Unstable periodic orbits of a chaotic growth cycle model. Yoshitaka Saiki

Transcription

1 Unstable periodic orbits of a chaotic growth cycle model Yoshitaka Saiki Department of Mathematical Sciences University of Tokyo (with Ken-ichi Ishiyama) March 28, 26

2 Business cycle theory [2] Important topic in macro-economics Periodic fluctuation is essential characteristic Low dimensional dynamical systems Stable model (Periodic point) [Gap exists, even if dynamics look similar] Chaotic model How to bridge the gap?

3 Purpose [3] Clearly recognize chaotic behavior by unstable periodic orbits (UPOs) Chaotic behavior (Complicated object) UPOs (Simple objects)

4 Preparation [4] Chaos : No single definition Deterministic behavior with two properties; Instability Reccurence Chaotic Special solution Unstable periodic solution Chaos UPOs Chaotic attractor: Infinite number of UPOs are (densely) embedded Chaotic orbit: UPO of infinite period

5 Chaotic Analysis by UPOs [5] Detection of UPOs: Difficult A few studies Background UPO analysis of fluid turbulence (Kawahara & Kida(2), Kato & Yamada(23)) Small number of UPOs (Only one or two) Coherent structure, Turbulent statistics Our study Two country chaotic business cycle model Numerically detect sorts of UPOs [using PC cluster (4 CPUs)] Characterize various chaotic behaviors by UPOs

6 Plan [6]. Chaotic business cycle model 2. Chaotic Analysis based on UPOs Regime and Regime transitions Statistical properties Growth rate of the number of UPOs 3. Summary

7 Two country business cycle model [7] Skeleton of the model Three agents in each country Capitalists Workers Government with Keynesian fiscal policy Investment interaction between two countries The only difference between two countries is their Keynesian fiscal policies Six dimensional ODEs (i =, 2) Labor share rate u i Employment rate v i Expected rate of inflation π e i Mutual action parameter between countries: η (η = 3.5: usual)

8 Model (Keyenes Goodwin type) [8] The first country [weak Keynesian policy] ( ) du. dt =.5 + π e.5 u v dv dt =.(.5( u ) 5 + η(u 2 u ) 3 +.5u.875v.)v dπ e dt = v (.4π e +.2).4π e.6 v The second country [strong Keynesian policy] ( ) du 2. dt =.5 + π2 e.5 u 2 v 2 dv 2 dt =.(.5( u 2) 5 + η(u u 2 ) 3 +.5u 2 4.2v )v 2 dπ e 2 dt = v 2(.4π e 2 +.2).4π e 2.6 v 2 (Parameters Yoshida & Asada (2))

9 Chaotic attractor [9] Bounded & Positive Lyapunov exponent(.99) v v u e π.2.4 u e π2 -.5 Figure: Projections to each country (u, v, π e) (left st country, right 2nd country).95 CHAOS v2 v.8 CHAOS u Figure Projections onto (u, v)-plane left st country, right 2nd country Consistent with Havie(2) u2 the empirical research by

10 Classification of typical dynamics [] Regime: Time period in which local maxima in u are monotone increasing Regime n: Regime which has n oscillations Regime transition m n: Regime m Regime n.8 Regime7 Regime3.6 u Figure: Classification of Regime in the time development of u (Regime7, Regime3 and Regime transition 7 3) t η = Number of oscillations Figure: Regime appearance in a chaotic orbit (99% of the Regimes are classified into Regime to Regime 7)

11 Chaotic analysis by UPOs [] Detection Newton-Raphson-Mees method with modification 9 UPOs are numerically detected Remove indistinguishable UPOs sorts of UPOs are identified Most of UPOs with short period are covered!

12 UPOs with only one Regime [2] UP On: UPO with n oscillations making a single Regime n.8 UPO.8 UPO2.6.6 u u t t.8 UPO3.8 UPO4.6.6 u u t t.8 UPO5.8 UPO6.6.6 u u t t UPO7.8.6 u Figure: Time development of u along UP On (T 25n) t Seven types of UPOs (UP O,, UP O7) are detected (NO UP O8 is found) This corresponds to the main Regimes appearing in a chotic orbit

13 UPOs with several Regimes [3] Ex)UP On.m [composed of Regime n and Regime m].8 UPO3.5.6 u Figure: Time development of u along UP O3.5 (Regime transition 3 5) Ex)U P On.m.l t.8 UPO u Figure: Time development of u along UP O4.6.8 (Regime transition 4 6, 6 8, 8 4) t

14 Regime transitions of chaos and UPOs [4] From\To C U C U C U 2 C U C U C U 3 C U C U C U C U C U C U 4 C U C U C U C U C U C U C U 5 C U C U C U C U C U C U C U C U 6 C U C U C U C U C U C U C U C U 7 C U C U C U C U C U C U C U C U 8 C U C U C U 9 Table Regime transitions observed in a chaotic orbit (C) and UPOs (U) All Regime transitions in a chaotic orbit are recognized by UPOs Ex) Chaos: Existence of Regime transition 4 6, Regime transition 6 8 and Regime transition 8 4 consistent UPO: Existence of UP O4.6.8 Ex) Chaos: Non-existence of Regime transition 8 8 consistent UPO: Non-existence of UP O8

15 Bifurcations of periodic orbits [5] Change of interaction between two countries (η) Time average of u UPO2 UPO3 UPO UPO4 UPO5 UPO6.22 UPO η Figure: Time average of u along UPOs at each η (+ Time average along an orbit on the attractor) η = 3.5 (usual) UP O,, UP O7 exist η = 2.6 UP O,, UP O5 exist

16 UPOs embedded in the attractor and chaos (η = 2.6) [6].8 η = Number of oscillations Figure: Regime appearance in a chaotic orbit(η = 2.6). UPO UPO2 UPO3 UPO4 UPO5 distance.. η=2.6. Figure: Distance between a point on the UPO (x UP O ) and a chaotic orbit {φ t (X)} t (d(τ) = min t τ φ t (X) x UP O ) η = 2.6 Regime and 2 are mainly observed UP O,, UP O5 are detected. But only UP O, UP O2 are embedded τ

17 UPOs embedded in the attractor and chaos (η = 3.5: Usual) [7].8 η = Number of oscillations Figure: Regime appearance in a chaotic orbit distance.. UPO UPO2 UPO3 UPO4 UPO5 UPO6 UPO7. η=3.5. Figure: Distance between a point on the UPO (x UP O ) and a chaotic orbit {φ t (X)} t η = 3.5 Regime 7 are mainly observed UPO 7 are embedded in the attractor UPOs are to be embedded in the attractor for capturing chaotic behavior τ

18 UPO2 is embedded in the attractor at various η [8]. distance... Figure Distance between a point on the UPO2 and chaotic orbit (η =2.7, 2.9, 3., 3.3, 3.5) η 2.5 : UPO2 itself becomes the attractor UPO2 is embedded in the attractor at various η ( η 3.5) UPO2 constructs the skeleton of economic behavior at various η τ

19 Statistical property (Time average along UPO) [9] Time average of u Time average of u T (Period of UPO) T (Period of UPO) Time average of v T (Period of UPO) Time average of v T (Period of UPO).. Time average of π e Time average of π e Figure T (Period of UPO) T (Period of UPO) Time averages of six variables along UPOs with period T (Line: Chaotic orbit) We can roughly estimate the time average of a chaotic orbit by one of any UPOs

20 Statistical correspondence between chaotic orbits and UPO2 [2] 3. UPO2 Chaos 3. UPO2 Chaos u UPO2 Chaos u 2 UPO2 Chaos v UPO2 Chaos v 2 UPO2 Chaos e π e π 2 Figure: PDFs of six variables along a chaotic orbit (Dotted line) and UPO2 (Solid line) (left: st country, right: 2nd country) Statistics along a chaotic orbit and UPO2: Similar

21 Number of UPOs [2] Topological variations of chaotic orbits can be estimated by variations of UPOs Number of UPOs N T (PERIOD) Figure: Number of UPOs against number of oscillations (N T T/25) and.35 N T,.4 N T,.45 N T #{UPOs with N T -oscillations}.4 N T Topological entropy is log.4 Complexity of the model: Low

22 Conclusion [22] Two-country chaotic business cycle model Several characteristics of the model are captured by various UPOs. Typical structure (Regime, Regime transition) of a chaotic business cycle Statistical property (Time average) Complexity of chaos (Growth rate of number of UPOs) Macroscopic character is captured by only one of any UPOs (e.g. UPO2). UPO2 is still embedded in the attractor with weaker interactions between two countries ********************************************** UPO2 is the skeleton of the attractor over a range of η The macroscopic structure of business cycle is not affected by the change of interaction parameter. **********************************************

23 Remarks [23] Business cycle theory Important topic in macro-economics Periodic fluctuation is essential characteristic Low dimensional dynamical systems Stable model (Periodic point) [Gap exists, even if dynamics look similar] Chaotic model Our study Chaotic business cycle model Characterized by unstable periodic orbits Bridge the gap through periodic orbit [Details Please see our papers!]

24 Acknowledgements [24] Researchers Abraham C.-L. Chian (INPE, Brazil) Kei-ichi Ishioka (Kyoto University) Shuhei Hayashi (University of Tokyo) Genta Kawahara (Osaka University) Toru Maruyama (Keio University) Michio Yamada (Kyoto University) Financial support 2st century COE program at University of Tokyo Computer resource PC cluster at University of Tokyo

25 Future work [25] Construct and analyze more realistic economic model by UPOs Non-hyperbolic structure [Quasi-stationary state] Multiple attractor Chaotic no-attractor [Chaotic transient] (Chian et al. (26)) Collaboration Let s try UPO analyses about some chaotic model! Please me! saiki@ms.u-tokyo.ac.jp

26 Reference (/2) [26] Bowen, R., On axiom A diffeomorphisms, Regional conference series in mathematics, 24 (American Mathematical Society, The United States of America, 978). Chian, A. C.-L., E. L. Rempel and C. Rogers, Complex economic dynamics: Chaotic saddle, crisis and intermittency, to appear in Chaos, Solitons and Fractals (26). Devaney, R. L., An Introduction to Chaotic Dynamical Systems, (Westview Press, The United States of America, 986). Goodwin, R. M., A growth cycle, in: C. H. Feinstein, ed., Socialism, capitalism, and economic growth, (The United States of America, 967). Harvie, D., Testing Goodwin : Growth Cycles in Ten OECD countries, Cambridge Journal of Economics, 24(3) (2) Ishiyama, K. and Y. Saiki, Unstable periodic orbits embedded in a chaotic economic dynamics model, Applied Economics Letters, 2 (25a) Ishiyama, K. and Y. Saiki, Unstable periodic orbits and chaotic economic growth, Chaos, Solitons and Fractals, 26 (25b) Ishiyama, K. and Y. Saiki, Unstable periodic orbits and chaotic transitions among growth patterns of an economy, The Practical Fruits of Econophysics: Proceedings of the Third Nikkei Econophysics Symposium, (Springer, 26) Kato, S. and M. Yamada, Unstable periodic solutions embedded in a shell model turbulence, Physical Review E, 68(2) (23)

27 Reference (2/2) [27] Kawahara, G. and S. Kida, Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst, Journal of Fluid Mechanics, 449 (2) Mees, A. I., Dynamics of Feedback Systems (John Wiley and Sons, New York, N. Y., 98). Phillips, A. W., The relation between unemployment and the rate of change of money wage rates in the United Kingdom, , Economica, 25 (958) Pohjola, M. J., Stable and Chaotic Growth: the Dynamics of a Discrete Version of Goodwin s Growth Cycle Model, Journal of Economics / Zeitschrift für Nationalökonomie, 4 (98) Saiki, Y., Numerical study about a chaotic dynamical system based on unstable periodic orbits Business cycle model of macro-economics, PhD thesis, The University of Tokyo (26). Skott, P., Effective demand, class struggle and cyclical growth, International Economic Review, 3() (989) Sportelli, M. C., A Kolmogoroff generalized predator-prey model of Goodwin s growth cycle, Journal of Economics / Zeitschrift für Nationalökonomie, 6 (995) Wolfstetter, E., Fiscal policy and the classical growth cycle, Journal of Economics / Zeitschrift für Nationalökonomie, 42 (982) Yoshida, H. and T. Asada, Dynamic analysis of policy lag in a Keynes-Goodwin model: stability, instability, cycles and chaos, Nagoya Gakuin University Discussion Paper, 53 (2).

28 Validity of periodic orbits [A] Numerical error grows exponentially by time rounding off error δ = 6 stability exponent λ =.3 permitted error err = 8 Maximum value of t (t max ) satisfying is δe λt err () t max 63 (2) Chaotic orbit Detected periodic orbit Big advantage Numerically invalid in general Numericaly valid

29 Numerical method for detecting unstable periodic orbits [A2] (Newton Raphson Mees method + damping) {φ t (X)} t R : the orbit passing through X( R n ) at t = of dx dt = f(x) (x Rn ) (3) Periodic is identified by the zeros of H(X, T ) = φ T (X) X. n + unknowns: one point X + period T. Algorithm (Newton method): H(X, T ) D X H(X, T ) X + D T H(X, T ) T Determine X and T satisfying H(X, T ) + H(X, T ) =. (4) Additional constraint : Modified vector X is orthogonized by the orbit; < f(x), X >=. n + constraints Introducing damping coefficient m (X, T ) = (X, T ) + 2 m ( X, T ) (2 m /e λt (λ stability exponent))

30 Criterion of convergence [A3] φ T (i)(x (i) ) X (i) (Practical error) ( X (i), T (i) ) (Absolute value of modified vecor) are sufficiently small (order: 9, 7 ) Detecting unstable periodic orbit More than 9 periodic orbits are numerically detected (remove overlapped orbits) sorts of UPOs

31 Damping coefficient [A4] 4 m (Damping exponent) T (Period of UPO) Figure Minimal damping coefficient m( Z) needed to detect each periodic orbit and λt log 2 (λ =.2,.3) This corresponds to the fact that Floquet exponent of periodic orbits are between.2 and.3.

32 Validity of periodic orbit (time step) [A5] err T. e-4 e-6 e-8 e- e-2 e-4 e-6 e-8 e-2 e-4... t (Time step) Figure Error (err T ( t) := T ) about the period of UPO2 numerically integrated by various time steps ( t) err T ( t) ( t) 4 (.3 t.) Fourth order Runge-Kutta method Numerical time integration of UPO2: Valid (Cancelling effect: ( t.3))

33 Goodwin type growth cycle model [A6] Class conflict between capitalists and labors Goodwin(967) continuous, conservative (stable (center)) periodic orbit Desai(973) continuous, dissipative (limit cycle Expected rate of inflation) Pohjola(98) discrete (chaos) Wolfstetter(982) continuous, dissipative (fixed point, limti cycle Keynes type finantial policy) Sportelli(995) continuous, dissipative (limit cycle price rigidity) Yoshida & Asada(2) continuous, dissipative (Choas time lag in policy) Ishiyama & Saiki(25) continuous, dissipative (Chaos two countries) Empirical research (OECD ten coun- Harvie(2) tries)

34 Skelton of the model [A7] (Agents capitalists, labors government) Labor share rate u wl py Employment rate v L N Expected rate of inflation π e ( = w ) [ du pa dt ) [ dv ( = Y an ] = (ŵ (ˆp + â))u ] dt = (Ŷ (â + ˆN))v [ ] dπ e dt = θ (ˆp πe ) Variable w: Nomial wage rate L: Labor level p: Price level Y : Gross national output (=Gross national income=gross national expenditure) N: Labor supply [a( Y/L): Labor productivity] [ˆp( ṗ/p): Actual inflation rate] Constant α( â): Technical progress rate β( ˆN): Growth rate of labors θ: Adaptive speed of worker s expectation Y is effected by the mutual action between countries ˆx means the growth rate of x (ẋ/x)

35 Details of the model [A8] Wage bargaining ŵ i = f i (v i, πi e f i ), >, f i v i πi e >, Price fluctuation ˆp i = γ i (ŵ i α i ), (Elasticity coefficient γ i =.5, Technical progress rate α i =.2) ( ) Ii + G i + C i Y i Output plan Ŷ i = ε i, (Output adjustment coefficient ε i =.) Private investment I i = h i (u i, u j )Y i, Y i h i u i <, h i u j >, j =, 2, j i. Government expenses G i = δ i Y i + µ i (v v i )Y i (Income tax rate δ i =2/7, Reaction coefficient of the government µ =.25, µ 2 = 6.) ( ) db i Consumption C i = c i ( δ i )(( u i )Y i + r i B i ) q i + ( δ i )u i Y i (Consumption coefficient of capitalists c i =.3) dt ************************ ( ) Phillips curve f i (v i, πi e ) =. 4.8 v i + π e i Investment function h i (u i, v j ) =.5( u i ) 5 +η(u j u i ) 3, η >

36 Statistical value of UPO [A9] Time average of u Time average of u UPO (24.87,.246).246+(.52/)(x-24.87) Figure T (Period of UPO) UPO (223.83,.246).246+(.52/9)(x ) T (Period of UPO) Time average of u Time average of u UPO (24.35,.246).246+(.52/5)(x-24.35) T (Period of UPO) UPO (323.3,.246).246+(.52/3)(x-323.3) T (Period of UPO) Time average of u along UPO with period T (u ) and ( (.52/N T )(T 24.87N T )) Any UPOs seem to have macroscopically similar structure

37 New knowledges about the UPO analysis [A] Detection of UPO: Damping coeffiiceint is important [General] Various chaotic behavior: classified by UPOs [General (Hyperbolic)] UPO sometimes becomes out of the attractor by the change of parameter [General] Deformation of chaotic attractor corresponds to the change of variations of UPOs embedded in the attractor [General] Exponential growth rate of UPOs can be estimated by detected UPOs [Hyperbolic ( Single global structure )] Statistics of UPOs with the same Poincare map periods have smooth dependence on the period of UPO [Hyperbolic ( Single global structure )] Statistical values of chaos are approximated by one of any UPOs [Hyperbolic ( Single global structure )] UPO with long period does not necesarily well approximate the statistical value of chaos [General]