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Unformatted text preview: Chapter 14 Dynamical Systems The second order homogeneous equation for the damped oscillator
i+2y$+wgm = 0 (14.1)
can be written as a system of two ﬁrst—order equations: i=y, . a (142)
y = wart:  27y 
We can also write this system in matrix form as
9': 0 1 a:
= , (14.3)
1) WE *27 1/
or
X = AX , (14.4) cm 'ém {221,45 0%
where X is the column matrix and A is the 2 X 2 matrix appearing in 14.3).
In general, a dynamical system is a system of differential equations of the
form dX
dt where f : R” x R —+ R” is a continuously diﬁerentiable function with domain
and range as indicated. In applications in mechanics, the symbol t in the above
equation usually stands for time. If f (X, t) is independent of time, (14.5) be
comes : f(X,t) , (14.5) dX A dynamical system of the form given by (14.6) is called an autonomous
system. This kind of system can be written explicitly as a system of ﬁrst—order 65 66 CHAPTER 14. DYNAMICAL SYSTEMS differential equations as follows. dazl
Ezflcpla am”):
dm2 9
: f‘($13' ' ' 7m” 7
dt ) (14.7)
dis"
E = fn($1,...,$n) . The system (14.7) [as well as the more general system (14.5)] is in general non
linear. f(X) is called the vector ﬁeld of the dynamical system, whose components
with respect to the local coordinates ($1,...,m“) of a point X 6 R” are
f1, . . . , f”. A solution (:1:1 (t), . . . ,x“(t)) of (14.7) is called an integral curve of
the vector ﬁeld f(X). The space in which X lives (in our case R“) is called the
phase space of the dynamical system. The phase space of a dynamical system
is in general a differentiable manifold. The points X at which f (X) = 0 are of special importance in the theory
of dynamical systems. They are called equilibrium, ﬁxed, stationary, or
singular points of the dynamical system. For example, X : (:2:,y) = (0,0)
is an equilibrium point of the system (14.2). The stability of solutions near
equilibrium points will be of particular interest. To gain a qualitative under—
standing of the nature of solutions near equilibrium points, one often linearizes
the dynamical system by Taylor expanding f (X) around an equilibrium point,
say, X: f(X) ~ f(X) + Df(X)(X  X) = Df(X)(X — X) , (14.8) Where D f (X) is the derivative of f at the point X = X. This derivative is
actually EL linear map from R” to L(]R”,1R{”), the space of linear operators on
IR”. D f (X) has the following matrix representation: i]: 31“ . . . 5—1”
83;}, (91% 8357;
5f“ 5f‘ _ _ _ 3f“
DfCY‘) = W 52'? 8w" , (14.9)
8f“ 8f” _ . _ 8f“
82:1 89:2 8x" X=X where all the derivatives are evaluated at the point X = X. Near an equilibrium
point X = X, then, the behavior of the dynamical system (14.7) is described
by the linearized system
dX
dt
if we choose a set of local coordinates so that X : 0. The solutions of (14.10)
depend on the eigenvalues of the linear operator D f(X) : Df(X) X, (14.10) 67 We will digress to state (without proofs)a few important facts on the theory
of linear operators. Theorem 14.1. The solution to the system of equations dX/dt : AX, with A
being constant, is X(t) = eAt X(O) , (14.11)
where 00 n n
em = Z A”: . (14.12)
n=0
Theorem 14.2. (a) IfB : A‘IAA (a similarity transformation), then
eB : A‘leAA. (14.13) (b) If [T,S] E TS — ST = 0, then eS+T = e5 eT . (14.14) (c) For all linear operators S,
6—5 = (eS)_1 . (14.15) Deﬁnition 14.1. A matrix S is said to be diagonalizable (semisimple) if
there exists an invertible matria' A such that A‘lSA is diagonal. Deﬁnition 14.2. A maria N is said to be nilpotent if there exists a positive
integer in such that Nm = 0. Theorem 14.3. {The Primary Decomposition Theorem Any A E L(E7 E)
(the linear space of linear operators on E, where E is in general some complex
vector space E), has a unique decomposition A = S + N, such that [5, N] = O,
S is diagonalizable, and N is nilpotent. Theorem 14.4. If a —b
B = , a,b E R, (14.16)
b a
then,
cosb  sinb
e3 = e“ . (14.17)
sinb cosb Theorem 14.5. IfA is the diagonal matria' given by A: a (1418) 68 L CHAPTER 14. DYNAMICAL SYSTEMS then 6":  (14.19) With the above facts we can now proceed to give a complete classiﬁcation
of two—dimensional linear dynamical systems. This is based on the following
central result. Theorem 14.6. For any 2 X 2 real matrix A there exists an invertible matrix
A such that A‘lAA has one of the following forms: (1) l (2) , (3) ,
0 ,u b a 1 /\ where A,u,a,b E R. Case (I) is obtained when the two eigenvalues A and u are
real and the matrix; A is diagonalizable (A is always diagonalizable when /\ 75 u);
(2) is obtained when the eigenvalues are a i ib; {3) is obtained when the two
real eigenvalues are equal to each other and A is not diagonalizable. The forms (1),(2) and (3) in the above theorem are called the Jordan
canonical forms of the 2 X 2 matrix A. According to Theorem 14.1, the solutions X(t) to the equation dX/ alt : AX ,
With A given by the cases (1), (2) in Theorem 14.6, are given by: a: t e"t 0 IE 0 /\ O
X(t) : ( ) = ( ) , for A :
W) 0 6‘” 21(0) ‘ 0 M
(14.20)
as t cos bt —sinbt :L’ O a —b
X(t) : ( ) 2 eat ( ) , for A =
y(t) sin bt cos bt 31(0) b a
(14.21)
For case (3) we note that
A O
A = = s + N, (14.22) 69 where
/\ 0 0 0
S = , N = , (14.23)
0 A 1 0
With
[S,N]:0, N220. (14.24)
Thus, by Theorem 14.2(b),
/\ 0 At 0 O 0
exp t = exp exp
1 /\ 0 At t 0
(14.25)
= e” 1 + 0 O = e’\t 1 O
t O t 1
The solution with A given by case (3) is thus
:13 t 1 0 m 0 /\ 0
( ) = eAt ( ) , for A = . (14.26)
W) t 1 11(0) 1 /\ The solutions (14.20), (14.21) and (14.26) can be understood more intuitively
and geometrically by using phase portraits, which depict the integral curves
of the vector ﬁeld of the dynamical system in each case. Case {1)a. The eigenvalues /\ and u of A satisfy A < O < u. The point 0, 0) of phase space is a saddle point. g E
I 111
0 reel .ng mme/M YR“
Wwwamw a» )5
A W“ Mn ”I “WWW
M‘x‘» £51”
A < 0 / I” Qé“ i); Case (1)1). A < ,u < 0. The solutions satisfy limtnooXﬂ) : 0, and (0,0) is a
sink node. t 0
W A /“ A<m<a 70 CHAPTER 14. DYNAJVIICAL SYSTEMS 0</\< ;lim _00Xt :0; 0,0 isasourc node. (a
M t—> () l ) e :l Q‘m X05320
23'?“ “9
rm“ (“m Irﬂmw ﬂwwwgtw..mwmmmjg§mwm . ,Wwwgﬂsmw _.‘MW_.__..§____ V x’ 0 A /M' ”W“. 64/14/42 Mr ”‘5‘“?
we we rs mi @E Cases (1)c. A diagonalizable. /\ = ,u < 0; lith00 X(t) = 0; (0,0) is a sink
focus. 0 < A = ,u; limt_,_oO X(t) = 0; (0,0) is a source focus. M we x MW“? 35
1’1 ‘z x “mm... ,
gmk 935% MW Xﬁﬂra gimme 3%;ng
J ~~_! «,3 a a??? m
.V Wig, WW w M MWW MWWWW WWJFW‘WWWw.‘ W . W mm mtg
Amyqo O O 0<Ajﬂ Cases (EM. (1 = 0, b > 0 or b < 0. The integral curves are closed curves
representing periodic solutions, which satisfy X (t + 27r/b) = X (t) These
curves (circles) are given by the equation 9:2 +212 = (a:(0))2 + (34(0))? The point (0,0) is called a center.
’3’ ’3; Kurg m(&rxo+{gzw§ Kites? sxﬁg Cases (2)1). a < 0, b > 0 and a < 0,12 < 0. For both cases we have
limt_,00 X(t) : 0. The point (0,0) is called a sink spiral. @be a<sgb>5
f ! WWW
5
‘ X
évtb Cases (2)6. a > 0,1) > 0 and a > 0,12 < O. The solutions satisfy lirnt_,_00 X(t) —
0. The integral curves are the same as those of Case (2)b, but with the directions
of the arrows reversed. The point (0,0) is called a source spiral.
a > 0 ) 30 4 o Case (3). /\ = a < 0 or 0 < /\ = a; A not diagonalizable. For negative
degenerate eigenvalues, lirn,g_>00 X(t) : 0 and the point (0,0) is called a sink
with improper node. For positive degenerate eigenvalues, lirnlts.DO X (t) = 0
and the point (0,0) is called a source with improper node. For both cases the phase (integral) curves are given by eM(t:1;(0) + 31(0)) . (14.27) 5.25% \X l r it x A.
\i ‘ fix ::/’” ‘
.q a s “x.“
1‘ 1 ‘ ‘X H ‘
V v.3 When we transform the equation dX’ = BX’ , (14.28) dt with B being a canonical form back to the original coordinates to obtain dX
117 _ AX , (14.29)
where B = X‘lAA for some invertible A, the above phase curves will in gen
eral be distorted but all the qualitative features will remain the same. This
follows from the fact that eigenvalues of a matrix are invariant under similarity transformations. X a. wit: KM”? 12M were “g’,~¥e~r,0 F925;; {6 diagram ‘3‘: my,
a s: 2 m shiazf’g new (f;
(Mung c’ 2;? he , V but?” aft“;
an 9' mm; m ire. ”sari S auras. mitt: EémfwaEazz s” " (ﬁr/5e. 72 CHAPTER 14. DYNAIVIICAL SYSTEZVIS ...
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 Spring '08
 LAM
 mechanics

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