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Control of Chaotic Population Dynamics Using OPCL Method
Copyright ~ IFAC Large Scale Systems: Theory and Applications. Bucharest. Romania. 2001
CONTROL OF CHAOTIC POPULATION DYNAMICS USING OPCL METHOD loan Grosu * Dragos Arotaritei ** Constantin Corduneanu ***
o
Bioengineering, University of Medicine and Pharmacy "Cr. T. Popa", RO-6600, Ia§i, Romania 00 Computer Science, Aalborg University Esbergerg, Denmark 0** Bioengineering, Univ. of Medicine and Pharmacy "Cr. T.Popa" Ia§i, Romania
Abstract: The OPCL method is applied to reproducing a recorded chaotic dynamics and for controlling a chaotic dynamics. Copyright ~20011FAC Keywords: closed-loop, .chaos, discrete systems, dynamics, open-loop
1. INTRODUCTION
OPCL (Chen, 1996; Chen and Liu, 1998; Tian, et al., 2000; Wang and Wang, 1999; Xiaofeng and Lai, 2000; Zonghua and Shigang, 1997).
The last decade has known a considerable interest for control of chaotic systems (Chen and Dong, 1998; Chen, 1999; Schuster, 2000; Boccaletti, 2(00). Chaotic systems have the distinguishing feature of unpredictability that is much undesirable in engineering. Among several methods (Chen and Dong, 1998; Chen, 1999; Schuster, 2000; Boccaletti, 2(00) the OPCL (OpenPlus-Closed-Loop) method (Jackson and Grosu, 1995a; Grosu, 1997; Chen and Dong, 1998) can be mentioned. This method is used for model based systems. It offers a driving in order to obtain any desired dynamics. OPCL is mathematically justified and is very robust against noise (Grosu, 1997). It works for continuous systems described by a system of ordinary differential equations and for discrete systems described by a system of nonlinear difference equations. As a particular case of this control is the synchronization of a dynamics with a prerecorded one. So the problem of unpredictability is solved in terms of OPCL (Grosu, 1997). This was shown for Lorenz system and for logistic map dynamics. Also we used OPCL for synchronization of two Chua circuits and for synchronization of a chain of neural oscillators (Dragoi and Grosu, 1998; Jackson and Grosu, 1995b). Others used or modified or improved
2. OPCL METHOD The OPCL method (Jackson and Grosu, 1995a; Grosu, 1997; Chen and Dong, 1998) is a modelbased method. Let's consider a discrete system: (1)
We want to drive this system to a goal dynamics gn' The driving term is:
D(Xn,gn,gn+d = gn+l - F(gn) +(B -
~(gn) )(xn gn
gn)
+ (2)
where B is a constant matrix with all eigenvalues of subunitary modulus. The drived system:
assures the convergence of Xn to gn' Based on this result a method of synchronization has been obtained (Grosu, 1997). If the dynamics of (1) is recorded let's say it is Xn and we want to 339
reproduce it we can drive the system (1) with the driving term:
(4)
In addition this synchronization is robust against noise. This means that if the driving term is affected by an additive noise the synchronization is still achieved. Recently a new method of control of chaotic systems was proposed (Grosu, 2000) by stabilization of unstable periodic orbits. In this case the driving term is:
'
100
..
200
300
400
500
600
n
Fig. 2. Control of the system (7), r = 3. For n < 320 no coupling, n > 320 coupling ON. Numerical results are shown in Fig. 1 with p 0.1 and r = 3.5.
where T is an integer and is the desired period.
If we want to have a periodic dynamics of (6) we need to use a driving of the type (5).
3. SYNCHRONIZATION AND CONTROL OF A POPULATION DYNAMICS
Numerical results are shown in Fig. 2. (r and T = 3) .
The above results can be applied to the synchronization of the dynamics of a population model (May, 1993; Stone, 1993): (6)
XnH = xnexp(r(l - xn))
Boccaletti, S. et al. (2000). The Control of chaos: Theory and Applications. Physics Reports, 392, 103-197. Chen, C-C (1995). Direct chaotic dynamics to any desired orbits via dosed-loop control. Physics Letters, A 213, 148 - 154.
(7)
Chen, G., X. Dong (1998). Prom Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore.
With p subunitary. + +
I
I
I I I
xn -
I
-
I
I
I
1-
..!~I - _
r
. - 1_
t-
~~I
....
~
""
++
I _I I
_-' I -
I
__
I
-
Chen, G.(ed) (1999) . Controlling Chaos and Bifurcations in Engineering Systems. CRC Press.
++
+.:
~
++ ++
+
+
+
+ ++ ++ + +
~
Chen L-Q, Y-Z Liu (1998). A modified openplus-dosed-loop approach to Control chaos in nonlinear oscillations. Physics Letters, A 245, 8790.
+ .. + + ++'\.
-+', _ I + "it- -n!I_,.!,I-" __ II+++++ + I + +..... ~+..' + 1_ _.J, I '*!:!'--:. + + + + l~ + J..
-,1I
.....-
I
I
I
I
11 ,I
_
_I -
-
!.i
...
-
11
'-
-
+ + ++ + + + + + +* + ++ +
I
-~ll
trf:-
T
I _ I
++
I
,t-.Jl-l-
II~-I+ --'I
I~\--""'
I
'+-./
+ *1" ++ +++ ++++ . . + ++. +
Dragoi, V ., I. Grosu (1998) . Synchronization of Locally Connected Neural Oscillators. Neural Processing Letters, 7, 199.
++
-..-:'?~
~~~ -It.
Ni;MIIUClJ1iI*"'..,.. ~.......l1at
100
200
300
400
500
600
= 3.0
REFERENCES
Let 's consider the dynamics of (6) Xn that we have it recorded. To reproduce it we have to drive the system (6) with the term:
I
=
Grosu, I. (1997). Robust Synchronization. Physical Review, E 56, 3709.
n
Grosu, I. (2000). Stabilization of Arbitrary Unstable Periodic Orbits of Nonlinear Systems, submitted Dec. 18.
Fig. 1. Xn -, Xn I. For n < 320 uncoupled, n > 320 coupled and synchronization achieved. 340
Jackson, E. A., I. Grosu (1995a). An Open-PlusClosed-Loop (OPCL) Control of Complex Systems. Physica, D 85, l. Jackson, E. A., I. Grosu (1995b). Toward Experimental Implementation of Migration Control. International Journal of Bifurcation and Chaos, 5,1767. May, RM. (1976). Simple mathematical models with very complicated dynamics. Nature. 261, 459. Schuster, H.G.(ed) (1999). Handbook of Chaos Control. Wiley-VCH Verlag. Stone, L. (1993) . Period doubling reversals and chaos in simple ecological models. Nature, 365, 617. Tian, Y-C, M.O. Tode, J. Tang (2000). Nonlinear open-plus-closed-Ioop (NOPCL) Control of dynamic systems. Chaos, Solitons and Practals, 11, 1029-1035. Wang, J ., X. Wang (1999). A global control of polynomial chaotic systems. International Journal of Control, 72, 911-918. Xiaofeng, G., CH Lai (2000). On the synchronization of different chaotic oscillators. Chaos, Solitons and Practals, 11, 1231-1235. Zonghua, L., C. Shigang (1997). Directing nonlinear dynamics systems to any desired orbit. Physical Review, E 55, 199-204.
341
CONTROL OF CHAOTIC POPULATION DYNAMICS USING OPCL METHOD loan Grosu * Dragos Arotaritei ** Constantin Corduneanu ***
o
Bioengineering, University of Medicine and Pharmacy "Cr. T. Popa", RO-6600, Ia§i, Romania 00 Computer Science, Aalborg University Esbergerg, Denmark 0** Bioengineering, Univ. of Medicine and Pharmacy "Cr. T.Popa" Ia§i, Romania
Abstract: The OPCL method is applied to reproducing a recorded chaotic dynamics and for controlling a chaotic dynamics. Copyright ~20011FAC Keywords: closed-loop, .chaos, discrete systems, dynamics, open-loop
1. INTRODUCTION
OPCL (Chen, 1996; Chen and Liu, 1998; Tian, et al., 2000; Wang and Wang, 1999; Xiaofeng and Lai, 2000; Zonghua and Shigang, 1997).
The last decade has known a considerable interest for control of chaotic systems (Chen and Dong, 1998; Chen, 1999; Schuster, 2000; Boccaletti, 2(00). Chaotic systems have the distinguishing feature of unpredictability that is much undesirable in engineering. Among several methods (Chen and Dong, 1998; Chen, 1999; Schuster, 2000; Boccaletti, 2(00) the OPCL (OpenPlus-Closed-Loop) method (Jackson and Grosu, 1995a; Grosu, 1997; Chen and Dong, 1998) can be mentioned. This method is used for model based systems. It offers a driving in order to obtain any desired dynamics. OPCL is mathematically justified and is very robust against noise (Grosu, 1997). It works for continuous systems described by a system of ordinary differential equations and for discrete systems described by a system of nonlinear difference equations. As a particular case of this control is the synchronization of a dynamics with a prerecorded one. So the problem of unpredictability is solved in terms of OPCL (Grosu, 1997). This was shown for Lorenz system and for logistic map dynamics. Also we used OPCL for synchronization of two Chua circuits and for synchronization of a chain of neural oscillators (Dragoi and Grosu, 1998; Jackson and Grosu, 1995b). Others used or modified or improved
2. OPCL METHOD The OPCL method (Jackson and Grosu, 1995a; Grosu, 1997; Chen and Dong, 1998) is a modelbased method. Let's consider a discrete system: (1)
We want to drive this system to a goal dynamics gn' The driving term is:
D(Xn,gn,gn+d = gn+l - F(gn) +(B -
~(gn) )(xn gn
gn)
+ (2)
where B is a constant matrix with all eigenvalues of subunitary modulus. The drived system:
assures the convergence of Xn to gn' Based on this result a method of synchronization has been obtained (Grosu, 1997). If the dynamics of (1) is recorded let's say it is Xn and we want to 339
reproduce it we can drive the system (1) with the driving term:
(4)
In addition this synchronization is robust against noise. This means that if the driving term is affected by an additive noise the synchronization is still achieved. Recently a new method of control of chaotic systems was proposed (Grosu, 2000) by stabilization of unstable periodic orbits. In this case the driving term is:
'
100
..
200
300
400
500
600
n
Fig. 2. Control of the system (7), r = 3. For n < 320 no coupling, n > 320 coupling ON. Numerical results are shown in Fig. 1 with p 0.1 and r = 3.5.
where T is an integer and is the desired period.
If we want to have a periodic dynamics of (6) we need to use a driving of the type (5).
3. SYNCHRONIZATION AND CONTROL OF A POPULATION DYNAMICS
Numerical results are shown in Fig. 2. (r and T = 3) .
The above results can be applied to the synchronization of the dynamics of a population model (May, 1993; Stone, 1993): (6)
XnH = xnexp(r(l - xn))
Boccaletti, S. et al. (2000). The Control of chaos: Theory and Applications. Physics Reports, 392, 103-197. Chen, C-C (1995). Direct chaotic dynamics to any desired orbits via dosed-loop control. Physics Letters, A 213, 148 - 154.
(7)
Chen, G., X. Dong (1998). Prom Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore.
With p subunitary. + +
I
I
I I I
xn -
I
-
I
I
I
1-
..!~I - _
r
. - 1_
t-
~~I
....
~
""
++
I _I I
_-' I -
I
__
I
-
Chen, G.(ed) (1999) . Controlling Chaos and Bifurcations in Engineering Systems. CRC Press.
++
+.:
~
++ ++
+
+
+
+ ++ ++ + +
~
Chen L-Q, Y-Z Liu (1998). A modified openplus-dosed-loop approach to Control chaos in nonlinear oscillations. Physics Letters, A 245, 8790.
+ .. + + ++'\.
-+', _ I + "it- -n!I_,.!,I-" __ II+++++ + I + +..... ~+..' + 1_ _.J, I '*!:!'--:. + + + + l~ + J..
-,1I
.....-
I
I
I
I
11 ,I
_
_I -
-
!.i
...
-
11
'-
-
+ + ++ + + + + + +* + ++ +
I
-~ll
trf:-
T
I _ I
++
I
,t-.Jl-l-
II~-I+ --'I
I~\--""'
I
'+-./
+ *1" ++ +++ ++++ . . + ++. +
Dragoi, V ., I. Grosu (1998) . Synchronization of Locally Connected Neural Oscillators. Neural Processing Letters, 7, 199.
++
-..-:'?~
~~~ -It.
Ni;MIIUClJ1iI*"'..,.. ~.......l1at
100
200
300
400
500
600
= 3.0
REFERENCES
Let 's consider the dynamics of (6) Xn that we have it recorded. To reproduce it we have to drive the system (6) with the term:
I
=
Grosu, I. (1997). Robust Synchronization. Physical Review, E 56, 3709.
n
Grosu, I. (2000). Stabilization of Arbitrary Unstable Periodic Orbits of Nonlinear Systems, submitted Dec. 18.
Fig. 1. Xn -, Xn I. For n < 320 uncoupled, n > 320 coupled and synchronization achieved. 340
Jackson, E. A., I. Grosu (1995a). An Open-PlusClosed-Loop (OPCL) Control of Complex Systems. Physica, D 85, l. Jackson, E. A., I. Grosu (1995b). Toward Experimental Implementation of Migration Control. International Journal of Bifurcation and Chaos, 5,1767. May, RM. (1976). Simple mathematical models with very complicated dynamics. Nature. 261, 459. Schuster, H.G.(ed) (1999). Handbook of Chaos Control. Wiley-VCH Verlag. Stone, L. (1993) . Period doubling reversals and chaos in simple ecological models. Nature, 365, 617. Tian, Y-C, M.O. Tode, J. Tang (2000). Nonlinear open-plus-closed-Ioop (NOPCL) Control of dynamic systems. Chaos, Solitons and Practals, 11, 1029-1035. Wang, J ., X. Wang (1999). A global control of polynomial chaotic systems. International Journal of Control, 72, 911-918. Xiaofeng, G., CH Lai (2000). On the synchronization of different chaotic oscillators. Chaos, Solitons and Practals, 11, 1231-1235. Zonghua, L., C. Shigang (1997). Directing nonlinear dynamics systems to any desired orbit. Physical Review, E 55, 199-204.
341