Transcript Nessun titolo diapositiva

Basic ingredients of a dynamical system
State variables : x = (x1, x2, …, xn)
Evolution operator
xi (t )  i (t; x1 (t0 ),...,xn (t0 ))
i  1,...,n
t 0
Initial condition xi(t0) i=1,…n + Local evolution laws
In continuous time: tIR, differential equations
dxi (t )
 f i x1 (t ), x 2 (t ),...,x n (t ) 
dt
i  1,...,n
These are sistems of
first order autonomous
evolution equations.
We shall see how higher
order equations, as well
In discrete time: tIN, difference equations (iterated maps) as nonautonomous
equatons can be reduced
to this kind of evolution
xi (t  1)  f i x1 (t ), x2 (t ),...,xn (t ) i  1,...n
equations


From continuous time to discrete time:
dxi
xi
 g ( x) 
 g ( x) 
dt
t
xi (t  t )  xi (t )
 g ( x)
t
assum e t  1 as unit tim e:
Then we get x i (t  1)  xi (t )  g ( x(t )), i.e. x i , t 1  f ( x t ) t  IN
However, often state changes in economic or social systems are driven
by decisions, i.e. events occurring in discrete time, that cannot be
continuously revised
Cobweb Model: production lag
An output at time t has unit price pt
Consumer demand at time t : Qd = D ( pt )
(D demand function)
Producer supply at time t : Qs = S (pet)
(S supply function)
where pet is the price expected by producers at time t on the basis of the information set
they have at the time t t , t being the production lag
Let D(p) be an invertible function (e.g. continuous decreasing):
D(pt) = S (pet) gives pt = D-1S(pet) = f (pet)
Assume naive expectations pet = pt-t and let pruduction lag t = 1
Then pt = f (pt-1)
If demand and supply are linear:
D(p) = a  bp ;
S(p) = c + dp
then:
pt  
d
ac
pt 1 
b
b
OLG models
Consider an economy with individuals and firms. The individual life is divided into two
periods. An individual born in period t consume c1t, in its first period and c2t in the second
one, with utility
u  c   1    u  c 
1t
2t
During the first period, he works, having a wage wt, and he consumes a portion of wt
saving the remaining for the next period. The population increases at a constant rate n.
The firms work in a perfect competition framework and have a production function
F(K,L) with constant scale returns. The output for worker is Y L  y  f  k 
Problem of the individuals:
c1t  st  wt
s.t. 
 c2t  1  rt 1  st
max u  c1t   1    u  c2t 
st  s  wt , rt 1 
Problem of the firms
f  kt   kt f '  kt   wt
Equilibrium in the good market
;
f ' kt   rt 1
Kt 1  Lt s  wt , rt 1 
which can be expressed in terms of capital/labour as
1  n  kt 1  s  f  kt   kt f '  kt  , f '  kt  
Discrete dynamical systems
Event driven time:
Set of dynamic times {t0 , t1 , t2 , …, tn , …}
Simulated time
t=0,1, 2,…, n,…
Given:
 x1 (t ) 
 x (t ) 
x(t )   2   IRn
.



x
(
t
)
 n 
= IN
 x t 1  f ( x t )
with 
 x 0 given
Repeated application of map f (i.e. composed with itself)
x1 = f(x0)
x2 = f(x1) = f(f(x0))= f○f (x0) = f 2(x0)
xn = f(xn-1) = f n(x0)
inductively defines a trajectory: t (x0) = {xt = f t (x0)}
i.e. (t, x0 ) : IN  IRn  f t ( x0 )
xt+1 = f ( xt )
x0 given
f
xt
xt + 1
Inductively, i.e. by iteration of map f ...
x0
f
x1
f
x2 ...
xt
… a trajectory is obtained
x1 = f (x0) x2 = f (x1) = f (f (x0) = f 2 (x0) … xt = f t (x0)
f
xt+1 ...
Linear maps: f ( x ) = a x.
x1 = a x0
x2 = a x1 = a ( a x0 ) = a² x0
x3 = a x2 = a ( a² x0 ) = a³ x0
…
xn = a xn1 = a ( a n-1 ) x0 = anx0
Multiplier  = a
If ||<1 contraction
If ||>1 expansion
0   1
1    0
 1
Solution in closed form:
  1
xt  x0 a
t
= 1 xt = x0 constant
= 1 xt = (-1)t x0 alternating
bifurcation values
Example: compound interest i%
Let r = i/100
Ct+1 = Ct + r Ct = (1+r) Ct
Solution: Ct = C0 (1+r)t
Affine (linear non homogeneous) xt 1  f ( xt )  axt  b
Can be reduced to the homogeneous case by a change of
variable (a translation)
Equilibrium (or steady state): xt+1 = xt
is a fixed point of the map, i.e.: f(x) = x
Solution:
b
x 
1 a
*
Let zt = xtx* i.e. xt = zt + b/(1a)
Then
zt+1 = a zt  zt = z0 a t hence
b  t
b

xt   x0 
a 
1 a 
1 a

Liner Cobweb with naive expectations
The model we considered is
d
ac
pt   pt 1 
b
b
i.e., a first order autonomous linear difference equation.
Then the generic solution is
n

d
 1
b
ac
 d
pt  K    
 b bd

d
 1
b
Stability of the equilibrium points
• An equilibrium point x* is stable if for any neighborhood
U of x* there esists a neighborhood V  U such that any
solution starting in V belongs to U for any t.
• Moreove, if V can be chosen such that
x  t   x* ,
t  
x* is said asymptotically stable
• An equilibrium point is unstable if it is not stable
• If x* is an asymptotically stable equilibrium point, the set
of the initial condition giving rise to the trajectories
converging to x* is the basin of attraction of x*
• If the basin of attraction of x* coincides with the whole
state space W then x* is globally asymptotically stable.
Stability conditions for a discrete linear system of dim. 1
with multiplier 
|| < 1
i.e. -1 <  < 1
If the unique equilibrium of a linear system is stable then it is always
globally stable, i.e. local stability is equivalent to glabal stability
Things are different for nonlinear systems
However their study always starts with their linear approximation
around equilibrium points
Let us introduce a non-linearity
xt 1  axt  sx
2
t
A tax propostional to the square of capital,
a population growing in an environment with limited resources
By the following linear (hence invertible) change of variable
z=(s/a)x
we get the so called “standard logistic map”
zt 1  azt (1  zt )
degree 2
z1  az0 (1  z0 )
z2  az1 (1  z1 )  a[az0 (1  z0 )][1  az0 (1  z0 )] degree 22=4
z3  az2 (1  z2 ) 
degree 23=8
 a 2 [az0 (1  z0 )][1  az0 (1  z0 )]1  a[az0 (1  z0 )][1  az0 (1  z0 )]
.
.
.
z10 = ……… degree 210 = 1024 !!!!
x t + 1 = f ( xt )
If f (xt) > xt then xt+1> xt
If f (xt) < xt then xt+1 < xt
Law of evolution:
xt+1 = xt
If f (xt) = xt then
Steady state
x0
x1
x2
x1
x1
x1 = f (x0)
x1
x3
x4
x2
x0
x0
x0
Stability
-1<f’(x*)<0
0<f’(x*)<1
Instability
f’(x*)>1
f’(x*)< -1
Local stability at an equilibrium point x*= f (x*)
Linear approximation around the equilibrium:
xt+1= f(xt)= f (x*) + f ’(x*)(xtx*) + o(xt x*)
Hence:
xt+1 x*  f ’(x*)(xtx*)
Xt+1 =  Xt where Xt = xtx* displacement from equilibrium
x* is locally asymptotically stable if || = | f ’(x*) | <1
x* is said to be hyperbolic if || = | f’(x*) | 1
Hartman-Grobman theorem (1959-1960). Let x* be a hyperbolic fixed
point of xt+1=f(xt), with f differentiable. Then a neignborhood of x*
exists where the map is topologically conjugate to its linear
approximation Xt+1 = f’(x*)Xt
For the logistic map q*=0 and p* = (a-1)/a are the two equilibria
f’ (x) = a(1-2x). Hence f’(q*) = a, f’(p*) = 2-a
q* stable for -1<a<1 ; p* stable for 1<a<3
Logistic map
xt 1   xt 1  xt 
Bifurcation diagram:
sequence of period
doubling bifurcation
leading to chaotic
dynamics.
But much more can be
said
logistic
Structural stability, Bifurcations
Consider an dynamical system depending on some parameters.
When a parameter undergoes a small variation, the phase portrait is
modified as well:
– if the new phase portrait is topologically conjugated to the old
one, we said that the system is structurally stable with respect
to the parameter variation
– if not, we said that a bifurcation has occurred
• The parameter values causing a bifurcation are called bifurcation
values
• A bifurcation is said local when it can be detected from the
linearised system.
Multiplier  = f ’ (x*) through value 1
• Fold bifurcation:
– two fixed points appear, one stable and one unstable
x
x
x
Bifurcation diagram
Normal form:
f(x,) =  + x  x2
Multiplier  = f ’ (x*) through value 1
• Transcritical bifurcation (or stability exchange):
– two fixed points merge, exchanging their stability
x
x
x
Bifurcation diagram
Normal form:
f(x,) =  x + x  x2
Multiplier  = f ’ (x*) through value 1
• Pitchfork bifurcation
– a fixed point becomes unstable (stable) and two further fixed points
appear, both stable (unstable)
x
x
subcritical
x
supercritical
Normal form:
f(x,) =
 x + x  x3
Multiplier  = f ’ (x*) through value 1
• Flip bifurcation (period doubling bifurcation):
– the fixed point becomes unstable and a stable period 2 cycle appears,
surrounding it. It corresponds to a pitchfork bifurcation of the second
iterated of the map.
Normal form:
f(x,) = 1x + x3
subcritical
supercritical
x
alfa
y
a = 2.5
a=2
x
x
y
3
5
x
4
x*
x
3
x
*
2
x
x
x
1
p*
2
x
0.5
0
1
x
x
1
p*
a=2
x
0

a = 3.1
y



f(x)
2
f(x)
0
x
0
0.5
F(x)=f (x)
1
x
a=3.1
1
2
F(x)=f (x)
x*
x*
0.5
a=2.5
x
xn
x0
n
yn
y0= x0+10 -6
n
|xn - yn|
n
Deterministic chaos
We may say that chaotic dynamics exist if there is:
• (1) Sensitivity to the initial conditions
two trajectories starting from different, but arbitrarly close, remains
bounded but their reciprocal distance exponentially increases and,
in a finite time, becomes as large as the the state variables.
• (2) Transitivity (or mixing):
the points of a trajectory obtained starting from a generic initial
condition densely cover a zone of the phase space, i.e. any point of the
trajectory is an accumulation point of the trajectory itself
• (3) Existence of an infinite number of repelling cycles
and the periodic points are dense in the region occupied by the chaotic
trajectories.
Remark: (2) and (3) imply (1)
a = 3.61
a = 3.678574
c
c2
c2=c3=x*
c3
c1
a 2  a  a1
c
c1
a  a1
Self-similarity
J
c
c2
I
c1
c3
c
c2=f(c1)
c3=f(c2)
c1=f(c)
The geometry of chaos: Stretching & Folding
0.875
Kneading of the dough
Invariant sets
• Equilibria: constant solutions
• Cycles: not costant periodic solutions
– finite number of points  x1, x2 ,..., xn  such that f  xi   xi 1
and f  xn   x1
• Equilibria and cycles are particular invariant sets, i.e., sets S such
that the orbits starting in S belong to S. The stability definition can
be extended to the invariant sets:
– An invariant set S0 is stable if for any open set U containing S0
there exists an open set V containing S0 such that any solution
with initial condition in V belongs to U for each t.
– Moreover, if V can be chosen so that
dist  x  t  , S0   0 per t  
then S0 is asymptotically stable
• Attractors: asymptotically stable invariant sets.
Let C = {c1, c2, …, ck} be a k-cycle of xt+1 = f(xt)
i.e. cic1 , i=2,…,k ; f(ci) = f(ci+1) , i=1, …, k-1, and f(ck)=c1
In other words:
C = {c1, f(c1),f 2(c1), …, f k-1(c1)} and f k(c1) = c1
Then c1 is a fixed point of f k (but it is not a fixed point of fi with i<k.
Indeed, any cj, j=1,…,k, is a fixed point of f k .
By the chain rule it is easy to compute the multiplier of C:
C = Dfk (ci) = f ′ (c1) ∙ f ′ (c2) ∙… ∙ f ′ (ck) =
C is stable if |C| < 1
k
 f ' (c )
j
j 1
What we said for the fixed points of f , on their stability
and local bifurcations etc. can be applied to k-cycles, seen
as fixed points of f k
In particular:
A couple of k-cycles (one stable and one unstable) can be
created by a fold bifurcation of f k
A k cycle can give rise to a 2k-ycle via a flip bifurcation of
fk
Sharkovsky Theorem (1964).
If a k-cycle exists for f : II, then at least a p-cycle exists for
each number p that follows k in the following total ranking of
natural numbers:
3, 5, 7, 9, …, 3∙2, 5∙2, 7∙2, …, 3∙22, 5∙22, …, ….24, 23, 22, 2, 1
Li & Yorke Theorem (1975): Period 3 implies chaos
If f: II has a 3-cycle then:
An uncountable set of points S  I exists that does not include any
cycle and has the following properties:
i) For any p, q  S, pq,
max lim f n ( p)  f n (q)  0 and min lim f n ( p)  f n (q)  0
n
n
(ii) For any q  S and any periodic point p  I
max lim f n ( p)  f n (q)  0
n 
The trajectories starting from an i.c. in S (scrambled set) are
chaotic, i.e. they have the 3 properies that characterize
deterministic chaos
Remark: it may occur mes(S) = 0 (invisible chaos)
f ' ' ' ( x) 3  f ' ' ( x) 

 
Let S(f) =
f ' ( x) 2  f ' ( x) 
2
Schwarzian derivative
Theorem of Singer (1978)
Let f : II of class C(3) have a finite number of critical points x1,…,xp
and S(f)<0 in I \ {x1,…,xp}.
Let C={c1,…,ck} be a stable k-cycle of f.
Then at least a critical point exists whose trajectory converges to C
In other words, any basin of a stable cycle must include at least a
critical point.
Then, the maximum number of stable cycles cannot exceed the number
of critical points
6a 2
S( f )  
0
2
(2ax  b)
Example: f(x) = ax2 + bx + c has
And is unimodal (1 critical point). Then no more than 1 stable cycle
Nonlinear autonomous
dynamical systems
• Two dynamical systems are topologically conjugated if there
exists a homeomorphsm h mapping orbits of the first system
onto orbits of the second one, preserving the direction of time.
• Let us consider an autonomous system in normal form and f (x)
be its second member, defined in W and C1 with f (0) = 0.
Moreover, let Df (0) be the jacobian matrix of f at 0, assumed
non singular.
– The linear dynamical system
x '  Df  0 x
o
xt 1  Df  0 xt
is called linearised (at x = 0) dynamical system.
• It is possible to prove that a nonlinear dynamical system and the
linearizated one are topologically conjugated in a neighborhood
of 0.
Topologically conjugate maps
xt 1  f ( xt )
yt 1  g ( yt )
y = h(x) where h is continuous and invertible.
x = h -1 (y) is the inverse transformation
Conjugate if g ○h = g(h(y)) = h○f = h(f(y))
Then the two maps have the same qualitative dynamics
y0  h( x0 ), y1  h( x1),..., yn  h( xn )
Basins
basins in 1- dimensional discrete dynamical systems
- generated by invertible maps
- generated by noninvertible maps
contact bifurcations and non connected basins
Continuous and increasing maps
•The only invariant sets are the fixed points.
•When many fixed points exist they are alternatingly stable and unstable:
the unstable fixed points are the boundaries that separate the basins of
the stable ones.
• Starting from an initial condition where the graph is above the diagonal,
i.e. f(x0)>x0, the trajectory is increasing, whereas if f(x0)<x0 the trajectory
is decreasing
r*
r*
p*
q*
p*
q*
f(x) = a arctan (x-1)
basin
boundary
a = 0.5
a=3
fold bifurcation
a=1
Continuous and decreasing maps
The only possible invariant sets are one fixed point and cycles of period 2, being f2=f°f
increasing
The periodic points of the 2-cycles are located at opposite sides with respect to the
unique fixed point, the unstable ones being boundaries of the basins of the stable ones.
If the fixed point is stable and no cycles exist, then it is globally stable.
f(x) = – ax3 + 1
a = 0.2
a = 0.5
a = 0.7
Nononvertible maps. Several preimages
Z0
c
q-1
p
p
Z2
q
q
c-1
r
Example: x’ = f(x) = ax (1-x)
Z0 - Z2 map:
if x’ < a/4 then
f 1 ( x' )  f11 ( x' )  f 21 ( x' )  x1, x2 
where:
aa  4 x'
1
x1  f ( x' )  
2
2a
aa  4 x'
1
1
x 2  f 2 ( x' )  
2
2a
1
1
Folding by f
c-1
Unfolding by f-1
critical point c = a/4
c1  f 21 (c)  f 21 (c) 
1
2
Remark: Df(c-1) = 0 and c = f(c-1)
Noninvertible map: f (x) = a x (1– x)
y
Z0
c=a/4
Z2
0
c-1= 1/2
1
x
z
Z1
cmax
p
Z3
cmin
Z1
r
q
z
Z1
q-12
q-11
cmax
p
Z3
cmin
q
r
Z1
c-1
After the examples some definitions
The basin of an attractor A is the set of all points that generate
trajectories converging to it:
B(A)= {x| f t(x)  A as t +}
Let U(A) be a neighborhood of A whose points converge to it. Then
U(A)  B(A), and also the points that are mapped into U after a finite
number of iterations belong to B(A):

B  A 
f  n (U ( A))
n 0
-n(x)
where f
represents the set of the rank-n preimages of x.
From the definition it follows that points of B are mapped into B both
under forward and backward iteration of T
f(B)  B, f -1(B) = B
;
f (B) B, f -1(B)= B
This implies that if an unstable fixed point or cycle belongs to B then
B must also contain all of its preimages of any rank.