Dynamical Systems

Dynamical systems

, Chapter I


&




Lecture 1|2
First order differential equations CODE)
first derivative appears
only the

f(x
· where
X(0) Xo (initial
X is a state a
variabl source : f'(x1s0 ,

tend away from
solutions
.
eq

=
condition)

Xn + 1 =
f(xn) ,
n = 0 ,7 ,2 . sink : f'(x)0 ,
solutions

tend to equilibrium
- =
aX
,
afR = x XX
= =
X(t) is an unknown real-valued function of variable to

& for each value a we
,
have a different differential equation


solution Keat
general :
X(t) =

,
KERR and K = x10) initial condition

x(t) akyat= =
aX(t)

* A no other solutions



initial value problem ([vP) : X' =
ax X 10) No =




the solution X(t) to IVP has to "solve the differential equation
2
take the valueno at t 0
=




· when K 0 the solution the constant XC = 0. > this is called the
equilibrium solution/point.
-
=
is
,




phase line
X(t) X(t)


when a changes the solutions change :




if

S
f
· a 0 :
limx( =
00 K Xo =
to
source
-

00 if K =
X 100


O if K =
x(0) =
0
~
equilibrium
* all nonzero solutions away from equili .


equilibrium




.
2 If a =
0 :
X (+) =
heat=constant X(t) X(t)

V




.
3 If aco :
lim x (H) =
0 to Sink
t+0




* all nonzero solutions tend to equili
* phase line :
XCH is a function of time ,
we can view it as a particle moving along the real line

at the
equilibrium it remains at rest (dot) ,
for any other solution it moves up or down (arrows)




the replace behavior (behavior of graph) doesn't
X stable when ato , whatever b (with a with the
qualitive change.
·
a
ax is constant same
sign as we
=
, ,




when a =
o
,
the slightest change in a leads to radical change in the behavior of solutions




Bifurcation changes of the
qualitive behavior of . A
solutions change from to this changes the equilibrium from
positive negative source
: m




to sink.

·
we have a bifurcation at a =o in the one-parameter family of equations x =
ax

, The logistic population model

For aso ,
we consider v =
ax as a simple model of population growth
.
·
x (t) measures the population of a species at time +

rate of
growth of population (directly proportional to the of population
a
·
size




logistic population growth model :


d = x =


ax/1) where a = rate of population growth when X N(very small
N
carrying capacity
=




* we consider N =
1 ~ =
ax( first order autonomous nonlinear differential equation
-S
right-side depends only on X
,
not to


solution : dx-ade
Std-Jadt
= Sa

eat For
=>
X(H)
vol
=
to we have
,
1 + k




so the solution becomes X10eat
eat
X (t) 0 X(t) 1 equilibrium
1 X10) + X10)
-




are points


#
= =

,


·
sink

f(x) =
X(1 x) -
R
source




·
solutions tent
to 1
X-0




to
00 -
tend


* > X



increasing
increasing




Constant harvesting and Bifurcations

·
Harvesting represents nation of population


Let a 1 =
then x =
X(1-x) (N 1) and we
=
consider that the population is harvested at constant rate 30
.


~ X =
X(1 x) - -

1

fn(x) bifurcation diagram :




X
1/2
X




a
U
source sink
h
em
[ > <

1/4

u
n
·
-
n