Stability constraints for two general forms of coupled systems of second order nonlinear differential equation

Aggarwal, J. K.

Bybee, H. H.

English

NASA Technical Reports Server

ON THE STABILITY CONSTRAINTS AND OSCILLATORY 
BEHAVIOR OF COUPLED SYSTEMS 
J. K. Aggarwal and H. H. Bybee 
Preprinted for 
lOTH MIDWEST SYMPOSIUM 
ON CIRCUIT THEORY 
May 18-19, 1967 
Purdue University 
Lafayette, Indiana 
LABORATORIES FOR ELECTRONICS AND 
RELATED SCIENCE RESEARCH 
College of Engineering 
THE UNIVERSITY OF TEXAS, AUSTIN 78712 
https://ntrs.nasa.gov/search.jsp?R=19680002711 2020-03-12T11:49:07+00:00Z
On the Stability Constraints and 
Oscillatory Behavior of Coupled Systems 
J. K. Aggarwal and H . H . Bybee 
Department of Electrical Engineering 
The University of Texas 
Austin, Texas 
Abstract: 
The system described b y  the differential equation 
% + g(k) + h(x) = 0 
has  been extensively investigated both from the stabil i ty as well as 
oscil latory points of view. The present paper d iscusses  some properties 
of a class of higher order systems which are  obtained by coupling second 
order systems of the above form. Sufficient conditions a re  derived for 
such  a system to be s table  and the  effects of introducing additional non- 
l inear i t ies  in  t h e  feedback path are discussed.  The oscillatory behavior 
is discussed  qualitatively and some new resul ts  are  presented concerning 
(1) t h e  coupling of an  oscillatory system to a stable system, and (2) the 
coupling of two oscil latory systems. 
1. Introduction 
Much work has  been done on nonlinear second order differential 
systems from both stabil i ty and periodic behavior standpoints because of 
the ease of applying analytic and topological methods to their solution. 
An extensive bibliography is found in Cesar i  [l]. Two of the more 
. 
XII-1-1 
general bas ic  second order forms are,  
and 
Constraints must be  placed on the nonlinear functions t o  assure asymp- 
totic stabil i ty in  the large, instability, or the existence of a l i m i t  cycle .  
For the  sake of completeness two typical results are  given below for the 
system (1). 
System (1) is asymptotically s tab le ' in  the large if  
i) xh(x) > 0 ,  x # 0; h(0) = 0 
ii) kg(k) > 0, k # 0; g(0) = o 
iii) J'" h(u)du -, m as 1x1 -, oJ 
This resul t  is proved by  using the Liapunov function 
0 
2 
(3) V(x,A) = 1/2 A + f x  h(u)du 
0 
and applying LaSalle' s Theorem E21 for which (i) and (ii) above assure  
that  the origin is the only solution along which 3 = 0.  
If g(k) = Agl(k) and h(x) i n  system (1) are continuous along with 
their  first derivatives,  then there exists at least one (nonconstant) 
periodic solution, x(t), provided there a r e  positive constants a ,  m,  and 
M such  that  
i) g,(jr) > m  for IAl>a 
ii) g (A) > -M for a l l  A 1 
X I I - 1 - 2  
iii) 
iv) 
g (k) < 0 in  some neighborhood of S O  excluding S O ,  
xh(x) > 0 ,  x #  0,  h(0) = 0 
1 
v) 
The proof of this  result uses the positive definite auxiliary func- 
Ih(x)l -)a and h (x ) / lX  h(u)du-( 0 a s  1x1 -)a. 
0 
tion (3) to show that all trajectories, except the origin, started within 
a s m a l l  neighborhood of the origin leave that neighborhood and all tra- 
jectories started sufficiently far from the origin are shown to enter a 
special  curve constructed from V. (For proof see [l]). 
For system (2) the Liapunov function (3) gives r ise  t o  a stability 
result  similar t o  (*) and Wax [3] has  recently proved the l i m i t  cycle 
theorem for a subclass of this  system. 
Higher order systems may be constructed by coupling second 
order subsystems as  follows: 
K ,  + g,(k.) + h.(x.) + k.(x) = 0 
1 1 1  1 1  1 
i =  l , Z , .  . . , n  
(4) 
- 
x = (X1'X2I.  . . /x,) 
The result  on asymptotic stability may be extended t o  include the above 
higher order systems. 
The 2n-order system (4) is asymptotically stable in  the large 
i f for  i =  1 , 2 , . . . , n  
(***) . 
XII-1-3 
iii) 
iv) JXi h.(u)du -.) aD a s  lxil and 
v) 
ki(G) = a U(z) where U is a positive semidefinite function 
a xi 
1 
0 
the  set of equations hi(xi> + kd;) = 0, i = 1,2 I . . . n, has  
only the trivial solution x = 0. 
- -  
One can prove th i s  resul t  using LaSalle' s Theorem [Z] where the 
Liapunov function is 
V(G,$) = 1/2 c n xi .2+ c" Jxi hi(u)du -+ U(';, 
i= 1 i=l o 
(5) 
for which 
n 
i= 1 
ir(XlZ1 = - c f.g.(Ai). 
1 1  
N o  resul ts  similar to (**) above exist for the system (4 ) .  However, 
very restricted resul ts  on amplitude bounds are  avai lable .  
Aggarwal [41 has  shown for the  system 
3 
% + A ( f  /3 - A) + Bx= -4x-Y) 
the existence of a n  "inner" amplitude bound and th i s  resul t  may be 
generalized t o  the situation where more than two oscil lators are  coupled. 
This paper d i scusses  the stabil i ty constraints and oscillatory 
behavior of system (4) and its extensions.  Also, resul ts  on the oscilla- 
tory behavior of the coupled systems, when one subsystem is stable  
and the  other oscillatory, or when both subsystems are  oscillatory are  
presented. 
XII-1-4  
2 .  Stability of Coupled Systems 
An extension of the System (4) with a single dissipative element 
has  the form 
- 
i =  1 ' 2 ,  ..., n 
where 
[t 1 
e = { O  depending upon the system. 
i - 1  
m 
In the above system the  coupling terms are  
n 
ki(G) + e,f( c 
m= 1 
e k ) 1 m m  
i = 1,2,. . . , n .  
System (7) is asymptoticall stable in the larg- if in additi  )n 
to the conditions (***) the following condition on f holds: 
Uf(u) > 0, u #  0,  f(0) = 0. 
Again th i s  result follows from the use  of LaSalle' s Theorem [Z] where 
the  Liapunov function is (5). This result  m a y  eas i ly  be extended t o  
systems with several dissipative elements. 
Another generalization of the System (4) has  the form 
% + fi(+ + = 0 (8) 
i =  1 , 2 , .  ..,n 
where 
*i = q k i )  + hi(Xi). 
XII-1-5 
~ 
The 2n-order system (8) i s  asymptotically stable in the large if  
for i =  1 , 2 ,  ..., n 
i) 
ii) 
iii) 
iv) 
v) 
vi) 
vi i) 
re U is a posi ive semidefinite function 
the  se t  of equations f .(a.)  + ki(o) = 0 has  only the trivial 
solution 0 = 0. 
1 1  - -  
This theorem is proved by applying the Liapunov function 
n n 
V(E,z)  = J"i f,(u)du + C Jxi h:(x,)k.dk, + U(,) 
1 1 1 1 1  i=l 0 i=l o 
for which 
3 .  Dissipatively Coupled van der Pol Oscillators 
In  the case of the conservatively coupled van der Pol oscil lators 
a n  inner amplitude bound always exis ts ,  however, for the c a s e  of dis- 
sipatively coupled van der Pol oscillators, a n  inner amplitude bound 
may be found only for a certain range of resistance.  By the inner ampli- 
tude bound is meant a closed surface in the phase space such that a l l  
XII-1-6 
trajectories starting within the surface leave the surface and no tra- 
jector ies  enter  the surface (except the origin). For the system 
3 
1 2 j z  .. 
X -I- e1 (3- 'E1) -I- w X = -C(x1-x2) -R(k -A ) 1 1  1 2  
3 
Ci e l ,  e 2 i  R >  0. 
using the  Liapunov function 
2 2 
2 2 w2 2 
X 
2 2 c  , A  / X  ,K ) =  1/2 (Al+jr )+-(x - x ' )  +-x +- v(xl 1 2 2 2 2 1 2  2 1  2 2 '  
one f inds tha t  
On converting I? = 0 to polar coordinates in  the (A , A  ) plane it is 
found tha t  if a n  R is taken such that 
1 2  
€ l e 2  
€1+€2 
O s R <  -
then  a n  inner amplitude bound exists.  The general inner amplitude 
bound is the  hypersurface V = a constant = 1/2 r where r is the radius 2 
of the  circle centered at the origin in the (5 jC ) plane that can be in- 1' 2 
scribed i n  the curve = 0,  that is 
2 2 2 
e l C O ~  0 + c s in  e - R(cos 8 - s in  8) 
elcos 8 + e 2 s i n  e 
11 2 2 4 4 r = min 13( 
O s e 5 2 ~ ~  
A s  a n  example, solutions were computed numerically for system (9) with 
el=l, e2=3, R =  0.5. 
The solutions do  not rapidly converge to a periodic solution. If R is 
increased to 0.8, the previous established condition on R is violated, 
and a n  inner amplitude bound may not be computed by this  method. 
The previous discussion has  shown that even though the exis- 
tence of periodic solutions cannot be proven theoretically one can  prove 
that given systems can  never decay to their singular point a t  the origin. 
4 .  Consenratively Coupled van der Pol Oscillator and a Stable System 
The behavior of the coupled system when a damped simple 
harmonic oscillator and a van der Pol oscil lator a r e  conservatively 
coupled has  been investigated. The system 
K + Al + x1 = (x2-x1) 1 .. 3 x + E(R2/3 - k2) + x2 = (x -x ) 2 1 2  
may be investigated using the positive definite function 
2 2  2 2 2  
1 2  1 2  1 2  V(Xl,A1,X2’k2) = 1/2 [ j c  + k  + (x -x ) + x  + x  3 .  
4 
2 2  j c  
1 3 
.2 HereV = -x - C (  - - R2 ) 
and V = 0 can  be  seen  to go through the origin of the (k , A  ) plane a s  1 2  
goes  through zero. Thus no circle can  be  drawn in the (k A ) plane 
2 1’ 2 
around the origin and inside the V = 0 curve. This means that no inner 
amplitude bound may b e  found from V under consideration but for C =  10, 
XII-1 -8  
numerical solutions show that  the  two equations lock into a periodic 
solution. The above absence of resul ts  may be observed for systems 
of the form 
aU*(X, xz) 
axl  
K 1+ klgl(") + = o  
Z 2 + " ( 3 - -  2 
3 
j c  au*(x, I xz) 
k ) +  = o  
ax 2 
where U*(xl, xz) is a positive definite radially unbounded function and 
g (k ) > 0 for all k gl(-K1) = gl(K1) and g (0) is finite. 1 1  1' 1 
5. Discussion 
Stability constraints have been derived for two general forms of 
coupled systems of second order nonlinear differential equations. Exam- 
p l e s  of these  systems have been investigated for oscillatory behavior, 
and  general conditions have been derived to assure  that their solutions 
never decay even though the systems have dissapative coupling. Further 
work is being undertaken on oscillatory behavior of such systems so as 
to find a resul t  similar to (**) for higher order systems. The hypersurface 
computed to find a n  inner amplitude bound is the first step.  Numerical 
r e su l t s  are not presented here for lack of space.  
X I I - 1 - 9  
6.. Bibliography 
[11 L. Cesari ,  Asymptotic Behavior and Stability Problems in  
Ordinary Differential Equations, Academic Press ,  N.Y. , 1963. 
r21 J.  P. LaSalle, "Some Extensions of Liapunov' s Second Method, 
IRE Trans. on Circuit  Theory, vol. CT-7, pp. 520-527, Dec. 
1960. 
r31 N. Wax, "On Some Periodic Solutions of the  Lienard Equation, 'I 
IEEE Trans. on Circuit Theory, vol. CT-13, pp. 419-423, Dec. 
1966. 
r41 J. K.  Aggarwal and C . G .  Richie, "On Coupled van der Pol 
Oscil lators,  " IEEE Trans. on Circuit Theory, vol.  CT-13, 
pp. 465-466, Dec. 1966. 
The research reported herein was sponsored by JSEP Research Grant 
AF-AFOSR-766-65, NSF Grant GK 261 and NASA Grant NAS8-18120. 
XII-1-10 


On the stability constraints and oscillatory behavior of coupled systems

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