑖 + 𝐾𝑖 0 𝑖 1 2 3 4 5 6 (3.14)
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which is a negative definite function in first quadrant. The results are shown in Figs 3.7~3.16 3.20~3.23, it is obviously that the errors from traditional method are bigger than that34
from new strategy.
Table 3.1. Error data for traditional method from 99.90s to 99.99s after the action of controllers. obtained. All data are picked from 99.90 to 99.99s with sampling time 0.01 s. From these data, the superiority of new strategy is remarkable.
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3.5 Summary
In this thesis, a new synchronization, multiple symplectic derivative synchronization, to achieve chaos control by part ial pragm ati c stabilit y stability is proposed. By using the parti al region stability theory, the controllers become simple linear homogeneous functions of error states and have less simulation error data than traditional method. Furthermore, the multiple symplectic derivative synchronization of chaos can be used to increase the security of secret communication.
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Fig. 3.1 Phase portrait of Ge-Ku-Mathieu-Chen system.
Fig. 3.2 Time histories of Ge-Ku-Mathieu-Chen system.
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Fig. 3.3 Phase portrait of Sprott 19-F system.
Fig. 3.4 Time histories of Sprott 19-F system.
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Fig. 3.5 Phase portrait of Sprott G-H system.
Fig. 3.6 Time histories of Sprott G-H system.
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Fig. 3.7 Time histories of e1, e2, e3 before control.
Fig. 3.8 Time histories of e1, e2, e3 before and after control for new strategy.
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Fig. 3.9 Time histories of e4, e5, e6 before control.
Fig. 3.10 Time histories of e4, e5, e6 before and after control for new strategy.
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Fig. 3.11 Phase portrait of e1, e2 before control.
Fig. 3.12 Phase portrait of e1, e2 before and after control for new strategy.
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Fig. 3.13 Phase portrait of e3, e4 before control.
Fig. 3.14 Phase portrait of e3, e4 before and after control for new strategy.
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Fig. 3.15 Phase portrait of e5, e6 before control.
Fig. 3.16 Phase portrait of e5, e6 before and after control for new strategy..
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Fig. 3.17 Phase portrait of error before and after control for traditional method.
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Fig. 3.18 Time histories of e1, e2, e3 for traditional method.
Fig. 3.19 Time histories of e4, e5, e6 for traditional method.
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Fig. 3.20 Time histories of e1, e2, e3 for new strategy.
Fig. 3.21 Time histories of e1, e2, e3 for traditional method.
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Fig. 3.22 Time histories of e4, e5, e6 for new strategy.
Fig. 3.23 Time histories of e4, e5, e6 for traditional method.
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To learn more about the process of synchronization, two different chaotic systems are selected to obtain this synchronization. chaotic phase portraits and time histories of states are shown in Figs 4.5~4.8. is a function of t shown as: 2 .
Given ̇ ̇ ̇ ̈ ̈ ̈ nd ̇ ̇ ̇ ̈ ̈ ̈ as:
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always keeps in eighth quadrant. Its chaotic phase portrait and time histories of states are shown in Figs 4.9~4.10. The quadrants are defined as following:The sign symbols of the first quadrant is + + + .
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which is a negative definite function in eighth quadrant. The results are shown in Figs 4.12~4.1351
To comparison the new and traditional methods, error data from 99.90s to 99.99s are given in Table 4.1 and Table 4.2.
Table 4.2. Error data for new strategy from 99.90 to 99.99s after the action of controllers.
Error for new strategy
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smaller than the error data for traditional method. It is obvious that new strategy has better efficiency of convergence in Figs 4.16~4.17. From these synchronization results, the superiority of new strategy is remarkable.It can be proved that new strategy are better than traditional method using error data in section 4.3.
New strategy are superior than traditional method in
high convergence rate and low error value, it can be applied widely in the future.
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Fig. 4.1 Time histories of Ge-Ku-Mathieu system.
Fig. 4.2 Projections of phase portrait of Ge-Ku-Mathieu system.
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Fig. 4.3 Projections of phase portrait of Ge-Ku-Mathieu system.
Fig. 4.4 Phase portrait of Ge-Ku-Mathieu system.
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Fig. 4.5 Time histories of Chen system.
Fig. 4.6 Projections of phase portrait of Chen system.
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Fig. 4.7 Projections of phase portrait of Chen system.
Fig. 4.8 Phase portrait of Chen system.
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Fig. 4.9 Time histories of error function before control for new strategy.
Fig. 4.10 Phase portrait of error function before control for new strategy.
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Fig. 4.11 Definition of the eight quadrants.
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Fig. 4.12 Time histories of error function before and after control for new strategy.
Fig. 4.13 Phase portrait of error function before and after control for new strategy.
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Fig. 4.14 Time histories of error function before and after control for new strategy.
Fig. 4.15 Phase portrait of error function before and after control for new strategy.
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Fig. 4.16 Time histories of error function before and after control for new strategy.
Fig. 4.17 Phase portrait of error function before and after control for new strategy.
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Chapter 5 Gen Trigram Symplectic Derivative Chaos
Synchronization by Partial Region Stability Theory
5.1 Chaos of Gen Trigrams
In order to learn chaos of Gen trigram in Fig 5.1, Chen system, Sprott G system and Sprott H system are selected. The first line chaos is an "unbroken" line chaos. It can be shown by combining Yin and Yang Chen chaos. Yang Chen system is chosen in first line. (0.5, 0.26, 0.35) and parameters a=-35, b=-3 and c=-27.2. The Yin and Yang chaotic
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behaviors are shown in Figs 5.3~5.4 by time histories and phase portraits, where − : −∞ → +∞ .
The second line chaos is a "broken" line chaos, it can be shown by linear feedback synchronization Yin and Yang chaos. Sprott G system is chosen for second line.
The states of linear feedback synchronization for Yang Sprott G system and Yang Sprott G system are shown in Figs 5.5~5.6.
The third line chaos is "broken" line chaos, it can be shown by linear feedback synchronization Yin and Yang chaos. Sprott H system is chosen in first line.
Sprott H system
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The states of linear feedback synchronization for Yang Sprott H system and Yang Sprott H system are shown in Figs 5.7~5.8.
5.2 Synchronization Result for Gen Trigram
By multiple symplectic derivative synchronization of three lines, Gen trigram synchronization can be achieved. First and second line chaos synchronization are used firstly.
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Second line and third line are "broken" line, it just can get Yang synchronization.
Given nd for Sprott G and Sprott H system as:
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and phase portraits, where − : −∞ → +∞ .
The second line chaos is an "unbroken" line chaos. It can be shown by combining Yin and Yang chaos. Yang Sprott 19 system is chosen in second line.
Yang Sprott 19 system
The third line chaos is "broken" line chaos, it can be shown by synchronization Yin and Yang chaos. Sprott F system is chosen in third line
Yang Sprott F system
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The states of linear feedback synchronization for Yang Sprott H system and Yang Sprott H system are shown in Figs 5.15~5.16.
5.4 Synchronization Result for Xun Trigrams
By multiple symplectic derivative synchronization of three lines can get Xun trigram synchronization. First and second line chaos synchronization are used firstly.
Given nd as:
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Second line and third line chaos synchronization also use the same way.
Given nd for Yang Sprott 19 and Sprott F system as:
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By multiple symplectic derivative synchronization of Gen trigram and Xun trigram, Gen-Xun hexagram synchronization can be achieved.
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Combining Yin and Yang synchronization can get Gen-Xun hexagram synchronization. The results are shown in Figs. 5.19.
5.6 Summary
In this chapter, “Gen trigram” and “Xun trigram” system and its synchronization by six classic chaos systems, Chen, Sprott F, Sprott G, Ge-Ku-Mathieu, Sprott 19 and Sprott F system, are introduced firstly. Gen and Xun trigram are studied for past and future chaos and are compared with Yang and Yin systems via numerical simulations. The multiple derivative synchronizations of three line Gen and of three line Xun trigram are obtained by partial stability theory. Finally, six line Gen-Xun hexagram are produced by synchronizations of Gen trigrams and Xun trigrams.
We firstly attempt to study the “eight trigrams” in “The Book of Changes” by
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the chaos synchronization. “The Book of Changes” is the head of “the Five Classics”, it will give extensive researches in chaos synchronization in the future.
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Fig. 5.1 Gen trigrams and it’s chosen system.
Fig. 5.2 Xun trigrams and it’s chosen system.
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Fig.5.3 Time histories of Yin and Yang Chen system.
Fig. 5.4 Projections of phase portrait of Yin and Yang Chen system.
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Fig. 5.5 Time histories of Sprott G system before and after control.
Fig. 5.6 Projections of phase portrait of Yin and Yang Sprott G system.
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Fig. 5.7 Time histories of Sprott H system before and after control.
Fig. 5.8 Projections of phase portrait of Yin and Yang Sprott H system.
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Fig. 5.9 Time histories of error in Chen and Sprott G systems.
Fig. 5.10 Time histories of error in Sprott G and Sprott H systems.
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Fig. 5.11 Time histories of Yin and Yang Ge-Ku-Mathieu system.
Fig. 5.12 Projections of phase portrait of Yin and Yang Ge-Ku-Mathieu system.
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Fig. 5.13 Time histories of Yin and Yang Sprott 19 system.
Fig. 5.14 Projections of phase portrait of Yin and Yang Sprott 19 system.
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Fig. 5.15 Time histories of Sprott F system before and after control.
Fig. 5.16 Projections of phase portrait of Yin and Yang Sprott F system.
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Fig. 5.17 Time histories of error in Ge-Ku-Mathieu and Sprott 19 systems.
Fig. 5.18 Time histories of error in Sprott 19 and Sprott F systems.
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Fig. 5.19 Time histories of error in Gen-Xun hexagram system.
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Chapter 6 Conclusions
In this thesis, Chen system is presented into Yang, Yin and “Tai Ji” systems and the “Tai Ji” Chen system is introduced firstly. Chaos of Yang system is studied by time histories, phase portraits, bifurcation diagrams and Lyapunov exponents. Next, the past of the system, Yin system, is studied. Finally, “Tai Ji” system is studied for past and future chaos and is compared with Yang and Yin systems via numerical simulations.
A new synchronization, multiple symplectic derivative synchronization, to achieve chaos control by parti al pragm ati c stabilit y stability is proposed. By using the parti al region stability theory, the controllers become simple linear homogeneous functions of error states and have less simulation error data than traditional method. Furthermore, the multiple symplectic derivative synchronization of chaos can be used to increase the security of secret communication.
In chapter 4, new synchronization and new strategy is presented. New synchronization have variable time scale τ and new strategy use variable quadrant to arrive synchronization. By using the partial region stability theory, the controllers become simple functions. The simulation results are given.
It can be proved that new strategy are better than traditional method using error data in section 4.3.
New strategy are superior than traditional method in high convergence rate and low error value, it can be applied widely in the future.
In chapter 5, “Gen trigram” and “Xun trigram” system and its synchronization by six classic chaos systems, Chen, Sprott F, Sprott G, Ge-Ku-Mathieu, Sprott 19 and Sprott F system, are introduced firstly. Gen and Xun trigram are studied for past and future chaos and are compared with Yang and Yin systems via numerical simulations.
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The multiple derivative synchronizations of three line Gen and of three line Xun trigram are obtained by partial stability theory. Finally, six line Gen-Xun hexagram are produced by synchronizations of Gen trigrams and Xun trigrams.
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