MDEF 2008MDEF 2008Urbino (Italy)Urbino (Italy)

September 25 - 27, 2008September 25 - 27, 2008

Bifurcation Curve Structure Bifurcation Curve Structure in a family of in a family of

Linear Discontinuous MapsLinear Discontinuous Maps

Anna Agliari & Fernando BignamiDept. of Economic and Social SciencesCatholic University, Piacenza (Italy)anna.agliari@unicatt.it ; fernando.bignami@unicatt.it

OUTLINEOUTLINE

ProblemProblem: The investigation of the bifurcation curves bounding the periodicity region in a piecewise linear discontinuous map.

• InitialInitial motivation of the studymotivation of the study:: Economic model describing the income distribution.

• Simplified mapSimplified map topologically conjugated to the model.

• Border collision bifurcationBorder collision bifurcation: The bifurcation curves of the different cycles are associated with the merging of a periodic point with the border point.

• TonguesTongues of first and second level: of first and second level: Analytical bifurcation curves

• Excursion beyond the economic model: Excursion beyond the economic model: coexistence of cycles of different period.

Basic modelBasic model

Solow (1956)

A single aggregate output, which can be used for consumption and investement purposes, is produced from capital and labor;Aggregate labor is exogeneous

Saving propensity is exogeneous: Ft(Lt , Kt) - Ct = s Ft(Lt , Kt) Production function is homogeneous, with intensive form f(kt)

where the state variable is the capital intensity kt

1 1

1 ,t t t

t t t t

K K I

K F L K C

11

11 t ttk k sf k

n

GeneralizationsGeneralizations

Kaldor (1956, 1957)The capital accumulation is generated by the savings behavior of two income groups: shareholders and workers. Shareholders drawing income from capital only and have saving propensity sc. Workers receive income from labor and have saving

propensity sw

Pasinetti (1962)In the Kaldor model the workers do not receive any capital income in spite of the fact that they contribute to capital formation with their savings. Workers receive wage income from labor as well as capital income as a return on their accumulated savings

The economic modelThe economic modelBöhm & Agliari (2007)Workers may have different savings propensities

from wage they save

from capital revenues they save

where

'ps f k kf k

'wws k f k

1

1

11 '

11

1 ' '1

c c ct t c t t

w w wt t w t t t p t t

k k s k f kn

k k s f k k f k s k f kn

c wk k k

Parameters: n>0 population increasing rate; δ, with 0<δ<1, capital depreciation

rate; sc, with 0< sc <1, saving propensity of shareholders; sw, with 0< sw <1, saving

propensity on wage of workers; sp, with 0< sp <1, saving propensity on income revenue

of workers.

TechnologyTechnology

Leontief production function:

min ,f k Bk A where A, B > 0

1

1

1

1

1

1 if

1

1:1

1 if 1

1 1

c cct t

c wt t

pw wt t

c ct t

c wt t

w w wt t

Bsk k

Ank k

Bs Bk k

nT

k kAn k k

As Bk k

n n

The axis is trapping: 0 0c cT k k 0ck

The one-dimensional mapThe one-dimensional map

1 if

11

if 1 1

p

w

Bs Ay y

n BF yAs A

y yn n B

The map F(y) is discontinuous, we can prove that it is topologically conjugated to

if 0

1 if 0

w p

w p

x s s xf x

x s s x

where 11

, , 1 1

p w

w p

Bs Bs n

n n B s s

Proof: Making use of the homeomorphism 1

w p

n Ay y

BA s s

Note: 0<<1, >0

Case Case ssw w > > sspp

if 0

1 if 0

x xf x

x x

*Rx

*Lx

Increasing mapUp to two fixed points: * * 1

; 1 1R Lx x

o if < 0 and • < 1 : left fixed point globally stable•≥ 1 : divergence

o if 0<< 1 and •<1 : coexistence of two stable fixed points (the border x = 0 separates the basins)• ≥1 : right fixed point globally stable

o if >1 and• ≤1 : right fixed point globally attracting•>1 : left fixed point stable with basin {x*

L 0} and divergence in {x*L 0}

o if = 0, the border x=0 stable fixed point with basin {x 0} and

•<1 : left fixed point stable with basin{x 0}

• ≥1 : divergence in {x < 0}o if =1, the right fixed point is locally stable with basin{x 0} and

• <1 : the border x=0 stable fixed point with basin {x 0}

• >1 : divergence in {x < 0}, the border x=0 being unstable• =1 : infinitely many fixed points exist.

Case Case ssww < < sspp

if 0; , ,

1 if 0

x xf x

x x

*Rx

Noninvertible mapUp to two fixed points: * * 1

; 1 1L Rx x

; , , ; , ,1f x f x 0,1 0, 0,

1

1 Right fixed point globally stable

1 divergence

0 < 0 < < 1 < 1

The right fixed point exists if > 1, and it is unstable.

If > 1 explosive trajectories may exist, and, in particular, whenthe generic trajectory is divergent .

If 0< 1 the trajectories are bounded

1

*Rx

1 1

Periodic orbits may exist

Case Case = 0 = 0

x

0

Period adding bifurcations

*0 , 1C

is a cycle of period 2 if 1

Border bifurcation

The cycles have only a periodic point on the left side:LR, LRR, LRRR, …They appear and disappear via border bifurcations.The border bifurcation values accumulate at 1

Cycle LRRCycle LRR

Appearance:The last point merges with the border

LR0

Disappearance:The first point merges with the border

0RR

Cycle LRCycle LRnn-1-1

Orbit:0

11

0

0

1 1

1 1

n nn

n

x

x x

Cycle condition: 1

0 1

1 1 1

1 1 1

n n

nx

It appears when xn-1 = 0: 2 1 21 1 1 1 0n n n

It disappears when x0 = 0: 11 1 0n n

Border bifurcation curves

Note that when = 0 the cycle of period k disappears simultaneously to the appearance of that of period k+1

Tongues of first levelTongues of first level

1

2

3

4

5

The tongues do not overlap: no coexistence of cycles is possible

The intersection points of two curve associated with a cycle belong to the straight line

1 0

On this line the multiplier of the cycle is 1: fold curve

If the parameters belong to this line, each point in the range (-1 , ) belong to a cycle.

Bifurcation diagramBifurcation diagram

1

2

3

4

5

LR

LRR

LRRR

5

7

0.7

Chaotic intervals

One-dim. bifurcation diagramOne-dim. bifurcation diagram

LR

LRLRRLRR LRRR

LRRLRRR

Cycle LRLRRCycle LRLRR

Appearance:The last point merges with the border

LRLR0

Disappearance:The third point merges with the border

LR0RR

Tongues of second levelTongues of second levelwith only two Lwith only two L

1

5

7

9Appearance: x2q+2=0

Cycle LRqLRq+1

1

2 2 1 11 1 0

1 1

q qq q

Disappearance: xq+1=02 1 1

2 1 1 1 10

1 1 1 1

q q q qq

Tongues of second levelTongues of second level

8

7

11

9

5

3

2

LRLRR

(LR)2LRR

LRRLR

LR(LRR)2

LR(LRR)3

Plane (Plane (, , ))

enlargement

2

3

4

5

2

3

50.7

Border bifurcation curvesBorder bifurcation curves

2

3

5

2

3

5

7

8

7

8

Beyond the economic model: Beyond the economic model: < 0< 0

Divergence: 1 1

1

22;3

3;43

Tongues overlap

Coexistence of cycles is a possible issue

divergence

Initial condition Initial condition

2

3

4

5

6

2

3

4

5

6

Flip bifurcation curves

Initial condition Initial condition

2

3

4

56

2

3

4

56

Main referencesMain references

• Pasinetti, L.L. (1962) “Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth”, Review of Economic Studies, 29, 267-279

• Samuelson, P.A. & Modigliani, F. (1966) “The Pasinetti Paradox in Neoclassical and More General Models”, Review of Economic Studies, 33, 269-301

• Böhm, V. & Kaas, L. (2000) “Differential Savings, Factor Shares, and Endogeneous Growth Cycles”, Journal of Economic Dynamics and Control, 24, 965-980

• Avrutin V. & Schanz M. (2006) “Multi-parametric bifurcations in a scalar piecewise-linear map” , Nonlinearity, 19, 531-552

• Avrutin V., Schanz M. & Banerjee S. (2006) “Multi-parametric bifurcations in a piecewise-linear discontinuous map”, Nonlinearity, 19, 1875-1906

• Leonov N.N. (1959) “Map of the line onto itself”, Radiofisica, 3(3), 942-956

• Leonov N.N. (1962) “Discontinuous map of the stright line”, Dohk. Ahad. Nauk. SSSR, 143(5), 1038-1041

• Mira C. (1987) “Chaotic dynamics” , World Scientific, Singapore