Chaotization of a New Ikeda-Mackey-Glass System by Chaos Signals as Parameters
4.3 Simulation Results
By using Eq.(4.1), the periodic motion of a new Ikeda-Mackey-Glass system is described as follows:
A periodic motion is obtained as shown by phase portraits in Fig.4.1, time histories in Fig.4.2 and Fig.4.3, and bifurcation diagram in Fig.4.4, where α1=25, β
=24.8 , k = 13.4 , 1 α2=4.7, b =1.2348,c=10, k =8, 2 τ1=5 and τ2=1.
By using Eq.(4.2), the chaotic motion of a new Ikeda-Mackey-Glass system is described as follows:
A chaotic motion is obtained as shown by phase portrait in Fig.4.5 and bifurcation
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where k =8. A new Ikeda-Mackey-Glass system(4.6) is a chaotic system by 2 parameter replacement method.
CASE I: p1=y1
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.6 and time histories in Fig.4.7 and Fig.4.8.
CASE II: p1=y2
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.9 and time histories in Fig.4.10 and Fig.4.11.
CASE III: p1=y 12
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.12 and time histories in Fig.4.13 and Fig.4.14.
CASE IV: p1=y 22
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.15 and time histories in Fig.4.16 and Fig.4.17.
CASE V:p1 = y1y2
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.18 and time histories in Fig.4.19 and Fig.4.20.
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CASE VI: p1 = y2y2(t−τ), where τ =30 sec.
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.21 and time histories in Fig.4.22 and Fig.4.23.
CASE VII: p1 =cosy1cosy2
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.24 and time histories in Fig.4.25 and Fig.4.26.
CASE VIII: p1 =R+y2, where R is the Rayleigh noise.
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.27 and time histories in Fig.4.28 and Fig.4.29
CASE IX: p1 = Ry2, where R is the Rayleigh noise.
The chaotic simulation results obtained is achieved in phase portrait in Fig.4.30 and time histories in Fig.4.31 and Fig.4.32.
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Fig.4.1. Phase portrait of an IMG system in period 2 when α1=25, β =24.8 ,K = 1 13.4 , α2=4.7, b =1.2348,c=10, K =8, 2 τ1=5 and τ2=1.
Fig.4.2. The time history of x1 of an IMG system in period 2 when α1=25, β =24.8 , K = 13.4 , 1 α2=4.7, b =1.2348,c=10, K =8, 2 τ1=5 and τ2=1.
x1
x 2
x 1
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Fig.4.3. The time history of x2 of an IMG system in period 2 when α1=25, β =24.8 , K = 13.4 , 1 α2=4.7, b =1.2348,c=10, K =8, 2 τ1=5 and τ2=1.
.
Fig.4.4. The bifurcation diagram of an IMG system when α1=25,β =24.8, α2
=4.7, b =1.2348,c=10, K =8, 2 τ1=5 and τ2=1.
x 2
k1
x 2
23
Fig4.5. An IMG chaotic attractor when α1=25, β =24.8 ,K = 14.1 , 1 α2=4.7, b
=1.2348,c=10, K =8, 2 τ1=5 and τ2=1.
Fig4.6. An IMG chaotic attractor when parameter is a chaos signal for CASE I.
x1
x 2
z1
z 2
24
Fig4.7. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE I.
Fig.4.8. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE I.
. z 1
z 2
25
Fig.4.9. An IMG chaotic attractor when parameter is a chaos signal for CASE II.
Fig.4.10. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE II.
. z1
z 2
z 1
26
Fig.4.11. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE II.
Fig.4.12. An IMG chaotic attractor when parameter is a chaos signal for CASE III.
z 2
z1
z 2
27
Fig.4.13. The time history of z1 of an IMG system in chaotic behavior when
parameter is a chaos signal for CASE III.
Fig.4.14. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE III.
z 1
z 2
28
Fig.4.15. An IMG chaotic attractor when parameter is a chaos signal for CASE IV.
Fig.4.16. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE IV.
z1
z 2
z 1
29
Fig.4.17. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE IV.
Fig.4.18. An IMG chaotic attractor when parameter is a chaos signal for CASE V.
z 2
z1
z 2
30
Fig.4.19. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE V.
Fig.4.20. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE V.
z 1
z 2
31
Fig.4.21. An IMG chaotic attractor when parameter is a chaos signal for CASE VI.
Fig.4.22. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VI.
. z1
z 2
z 1
32
Fig.4.23. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VI.
Fig.4.24. An IMG chaotic attractor when parameter is a chaos signal for CASE VII.
z 2
z1
z 2
33
Fig.4.25. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VII.
Fig.4.26. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VII.
z 1
z 2
34
Fig.4.27. An IMG chaotic attractor when parameter is a chaos signal for CASE VIII.
Fig.4.28. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VIII.
z1
z 2
z 1
35
Fig.4.29. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal for CASE VIII.
Fig.4.30. An IMG chaotic attractor when parameter is a chaos signal for CASE IX.
z 2
z1
z 2
36
Fig.4.31. The time history of z1 of an IMG system in chaotic behavior when parameter is a chaos signal CASE IX.
Fig.4.32. The time history of z2 of an IMG system in chaotic behavior when parameter is a chaos signal CASE IX.
z 1
z 2
37