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

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

3. 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: F t (L t, K t ) - C t = s F t (L t, K t ) Production function is homogeneous, with intensive form f(k t ) where the state variable is the capital intensity k t

4. 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 s c. Workers receive income from labor and have saving propensity s w 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

5. The economic model Böhm & Agliari (2007) Workers may have different savings propensities from wage they save from capital revenues they save where Parameters : n>0 population increasing rate; δ, with 0<δ<1, capital depreciation rate; s c, with 0< s c <1, saving propensity of shareholders; s w, with 0< s w <1, saving propensity on wage of workers; s p, with 0< s p <1, saving propensity on income revenue of workers.

6. Technology Leontief production function: where A, B > 0 The axis is trapping:

7. 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: 0 0

8. Case s w > s p   Increasing map Up to two fixed points: 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.

9. Case s w < s p   Noninvertible map Up to two fixed points:   Right fixed point globally stable divergence

10. 0 <  < 1   The right fixed point exists if  > 1, and it is unstable. If  > 1 explosive trajectories may exist, and, in particular, when the generic trajectory is divergent. If 0  < 1 the trajectories are bounded   Periodic orbits may exist

11. Case  = 0 x  0 Period adding bifurcations is a cycle of period 2 if 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

12. Cycle LRR Appearance: The last point merges with the border LR0 Disappearance: The first point merges with the border 0RR

13. Cycle LR n-1 Orbit: Cycle condition: It appears when x n-1 = 0: It disappears when x 0 = 0: Border bifurcation curves Note that when  = 0 the cycle of period k disappears simultaneously to the appearance of that of period k+1

14. Tongues of first level 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 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.

15. Bifurcation diagram 2 3 4 5 LR LRR LRRR 5 7 Chaotic intervals

16. One-dim. bifurcation diagram LR LRLRR LRR LRRR LRRLRRR

17. Cycle LRLRR Appearance: The last point merges with the border LRLR0 Disappearance: The third point merges with the border LR0RR

18. Tongues of second level with only two L 5 7 9 Appearance: x 2q+2 =0 Cycle LR q LR q+1 Disappearance: x q+1 =0

19. Tongues of second level 8 7 11 9 5 3 2 LRLRR (LR) 2 LRR LRRLR LR(LRR) 2 LR(LRR) 3

20. Plane ( ,  ) enlargement 2 3 4 5 2 3 5

21. Border bifurcation curves 2 3 5 2 3 5 7 8 7 8

22. Beyond the economic model:  < 0   Divergence: 2 2;3 3;4 3 Tongues overlap Coexistence of cycles is a possible issue divergence

23. Initial condition  2 3 4 5 6 2 3 4 5 6 Flip bifurcation curves

24. Initial condition  2 3 4 5 6 2 3 4 5 6

25. Main 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