2001/02: Lecture 4. Non-linear differential equations
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Equilibrium point
Linear stability theory, two-dimensional case;
Types of equilibrium points: node, saddle, focus, centre;
Limit cycle, Poincare map, stability of limit cycle;
Phase portrait;
Sources and sinks, basins of attraction;
Examples of mathematical models and their phase portraits.
Nonlinear system (of nth first order differential equations):
y   f ( y), y(t )  R n , f ( y)  R n
(1)
Equilibrium (equilibrium point, steady state, stationary solution):
Point y   R n is an equilibrium point iff f ( y  )  0 ( y  y  y  0 - velocity at the
equilibrium point equals zero).
Example 1:
y   y  y 3 . Steady states: y  y 3  0,  y   0, y   1, y   1 .
For    y  1 the function f ( y)  y  y 3 is positive and any solution with initial
value from this interval grows up and tends to the equilibrium y   1 .
For  1  y  0 the function f ( y)  y  y 3 is negative and any solution with initial
value from this interval decreases and tends to the equilibrium y   1 .
For 0  y  1 the function f ( y)  y  y 3 is positive and any solution with initial
value from this interval grows up and tends to the equilibrium y   1 .
For  1  y   the function f ( y)  y  y 3 is negative and any solution with initial
value from this interval decreases and tends to the equilibrium y   1 .
Example 2:
 x  x( x  y  3)

 y  y ( x  y  1)
Equilibria: (0,0); (2,1); (0,1); (3,0)
Equilibrium point is stable if any trajectory with initial value nearby of the
equilibrium point stays near the equilibrium for all time.
Equilibrium point is asymptotically stable if any trajectory with initial value nearby
of the equilibrium point tends to the equilibrium if the time tends to infinity.
In the example 1 the equilibrium y   1 and the equilibrium y   1 are
asymptotically stable and the equilibrium y   0 is unstable.
1
Linear stability theory:
Let consider a system of two nonlinear equations
 x   f ( x, y )

 y   g ( x, y )
(1)
 f ( x  , y  )  0
and suggest that ( x  , y  ) is equilibrium. It means that 
.
 g ( x  , y  )  0
We are looking for the nearby local solutions (close to the equilibrium):
 x  x   u

 y  y   v
here u   , v   (small).
Substitution gives:


u   f ( x  u, y  v)

v  g ( x   u, y   v)
Expand the RHS in the Taylor series:

f ( x  , y  )
f ( x  , y  )



u

f
(
x
,
y
)

u

v  o(u 2 , v 2 , uv)

x
y






v  g ( x  , y  )  g ( x , y ) u  g ( x , y ) v  o(u 2 , v 2 , uv)

x
y
u   au  bv

v  cu  dv

f ( x , y  )
f ( x , y  )
a

,
b


x
y

here 




c  g ( x , y ) , d  g ( x , y )

x
y
and we neglect the high order (>1) terms.
We can solve the linear system:
u   au  bv

v  cu  dv
(2)
The characteristic equation: 2  p  q  0 , where p  (a  d ), q  ad  bc .
(We are looking for solutions of the type u  Aet , v  Be t , . Substitution into
system (2) gives:
Aet  aAet  bBe t
A  aA  bB
(  a) A  bB  0
 
 


t
t
t
Be  cAe  dBe
B  cA  dB
 cA  (  d ) B  0
(  a)(  d )  cb  0
Real eigenvalues:
 p D
D  0  1, 2 
,
2
 2  (a  d )  (ad  bc)  0 ).
y(t )  Ae1t  Be 2t ,
2
If both 1  0 and 2  0 then the equilibrium is asymptotically stable and the type of
the equilibrium is a stable node (sink).
If both 1  0 and 2  0 then the equilibrium is unstable and the type of the
equilibrium is an unstable node (source).
If 1  0 and 2  0 then the equilibrium is unstable and the type of the equilibrium is
saddle. Incoming separatrix and outcoming separatrix.
To draw the phase portrait near the equilibrium for the cases node and saddle we need
to find stright-line trajectories. We seek trajectories of the form:
x(t )  e t z , here z  0 is some fixed vector. Substitute x(t) in equation and obtain
e t z  e t Az  Az  z .
Here z is eigenvector corresponding to the eigenvalue  .
u
u
a b
u
A    , z    , A     
v 
v 
c d 
v 
 a   b  u 

   0 .
d    v 
c
u 
( A  E)   0 ,
v 
b 



If a    0 than the vector  a    is eigenvector corresponding to

1 

the eigenvalue  .
The engenvector corresponding to the eigenvalue with a smaller absolute value is a
tangent for the bunch of trajectories near the node (please, explain!).
Important rule to draw a phase portrait: any two nearest trajectories have the same
direction and go in parallel.
Complex conjugate eigenvalues:
 pi D
D  0  1, 2 
   i .
y(t )  e at ( A cos t  B sin t ).
2
If   0 then the equilibrium is asymptotically stable and the type of the equilibrium
is a stable focus (sink). Solutions are spirals.
If   0 then the equilibrium is unstable and the type of the equilibrium is an
unstable focus (source). Solutions are spirals.
If   0 then the equilibrium is stable (but not asymptotically stable!) and the type of
the equilibrium is a centre. Solutions are circles.
Equal eigenvalues:
Special star-like node – stable or unstable.
Periodic solution (limit cycle):
 (t )   (t  T ) for any t and x   (t ) is a solution for system (1).
3
T is period of the limit cycle (period oscillations).
A limit cycle is stable if any trajectory with initial value nearby of the limit cycle
stays near the limit cycle for all time.
A limit cycle is asymptotically stable if any trajectory with initial value nearby of the
limit cycle tends to the limit if the time goes to infinity.
Stable limit cycle
Unstable limit cycle
Pouncare section, Pouncare map, stability of the limit cycle.
Example: Predator – Prey (Lotka-Volterra) model:
 x  ax  bxy

 y  cxy  dy
x(t)- density of the prey population at moment t.
y(t)- density of the predator population at moment t.
Predator-prey model with limitations:
 x  (1  x) x  bxy

 y  cxy  dy
Example: Van der Pol’s equation:
x  a( x 2  1) x   2 x  0
OR
 x  y

2
2
 y  a( x  1) y   x
4
Half-Stable limit cycle
1.4
1.2
1
Prey
0.8
0.6
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
1
Predator
1.2
1.4
1.6
1.8
2
0.24
0.04
0.22
0.03
0.2
0.02
0.18
0.01
Prey
Prey
0.16
0
0.14
-0.01
0.12
-0.02
0.1
-0.03
0.08
-0.04
0.06
-0.05
0
0.02
0.04
0.06
Predator
0.08
0.1
0.12
0.6
0.65
0.7
Predator
0.75
0.8
0.1
0.08
0.15
0.06
0.1
0.04
0.05
Prey
Prey
0.02
0
0
-0.02
-0.05
-0.04
-0.1
-0.06
-0.15
-0.08
0.1
0.15
0.2
Predator
0.25
-0.2
0.3
5
0.8
0.85
0.9
0.95
1
1.05
Predator
1.1
1.15
1.2
1.25