mdef 2008 urbino (italy) september 25 - 27, 2008 bifurcation curve structure in a family of linear...
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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)[email protected] ; [email protected]
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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.
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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
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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
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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.
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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
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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
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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.
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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
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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
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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
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Cycle LRRCycle LRR
Appearance:The last point merges with the border
LR0
Disappearance:The first point merges with the border
0RR
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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
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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.
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Bifurcation diagramBifurcation diagram
1
2
3
4
5
LR
LRR
LRRR
5
7
0.7
Chaotic intervals
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One-dim. bifurcation diagramOne-dim. bifurcation diagram
LR
LRLRRLRR LRRR
LRRLRRR
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Cycle LRLRRCycle LRLRR
Appearance:The last point merges with the border
LRLR0
Disappearance:The third point merges with the border
LR0RR
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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
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Tongues of second levelTongues of second level
8
7
11
9
5
3
2
LRLRR
(LR)2LRR
LRRLR
LR(LRR)2
LR(LRR)3
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Plane (Plane (, , ))
enlargement
2
3
4
5
2
3
50.7
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Border bifurcation curvesBorder bifurcation curves
2
3
5
2
3
5
7
8
7
8
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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
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Initial condition Initial condition
2
3
4
5
6
2
3
4
5
6
Flip bifurcation curves
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Initial condition Initial condition
2
3
4
56
2
3
4
56
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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