Hip´ otese αβ = 1

(β−1). (B.25)

Outro caso B = 0

Das condi¸c˜oes B.17, B.18, B.24 e B.25, conclu´ımos que n˜ao ´e poss´ıvel se chegar a nenhum conjunto de parˆametros de acoplamento que atendam as restri¸c˜oes exigidas se α·β 6= 1 e A >0 ou B >0.

Na verdade a an´alise de A < 0, igualmente nos leva a conclus˜ao de que n˜ao ´e poss´ıvel termos um acoplamento assim´etrico que respeite as restri¸c˜oes B.22, B.23 e de que pn+1, Pn+1 ≥0 se αβ 6= 1.

Assim temos que analisar o que acontece com as equa¸c˜oes B.15, B.16, B.22 e B.23 quando αβ = 1.

B.2.2 Hip´ otese αβ = 1

Nesta hip´otese, diretamente de B.24 conclu´ımos que:

B = −αA . (B.26)

Ou da mesma forma,

A = −βB . (B.27)

Mas como ainda temos a restri¸c˜ao de que tantopn+1, quantoPn+1sejam sempre positivos, conclu´ımos que,

SeA /∈ [−2,0], B /∈(−2,0] (B.28) SeA /∈ (−2,0], B /∈[−2,0].

Assim de B.18 vemos tamb´em que β <0 e de forma semelhanteα <0.

Em resumo temos que mapas assim´etricos ter˜ao a forma, qn+1 = (2 +A)qn−BQn

Qn+1 = (2 +B)Qn−Aqn (B.29)

pn+1 = 2 +B

K pn + A KPn

Pn+1 = 2 +A

K Pn + B Kpn.

Com K = 4 + 2A+ 2B, α <0, B =−αA,α 6=−1 (sen˜ao seria sim´etrico) e SeA /∈ [−2,0], B /∈(−2,0]

SeA /∈ (−2,0], B /∈[−2,0]. (B.30)

Bibliografia

[1] E. Ott. Chaos in Dynamical Systems. Cambridge University Press, 1994.

[2] H. Poincar´e. Sur les ´equations de la dynamique et le probl`eme de trois corps.

Acta Math, 13:1, 1890.

[3] H. Poincar´e. Les Methodes Nouvelles de la Mecanique Celeste. Gauthier-Villars, Paris, 1899.

[4] P. ˇSeba. Physical Review Letters, 64:1855, 1990.

[5] P. ˇSeba and K. ˙Zyczkowski. Physical Review A, 44:3457, 1991.

[6] M. C. Gutzwiller. Chaos in Classical and Quantum Physics. New York: Spring, 1990.

[7] H. Friedrich and D. Wintgen. Physics Reports, 183:37, 1989.

[8] K. Karremans, W. Vassen, and W. Hogervorst. Physical Review A, 60:2275, 1999.

[9] V. V. Flambaum, A. A. Gribakina, G. F. Gribakin, and I. V. Ponomarev. Phy-sica D, 131:205, 1999.

[10] C. M. Marcus, S. R. Patel, A. G. Huibers, S. M. Cronenwett, M. Switkes, I. H.

Chan, R. M. Clarke, J. A. Folk, S. F. Godijn, K. Campman, and A. Gossard.

Chaos, Solitons and Fractals, 8:1261, 1997.

[11] M. C. Gutzwiller. Journal of Mathematical Physics, 12:343, 1971.

[12] J. H. Van Vleck. Proc. Natl. Acad. Sci. USA, 14:178, 1928.

[13] H. J. Hannay and A. M. O. de Almeida. Periodic orbits and a Correlation-Function for the semiclassical density of states. Journal Physics A, 17:3429, 1984.

[14] A. M. O. de Almeida. Resonant periodic-orbits and the semiclassical energy-spectrum. Journal of Physics A, 20:5873, 1987.

[15] A. J. Lichtenberg and M. A. Lieberman.Regular and Stocastic Motion. Springer-Verlag, 1989.

[16] S. Wiggins. Introduction to Apllied Nonlinear Dynamical Systems and Chaos.

Springer-Verlag, 1990.

[17] M. Saraceno and A. Voros. Towards a semiclassical theory of the quantum baker’s map. Physica D, 79:206, 1994.

[18] N. L. Balazs and A. Voros. The Quantized Baker’s Transformation. Annals of Physics, 190:1, 1989.

[19] V. I. Arnold and A. Avez. Probl`eme ergodiques de la m´echanique classique.

Taythier-Villars, 1967.

[20] E. Hopf. Ergodentheorie. Springer, Berlim, 1937.

[21] J. H. Hannay and M. V. Berry. Quantization of Linear maps on a Torus - Fresnel diffraction bya a periodic grating. Physica D, 1 (3):267, 1980.

[22] N. L. Balazs and A. Voros. The Quantized Baker’s Transformation. Europhysics Letters, 4:1089, 1987.

[23] M. Saraceno. Classical Strutures in the Quantized Baker Transformation . An-nals of Physics, 199:37, 1990.

[24] E. J. Heller. Physical Review Letters, 53:1515, 1984.

[25] E. J. Heller, P. W. O’Connor, and J. Gehlen. Physica Scripta, 40:354, 1989.

[26] A. M. Ozorio de Almeida and M. Saraceno. Periodic Orbit Theory for the Quantized Baker’s Map. Annals of Physics, 1:1, 1990.

[27] M. G. E. da Luz and A. M. O. de Almeida. Path integral for the quantum baker’s map. Nonlinearity, 8:43, 1995.

[28] A. M. O. de Almeida and M. G. E. da Luz. Path integrals and edge corrections for torus maps. Physica D, 94:1, 1995.

[29] R. Rubin and N. Salwen. A Canonical Quantization of the Baker’s Map. Annals of Physics, 269:159–181, 1998.

[30] A. N. Soklakov and R. Schack. Classical limit in terms of symbolic dynamics for the quantum baker’s map. Physical Review E, 61:5108, 2000.

[31] P. W. O’Connor, S. Tomsovic, and E. J. Heller. Accuracy of Semiclassical Dynamics in the Presence of Chaos. Journal of Statistical Physics, 68:131, 1992.

[32] P. W. O’Connor, S. T., and E. J. Heller. Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier. Physica D, 55:340, 1992.

[33] K. Inoue, M. Ohya, and I. V. Volovich. Semiclassical properties and chaos degree for the quantum Baker’s map. Journal of Mathematical Physics, 43:734, 2002.

[34] A. Tanaka. Quantum mechanical entanglements with chaotic dynamics. Journal of Physics A, 29:5475, 1996.

[35] A. J. Scott and C. M. Caves. Entangling power of the quantum baker’s map.

Journal of Physics A, 36:9553, 2003.

[36] A. Lakshminarayan. Shuffling cards, factoring numbers and the quantum baker’s map. Journal of Physics A, 38:L597, 2005.

[37] J. H. Hannay, J. P. Keating, and A. M. Ozorio de Almeida. Optical Realization of the Bakers transformation. Nonlinearity, 7:1327, 1995.

[38] J. C. Howell and J. A. Yeazell. Linear optics simulations of the quantum baker’s map. Physical Review A, 61:012304, 1999.

[39] Y. S. Weinstein, S. Lloyd, J. Emerson, and D. G. Cory. Experimental Implemen-tation of the Quantum Baker’s Map. Physical Review Letters, 89-15:157902–1, 2002.

[40] T. A. Brun and R. Schack. Realizing the quantum baker’s map on a NMR quantum computer. Physical Review A, 59:2649, 1999.

[41] A. Lakshminarayan and N. L. Balazs. The Classical and Quantum-Mechanics of Lazy baker maps. Annals of Physics, 226:350, 1993.

[42] P. Pakonski, A. Ostruszka, and K. Zyczkowski. Quantum baker map on the sphere. Nonlinearity, 12:269, 1999.

[43] M. Saraceno and R. O. Vallejos. The quantized D-transformation. Chaos, 6:193, 1996.

[44] K. kaneko. Nonlinear Science Theory and Applications of coupled map lattices.

John Wiley & Sons, Ltd, England, 1993.

[45] H. Chate and P. Manneville. Physica D, 32:402, 1998.

[46] Y. Oono and S. Puri. Physical Review Letters, 58:836, 1987.

[47] J. Miller and D. A. Huse. Physical Review E, 48:2528, 1993.

[48] M. V. Vessen, P. C. Rech, M. W. Beims, J. A. Freire, M. G. E. da Luz, P. G.

Lind, and J. A. C. Gallas. The dynamics of complex-amplitude norm-preserving lattices of coupled oscillators. Physica A, 338:537, 2004.

[49] D. K. W´ojcik and J. R. Dorfman. Quantum multibaker maps: Extreme quantum regime. Physical Review E, 66:036110–1, 2002.

[50] Y. Elskens and R. Kapral. Reversible Dynamics and the Macroscopic Rate Law for a Solvable Kolmogorov System: The Three Bakers’ Reaction. Journal of Statistical Physics, 38:1027, 1984.

[51] A. Lakshminarayan and N. L. Balazs. Relaxation and Localization in Interacting Quantum Maps. Journal of Statistical Physics, 77:311, 1994.

[52] J. P. Keating. The cat maps: quantum mechanics and classical motion. Nonli-nearity, 4:309, 1991.

[53] P. C. Shields. The Theory of Bernoulli Shifts. University of Chicago Press, Chicago, Illinois, 1973.

[54] J. Guckenenheimer and P.Holmes. Nonlinear Oscilations, Dynamical Systems, and Bifurcation of Vector Fields. Springer-Verlag, 1983.

[55] K. Kaneko. Transition from Torus to Chaos Accompanied by Frequency Loc-kings with Symmetry Breaking- in connection with the Coupled-Logistic Map.

Progress of Theoretical Physics, 69:1427, 1983.

[56] T. Hogg and B. A. Huberman. Generic behavior of coupled oscillators. Physical Review A, 29:275, 1984.

[57] P. C. Rech, M. W. Beims, and J. A. C. Gallas. Naimark-Sacker bifurcations in linearly coupled quadratic maps. Physica A, 342:351, 2004.

[58] M. W. Beims, P. C. Rech, and J. A. C. Gallas. Fractal and riddled basins:

arithmetic signatures in the parameter space of two coupled quadratic maps.

Physica A, 295:276, 2001.

[59] G. Hu and Z. L. QuL. Controlling Spatiotemporal Chaos in Coupled Map Lattice Systems. Physical Review Letters, 72 (1):68, 1994.

[60] F. H. Willeboordse and K. Kaneko. Pattern Dynamics of a Coupled Map Lattice for Open Flow. Physica D, 86 (3):428, 1995.

[61] S. E. D. Pinto, J. T. Lunardi, A. M. Saleh, and A. M. Batista. Some aspects of the synchronization in coupled maps. Physical Review E, 72 (3), 2005.

[62] C. Anteneodo, S. E. D. Pinto, A. M. Batista, and R. L. Viana. Analytical results for coupled-map lattices with long-range interactions. Physical Review E, 68 (4), 2003.

[63] A. M. Ozorio de Almeida. Hamiltonian Systems: Chaos and Quantization.

Cambridge Univ. Press., 1988.

[64] H. Goldstein. Classical mechanics. Addison-Wesley, 1980.

[65] P. Ehrenfest. Z. Phys., 45:455, 1927.

[66] M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros. Quantum maps. Annals of Physics, 122:1:26, 1979.

[67] M. V. Vessen, M. C. Santos, B. K. Cheng, and M. G. E. da Luz. Origin of quantum chaos for two particles interacting by short-range potentials. Physical Review E, 64 (2):026201, 2001.

[68] M. M. Sano. Statistical properties of actions of periodic orbits. Chaos, 10:195, 2000.

[69] T. Guhr and H. A. Weidenmuller. Annals of Physics, 199:412, 1990.

[70] H. Lindgren. Geometric Dissections. Princeton, Van Nostrand, 1964.

[71] I. Stewart. Math Hysteria. Oxford University Press, 2005.

[72] F. Leyvraz, C. Schmit, and H. Seligman. Journal of Physics A, 29:L575, 1996.

[73] J. P. Keating and J. M. Robbins. Journal of Physics A, 30:L177, 1997.

[74] A. Gusso, M. G. E. da Luz, and L. G. C. Rego. Quantum chaos in nanoelectro-mechanical systems. Physical Review B, 73 (3):035436, 2006.

[75] C. Dembowski, H. D. Graff, A. Heine, H. Rehfeld andA. Richter, and C. Schmit.

Physical Review E, 62:R4516, 2000.

[76] C. Dembowski, B. Dietz, H. D. Graf, A. Heine, F. Leyvraz, M. Miski-Oglu, A. Richter, and T. H. Seligman. Physical Review Letters, 90:014102, 2003.

[77] R. Schafer, M. Barth, F. Leyvraz, M. Muller, T. H. Seligman, and H. J. Stock-mann. Physical Review E, 66:016202, 2002.

[78] E. P. Wigner. Proc. Fourth Canadian Math. Congress, page 174, 1957.

[79] F. J. Dyson and M. L. Mehta. Journal of Mathematical Physics, 4:489, 1957.