Basic ingredients of a dynamical system

State variables : x = (x1, x2, …, xn)

Evolution operator

Initial condition xi(t0) i=1,…n + Local evolution laws

In continuous time: tIR, differential equations

In discrete time: tIN, difference equations (iterated maps)

nitxtxtxftx nii ,...1)(),...,(),()1( 21

0,...,1))(),...,(;()( 001 tnitxtxttx nii

nitxtxtxfdt

tdxni

i ,...,1)(),...,(),()(

21

These are sistems of first order autonomous evolution equations.

We shall see how higher order equations, as well as nonautonomous equatons can be reduced to this kind of evolution equations

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

From continuous time to discrete time:

INtxfxeitxgtxtxgetweThen

timeunitastassume

xgt

txttxxg

t

xxg

dt

dx

ttiii

iiii

)(..)),(()()1(

:1

)()()(

)()(

1,

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(pe

t) = 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: b

cap

b

dp tt

1

Cobweb Model: production lagCobweb Model: production lag

OLG modelsOLG modelsConsider 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

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

Problem of the individuals:

Problem of the firms

Equilibrium in the good market

which can be expressed in terms of capital/labour as

1 21t tu c u c

Y L y f k

1

1 22 1

max 1 s.t.

1t t t

t tt t t

c s wu c u c

c r s

1,t t ts s w r

1' ; 't t t t t tf k k f k w f k r

1 1,t t t tK L s w r

11 ' , 't t t t tn k s f k k f k f k

)(:),( 00 xfIRINxt tn

n

n

IR

tx

tx

tx

tx

)(

.

)(

)(

)( 2

1

givenx

xfx tt

0

1 )(

Event driven time: Set of dynamic times {t0 , t1 , t2 , …, tn , …}Simulated time t = 0 , 1 , 2 , … , n , … = IN

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)

Discrete dynamical systems

inductively defines a trajectory: (x0) = {xt = f t (x0)}i.e.

withGiven:

fxt xt + 1

… a trajectory is obtained

xt+1 = f ( xt )x0 given

Inductively, i.e. by iteration of map f ...

x1 = f (x0) x2 = f (x1) = f (f (x0) = f 2 (x0) … xt = f t (x0)

x0 f x2 ... fxt xt+1 ...x1 f

0t

tx x a

0 1

1 0

1

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 xn1 = a ( a n-1 ) x0 = anx0

Solution in closed form:

Multiplier = aIf ||<1 contractionIf ||>1 expansion

= 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

baxxfx ttt )(1Affine (linear non homogeneous)

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:

Let zt = xtx* i.e. xt = zt + b/(1a)

Then zt+1 = a zt zt = z0 a t hence

a

bx

1*

a

ba

a

bxx t

t

110

The model we considered is

i.e., a first order autonomous linear difference equation.

Then the generic solution is

b

cap

b

dp tt

1

n

td a c

p Kb b d

1

d

b 1

d

b

Liner Cobweb with naive expectationsLiner Cobweb with naive expectations

Stability of the equilibrium pointsStability of the equilibrium points• An equilibrium point x* is stable if for any neighborhood U of x* there

esists a neighborhood such that any solution starting in V belongs to U for any t.

• Moreove, if V can be chosen such that

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 then x* is globally asymptotically stable.

V U

*, x t x t

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

Stability conditions for a discrete linear system of dim. 1 with multiplier

|| < 1i.e. -1 < < 1

Things are different for nonlinear systems

However their study always starts with their linear approximation around equilibrium points

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)xwe get the so called “standard logistic map”

21 ttt sxaxx Let us introduce a non-linearity

)1(1 ttt zazz

.

.

.z10 = ……… degree 210 = 1024 !!!!

)1( 001 zazz )]1(1)][1([)1( 0000112 zazzazazazz

degree 2

degree 22=4

)]1(1)][1([1)]1(1)][1([

)1(

000000002

223

zazzazazazzaza

zazz

degree 23=8

If f (xt) > xt then xt+1> xt

If f (xt) < xt then xt+1 < xt

If f (xt) = xt then xt+1 = xt

Steady state

x0

x1 = f (x0)

Law of evolution: x t + 1 = f ( xt )

x0

x1

x1

x0

x1

x2

x0

x1

x2

x3

x4

0<f’(x*)<1

-1<f’(x*)<0

f’(x*)>1 f’(x*)< -1

Stability

Instability

Local stability at an equilibrium point x*= f (x*)

Linear approximation around the equilibrium:

xt+1= f(xt)= f (x*) + f ’(x*)(xtx*) + o(xt x*)

Hence:

xt+1 x* f ’(x*)(xtx*)

Xt+1 = Xt where Xt = xtx* displacement from equilibrium

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

x* is locally asymptotically stable if | = | f ’(x*) | <1

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

1 1t t tx x x

Bifurcation diagram: sequence of period doubling bifurcation leading to chaotic dynamics.

But much more can be said

Logistic mapLogistic map

Structural stability, Bifurcations 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.

• Fold bifurcation:– two fixed points appear, one stable and one unstable

xxx

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 x

subcriticalsupercritical

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.

x

alfa

supercritical subcritical

Normal form:f(x,) = x + x3

Multiplier = f ’ (x*) through value 1

x

y

x*

a = 2

0 .5 1

p*x

0

x1

x2

x3

x

y

x*

a = 2 .5

0 .5 1p *

x0

x1

x2

x3

x4

x5

x

y

a= 3 .1

0 10 .5

F x = f x( ) ( )2

F x = f x( ) ( )2

f x( )

f x( )

x*

x0

x*

a = 2

a = 3.1

a = 2.5

x0

y0= x0+10 -6

|xn - yn|

xn

yn

n

n

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)

c1

c3

c2

c

c1

c

12 aaa 1aa

c2=c3=x*

a = 3.61 a = 3.678574

Self-similarity

c1

c3

c2

c

I

J

c1=f(c)

c2=f(c1)

c

c3=f(c2)

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

• 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

then S0 is asymptotically stable

• Attractors: asymptotically stable invariant sets.

1 2 1

1

, ,..., such that

and

n i i

n

x x x f x x

f x x

0, 0 per dist x t S t

Let C = {c1, c2, …, ck} be a k-cycle of xt+1 = f(xt)

i.e. cic1 , 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

jjcf

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 f k

Sharkovsky Theorem (1964).

If a k-cycle exists for f : II, 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: II 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, pq,

(ii) For any q S and any periodic point p I

0)()(limmin0)()(limmax

qfpfandqfpf nn

n

nn

n

0)()(limmax

qfpf nn

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)

Let S(f) = 2

)('

)(''

2

3

)('

)('''

xf

xf

xf

xfSchwarzian derivative

Theorem of Singer (1978)

Let f : II 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

Example: f(x) = ax2 + bx + c has

And is unimodal (1 critical point). Then no more than 1 stable cycle

0)2(

6)(

2

2

bax

afS

Nonlinear autonomous Nonlinear autonomous dynamical systemsdynamical 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 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

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.

1' 0 o 0t tx f x x f x D D

)(1 tt xfx )(1 tt ygy

)(),...,(),( 1100 nn xhyxhyxhy

Topologically conjugate maps

Then the two maps have the same qualitative dynamics

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))

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

p*

q*

r*

p*

q*

r*

f(x) = a arctan (x-1)

a = 3

a = 1

a = 0.5

basinboundary

fold bifurcation

a = 0.5a = 0.2

Continuous and decreasing mapsThe only possible invariant sets are one fixed point and cycles of period 2, being f2=f°f increasingThe 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 = 0.7

Z2

Z0

c

c-1

p

q

p

q

r

q-1

Nononvertible maps. Several preimages

x’ = f(x) = ax (1-x)

Z0 - Z2 map:if x’ < a/4 then

where:

a

xaaxfx

2

'4

2

1)'(1

11

211

21

11 ,)'()'()'( xxxfxfxf

a

xaaxfx

2

'4

2

1)'(1

22

critical point c = a/4

2

1)()( 1

21

21 cfcfc

Remark: Df(c-1) = 0 and c = f(c-1)

Example:

Folding by f

Unfolding by f-1

c-1

x

y

Z0

Z2

0 1c-1

Noninvertible map: f (x) = a x (1– x)

= 1/2

c=a/4

Z3

Z1

cmax

p

q

cmin

Z1

z

r

Z3

Z1

cmax

p

qcmin

Z1

z

r

c-1

q-11

q-12

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):

where f -n(x) 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.

0

( ( ))n

n

B A f U A