696 Note
Autonomous Dynamical Systems, Explicitly Time Dependent First Integrals, and Gradient Systems
Willi-Hans Steeb
International School for Scientific Computing, University of Johannesburg, Auckland Park 2006, South Africa
Reprint requests to W.-H. S.;
E-mail:steebwilli@gmail.com Z. Naturforsch.66a,696 – 697 (2011) DOI: 10.5560/ZNA.2011-0030
Received June 6, 2011 / revised July 1, 2011
We study first integrals and first-order autonomous sys- tems of differential equations. Some of the concepts for ex- plicitly time-independent first integrals are extended to ex- plicitly time-dependent first integrals. Several applications are given.
Key words:Dynamical System; Time Dependent First Integrals; Gradient Systems.
We investigate first-order autonomous systems of differential equations
duj
dt =fj(u), j=1,2, . . . ,n, (1) and first integrals in particular explicitly time-depend- ent first integrals [1]. It is assumed that the functions fj:Rn→Rare analytic. If the autonomous system (1) admits a first integralI, it can be written as [2]
du
dt =S(u)∇I(u), (2)
where ∇I(u) := (∂I/∂u1, . . . ,∂I/∂un)T, ST(u) =
−S(u)is a skew-symmetricn×nmatrix andTdenotes transpose. This representation is important when we discretize a dynamical system (1) and want to preserve the first integral. As an example consider
du1
dt =cu1+c23u2u3, (3a) du2
dt =cu2+c31u3u1, (3b) du3
dt =cu3+c12u1u2, (3c)
wherec23,c31,c126=0, andcdescribes the damping. If c=0, then the system admits the first integral I1= (c31u21−c23u22)/2 and the dynamical system can be written as
du1/dt du2/dt du3/dt
=
0 −u3 0
u3 0 c12u1/c23 0 −c12u1/c23 0
·
∂I1/∂u1
∂I1/∂u2
∂I1/∂u3
.
(4)
Note that the system also admits the independent first integral(c12u21−c23u23)/2, and the dynamical system can be reconstructed using Nambu mechanics. Ifc6=0, thenI1,I2are not first integrals anymore.
Now many dissipative dynamical systems such as the Lorenz model and the Rikitake-two disc dynamo admit, depending on the bifurcation parameters, ex- plicitly time-dependent first integrals. These type of first integrals are of the form [3–9]
f(u(t))eλt. (5)
The dynamical system given above withc6=0 admits first integrals of this form. Thus the concepts given above to cast the dynamical system (1) into the form (2) has to be extended. We show with an example that the concept to write system (1) if a first integral exists as a gradient system (2) can be extended to explicitly time-dependent first integrals of the type given by (5).
To study such cases we extend the autonomous system (1) to
du
dτ =f(u), dun+1
dτ =1. (6)
Thus we sett→un+1andun(τ=0) =0. The first in- tegrals (5) then take the formf(u1, . . . ,un)eλun+1.
As an example consider the autonomous system du1
dt =u1(1+au2+bu3), (7a) du2
dt =u2(1−au1+cu3), (7b) du3
dt =u3(1−bu1−cu2), (7c)
c
2011 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com
W.-H. Steeb·Dynamical Systems, Time-Dependent First Integrals, Gradient Systems 697 where a,b,c∈R with the first integral I(t,u(t)) =
(u1+u2+u3)e−t. Thus we consider the autonomous system
du1
dτ =u1(1+au2+bu3), (8a) du2
dτ =u2(1−au1+cu3), (8b) du3
dτ =u3(1−bu1−cu2), (8c) du4
dτ =1. (8d)
This system can be written in the form
du1/dτ du2/dτ du3/dτ du4/dτ
=
0 s12eu4 s13eu4 s14eu4
−s12eu4 0 s23eu4 s24eu4
−s13eu4 −s23eu4 0 s34eu4
−s14eu4 −s24eu4 −s34eu4 0
·
∂I/∂u1
∂I/∂u2
∂I/∂u3
∂I/∂u4
(9) withI(u) = (u1+u2+u3)e−u4,∂I/∂u1=∂I/∂u2=
∂I/∂u3=e−u4,∂I/∂u4=−(u1+u2+u3)e−u4, and s12=1
3u1−1
3u2+au1u2, (10a) s13=1
3u1−1
3u3+bu1u3, (10b) s23=1
3u2−1
3u3+cu2u3, (10c)
wheres14=s24=s34=−1/3. Thus from the example it is obvious how to extend the concept to explicitly time-dependent first integrals.
The approach can be extended if there are more than one explicitly first integral. Consider the dynamical system
du1
dt =cu1+c234u2u3u4, (11a) du2
dt =cu2+c134u1u3u4, (11b) du3
dt =cu3+c124u1u2u4, (11c) du4
dt =cu4+c123u1u2u3. (11d) This is an extension of system (3) to higher dimen- sions. This system has been studied for c=0 from a Lie algebraic and integrability point of view by Steeb [10]. Such a system forc=0 appears from the self-dual Yang–Mills equation by exact reduction [11].
Forc=0, we have the three first integrals c134u21−c234u22, c124u22−c134u23,
c123u23−c124u24.
(12)
Including dampingc6=0 provides the explicitly time- dependent first integrals
e−ct(c134u21−c234u22), e−ct(c124u22−c134u23), e−ct(c123u23−c124u24).
(13)
[1] W.-H. Steeb, Nonlinear Workbook, 5th edition. World Scientific, Singapore 2011.
[2] G. R. W. Quispel and G. S. Turner, J. Phys. A: Math.
Gen.29, L341 (1996).
[3] W.-H. Steeb, J. Phys. A: Math. Gen.15, L380 (1982).
[4] M. Kus, J. Phys. A: Math. Gen.16, L689 (1983).
[5] F. Schwarz and W.-H. Steeb, J. Phys. A: Math. Gen.17, L819 (1984).
[6] K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientific, Singapore 1991.
[7] M. A. Almeida and I. C. Moreira, J. Phys. A: Math.
Gen.25, L669 (1992).
[8] M. A. F. Sanju´an, J. L. Valero, and M. G. Velarde, Il Nuovo Cim.13D, 913 (1991).
[9] W.-H. Steeb, Z. Naturforsch.49a, 751 (1994).
[10] W.-H. Steeb, Int. J. Geom. Meth. Mod. Phys.7, 405 (2010).
[11] W.-H. Steeb and N. Euler, Il Nuovo Cim.106B, 1059 (1991).
