Bifurcation Curve Structure in a family of Linear Discontinuous Maps

1
Bifurcation Curve Structure in a family of
Linear Discontinuous Maps
MDEF 2008 Urbino (Italy) September 25 - 27, 2008
Anna Agliari Fernando Bignami Dept. of
Economic and Social Sciences Catholic University,
Piacenza (Italy) anna.agliari_at_unicatt.it
fernando.bignami_at_unicatt.it
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OUTLINE

• Problem The investigation of the bifurcation
  curves bounding the periodicity region in a
  piecewise linear discontinuous map.
• Initial motivation of the study Economic model
  describing the income distribution.
• Simplified map topologically conjugated to the
  model.
• Border collision bifurcation The bifurcation
  curves of the different cycles are associated
  with the merging of a periodic point with the
  border point.
• Tongues of first and second level Analytical
  bifurcation curves
• Excursion beyond the economic model coexistence
  of cycles of different period.

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Basic 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
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Generalizations
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 model
Böhm Agliari (2007) Workers may have different
savings propensities from wage they save from
capital revenues they save where
Parameters ngt0 population increasing rate d,
with 0ltdlt1, capital depreciation rate sc,
with 0lt sc lt1, saving propensity of shareholders
sw, with 0lt sw lt1, saving propensity on wage of
workers sp, with 0lt sp lt1, saving propensity on
income revenue of workers.
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Technology
Leontief production function
where A, B gt 0
The axis is trapping
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The one-dimensional map
The map F(y) is discontinuous, we can
prove that it is topologically conjugated to
where
Proof Making use of the homeomorphism
Note 0ltalt1, bgt0
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Case sw gt sp
Increasing map Up to two fixed points

• if m lt 0 and
• b lt 1 left fixed point globally stable
• b 1 divergence
• if 0lt m lt 1 and
• b lt1 coexistence of two stable fixed points
  (the border x 0 separates the basins)
• b 1 right fixed point globally stable
• if m gt1 and
• b 1 right fixed point globally attracting
• b gt1 left fixed point stable with basin xL
  gt 0 and divergence in xL lt 0
• if m 0, the border x0 stable fixed point with
  basin x gt 0 and
• b lt1 left fixed point stable with basinx lt
  0
• b 1 divergence in x lt 0
• if m 1, the right fixed point is locally stable
  with basinx gt 0 and
• b lt1 the border x0 stable fixed point with
  basin x lt 0
• b gt1 divergence in x lt 0, the border x0
  being unstable
• b 1 infinitely many fixed points exist.

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Case sw lt sp
Noninvertible map Up to two fixed points
m
m
m-1
m-1
Right fixed point globally stable
divergence
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0 lt m lt 1
The right fixed point exists if b gt 1, and it is
unstable. If b gt 1 explosive trajectories may
exist, and, in particular, when the generic
trajectory is divergent . If 0 lt b lt 1 the
trajectories are bounded
Periodic orbits may exist
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Case a 0
x
is a cycle of period 2 if
0
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
b
Period adding bifurcations
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Cycle LRR
Appearance The last point merges with the
border LR0
Disappearance The first point merges with the
border 0RR
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Cycle LRn-1
Orbit
Cycle condition
Border bifurcation curves
It appears when xn-1 0
It disappears when x0 0
Note that when a 0 the cycle of period k
disappears simultaneously to the appearance of
that of period k1
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Tongues of first level
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
On this line the multiplier of the cycle is 1
fold curve
If the parameters belong to this line, each point
in the range (m -1 , m) belong to a cycle.
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Bifurcation diagram
Chaotic intervals
LRRR
7
LRR
5
LR
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One-dim. bifurcation diagram
LR
LRR
LRRR
LRLRR
LRRLRRR
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Cycle 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 levelwith only two L
Cycle LRqLRq1
9
Appearance x2q20
7
5
Disappearance xq10
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Tongues of second level
3
11
8
5
7
9
LRLRR
LRR
LR
(LR)2LRR
LR(LRR)2
2
LR(LRR)3
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Plane (b , m)
4
3
5
enlargement
2
3
5
2
21
Border bifurcation curves
3
3
8
8
5
5
7
7
2
2
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Beyond the economic model a lt 0
divergence
Divergence
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3
23
2
Tongues overlap
Coexistence of cycles is a possible issue
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Initial condition m - 1
6
6
5
5
4
4
3
3
2
2
Flip bifurcation curves
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Initial condition m
6
6
5
5
4
4
3
3
2
2
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Main references

• 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

• 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