1. 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: tIR, differential equations
In discrete time: tIN, difference equations (iterated maps)
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

2. From continuous time to discrete time:
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

3. 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 Dt , Dt 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-Dt and let pruduction lag Dt = 1
Then pt = f (pt-1)
If demand and supply are linear:
D(p) = a  bp ; S(p) = c + dp then:

4. 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
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

5. Discrete dynamical systems
Event driven time:
Set of dynamic times {t0 , t1 , t2 , …, tn , …}
Simulated time t = 0 , 1 , 2 , … , n , … = IN
Given:
with
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.

6. f f Inductively, i.e. by iteration of map f ...
xt+1 = f ( xt )
x0 given f xt
xt + 1
Inductively, i.e. by iteration of map f ... x0 f
x2 ... xt
xt+1 ... x1
… a trajectory is obtained
x1 = f (x0) x2 = f (x1) = f (f (x0) = f 2 (x0) … xt = f t (x0)

7. 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 l = a
If |l|<1 contraction
If |l|>1 expansion
Solution in closed form:
= xt = x0 constant
= 1 xt = (-1)t x0 alternating
bifurcation values

8. Example: compound interest i%
Let r = i/100
Ct+1 = Ct + r Ct = (1+r) Ct
Solution: Ct = C0 (1+r)t

9. Affine (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 = xtx* i.e. xt = zt + b/(1a)
Then
zt+1 = a zt  zt = z0 a t hence

10. Liner Cobweb with naive expectations
The model we considered is
i.e., a first order autonomous linear difference equation.
Then the generic solution is

11. Stability 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 W then x* is globally asymptotically stable.

12. Stability conditions for a discrete linear system of dim. 1
with multiplier l
|l| < i.e < l < 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

13. z=(s/a)x Let us introduce a non-linearity
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”
degree 2
degree 22=4
degree 23=8 .
z10 = ………
degree 210 = !!!!

14. Law of evolution: x t + 1 = f ( xt )
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 x0 x1 x2 x3 x4 x1 x0 x2 x0
x1 = f (x0)

16. Stability
-1<f’(x*)<0
0<f’(x*)<1
Instability
f’(x*)>1
f’(x*)< -1

17. 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 = l Xt where Xt = xtx* displacement from equilibrium
x* is locally asymptotically stable if |l| = | f ’(x*) | <1
x* is said to be hyperbolic if |l| = | f’(x*) | 1
Hartman-Grobman theorem ( ). 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

18. 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

19. Logistic map
Bifurcation diagram: sequence of period doubling bifurcation leading to chaotic dynamics.
But much more can be said
logistic

20. 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.

21. Multiplier l = f ’ (x*) through value 1
Fold bifurcation:
two fixed points appear, one stable and one unstable
Bifurcation diagram
Normal form: f(x,a) = a + x - x2

22. Multiplier l = f ’ (x*) through value 1
Transcritical bifurcation (or stability exchange):
two fixed points merge, exchanging their stability
Bifurcation diagram
Normal form:
f(x,a) = a x + x - x2

23. Multiplier l = f ’ (x*) through value 1
Pitchfork bifurcation
a fixed point becomes unstable (stable) and two further fixed points appear, both stable (unstable)
subcritical
supercritical
Normal form:
f(x,a) =
a x + x - x3

24. Multiplier l = 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,a) = -(1+a)x + x3
subcritical
supercritical

25. a = 2.5
a = 2
a = 3.1

27. x0
y0= x
|xn - yn| xn yn n

28. 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)

29. c1 c3 c2 c
c2=c3=x*
a = 3.61
a =

30. Self-similarity

31. c1 c3 c2 c I J
c1=f(c)
c2=f(c1) c
c3=f(c2)
8.4

32. The geometry of chaos: Stretching & Folding
0.875

33. Kneading of the dough

34. 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.

35. 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:
lC = Dfk (ci) = f ′ (c1) ∙ f ′ (c2) ∙… ∙ f ′ (ck) =
C is stable if |lC| < 1

36. 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

37. 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

38. 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:
For any p, q  S, pq,
(ii) For any q  S and any periodic point p  I
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)

39. Let S(f) =
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
Example: f(x) = ax2 + bx + c has
And is unimodal (1 critical point). Then no more than 1 stable cycle

40. 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
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.

41. Then the two maps have the same qualitative dynamics
Topologically conjugate maps
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

42. Basins
basins in 1- dimensional discrete dynamical systems
- generated by invertible maps
- generated by noninvertible maps
contact bifurcations and non connected basins

43. 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* p* q* q*

44. f(x) = a arctan (x-1) a = 0.5 a = 3 a = 1 basin boundary
fold bifurcation
a = 1

45. 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

46. a = 0.7

47. Nononvertible maps. Several preimagesZ0 c
q-1 p p Z2 q q
c-1 r

48. Z0 - Z2 map: if x’ < a/4 then where: x’ = f(x) = ax (1-x)
Example:
x’ = f(x) = ax (1-x)
Z0 - Z2 map:
if x’ < a/4 then
where:
critical point c = a/4
Remark: Df(c-1) = 0 and c = f(c-1)
Folding by f
c-1
Unfolding by f-1

49. Noninvertible map: f (x) = a x (1– x)
c=a/4
= 1/2

50. z Z1
cmax Z3 p
cmin Z1 q r

51. z Z1
q-12
q-11
cmax p Z3
cmin q r Z1
c-1

53. 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.