# American Institute of Mathematical Sciences

August  2016, 9(4): 1201-1234. doi: 10.3934/dcdss.2016049

## Forced linear oscillators and the dynamics of Euclidean group extensions

 1 Department of Mathematics, Rutgers University, Camden NJ 08102, United States

Received  November 2015 Revised  February 2016 Published  August 2016

We study the generic dynamical behaviour of skew-product extensions generated by cocycles arising from equations of forced linear oscillators of special form. This work extends our earlier work on cocycles into compact Lie groups arising from differential equations of special form, (cf. [21]), to the case of non-compact fiber groups of Euclidean type. The earlier techniques do not work in the non-compact case. In the non-compact case one of the main obstacle is the lack of recurrence'. Thus, our approach to studying Euclidean group extensions is : (i) first, to use a twisted version' of the so called conjugation approximation method' and then (ii) to use geometric-control theoretic methods' developed in our earlier work (cf. [20] and [21]). Even then, our arguments only work for base flows that admit a global Poincaé section, (e.g. for the irrational rotation flows on tori and for certain nil flows). We apply these results to study generic spectral behaviour of the forced quantum harmonic oscillator with time dependent stationary force restricted to satisfy given constraints.
Citation: Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049
##### References:
 [1] A. Avila, G. Forni and C. Ulcigrai, Mixing for time changes of Heisenberg nil flows, J. of Diff Geometry, 89 (2011), 369-410. [2] D. Anosov and A. Katok, New examples in smooth ergodic theory, Trans. Moscow Math. Soc., 23 (1970), 1-35. [3] P. Ashwin and I. Melbourne, Non-compact drft for relative equillibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. doi: 10.1088/0951-7715/10/3/002. [4] P. Ashwin, I. Melbourne and M. Nicol, Euclidean extensions of dynamical systems, Nonlinearity, 14 (2001), 275-300. doi: 10.1088/0951-7715/14/2/306. [5] J. Bellisard, Stability and instability in quantum mechanics, Trends and developments in the eighties (Bielefeld, 1982/1983), World Sci. Publishing, Singapore, 1985, 1-106. [6] L. Bunimovich, H. Jauslin, J. Lebowitz, A. Pellegrinoti and P. Nilaba, Diffusive energy growth in classical and quantum driven oscillators, Journal of Statistical Physics, 62 (1991), 793-817. doi: 10.1007/BF01017984. [7] M. Combescure, Recurrent versus diffusive dynamics for a kicked quantum oscillator, Annales de l'Institute Henri Poincaré (A) Physique Theorique, 57 (1992), 67-87. [8] M. Fields, I. Melbourne and M. Nicol, Symmetric attractors for diffeomorphisms and flows, Proc. London Math. Soc., 72 (1996), 657-696. doi: 10.1112/plms/s3-72.3.657. [9] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34 (1979), 321-336. doi: 10.1007/BF02760611. [10] M. Herman, Construction de diffeomorphismes ergodiques, preprint. [11] R. Johnson and M. Nerurkar, On null Controllability of linear systems with recurrent coefficients and constrained controls, (jointly with R. Johnson), Journal of Dynamics and Differential Equations, 4 (1992), 259-273. doi: 10.1007/BF01049388. [12] H. Keynes and D. Newton, Ergodicity in $(G,\sigma )$ extensions, Springer Verlag Lecture Note in Math., 668 (1978), 173-178. [13] J. Lebowitz and H. Jauslin, Spectral and stability aspects of quantum chaos, Chaos, 1 (1991), 114-121. doi: 10.1063/1.165809. [14] E. Lesigne and D. Volny, Large deviations for generic stationary processes, Colloquium Mathematicum, 84/85 (2000), 75-82. [15] E. Merzbacher, Quantum Mechanics, 5th edition, Wiley, New York, 1965. [16] I. Melbourne, V. Nitica and A. Torok, Transitivity of Euclidean type extensions of hyperbolic systems, Ergodic Theory and Dynamical Systems, 29 (2009), 1582-1602. doi: 10.1017/S0143385708000886. [17] M. Nerurkar, On the construction of smooth ergodic skew products, Ergodic Theory and Dynamical Systems, 8 (1988), 311-326. doi: 10.1017/S0143385700004454. [18] M. Nerurkar, Spectral and stability questions regarding evolution of non-autonomous linear systems, J. of Discrete and Continuous Dynamical Systems, (2004), 114-120. [19] M. Nerurkar and H. Jauslin, Stability of oscillators driven by ergodic processes, J. of Math. physics, 35 (1994), 628-645. doi: 10.1063/1.530657. [20] M. Nerurkar and H. Sussmann, Construction of minimal cocycles arising from specific differential equations, (jointly with H. Sussmann), Israel Journal of Mathematics, 100 (1997), 309-326. doi: 10.1007/BF02773645. [21] M. Nerurkar and H. Sussmann, Construction of ergodic cocycles arising from linear differential equations of special form, Journal of Modern Dynamics, 1 (2007), 205-253. doi: 10.3934/jmd.2007.1.205. [22] V. Nitica and M. Pollicott, Transitivity of Euclidean group extensions of Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 257-269. doi: 10.1017/S0143385704000471. [23] K. Schmidt, Cocycles and Ergodic Transformation Groups, MacMillan of India, 1977.

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##### References:
 [1] A. Avila, G. Forni and C. Ulcigrai, Mixing for time changes of Heisenberg nil flows, J. of Diff Geometry, 89 (2011), 369-410. [2] D. Anosov and A. Katok, New examples in smooth ergodic theory, Trans. Moscow Math. Soc., 23 (1970), 1-35. [3] P. Ashwin and I. Melbourne, Non-compact drft for relative equillibria and relative periodic orbits, Nonlinearity, 10 (1997), 595-616. doi: 10.1088/0951-7715/10/3/002. [4] P. Ashwin, I. Melbourne and M. Nicol, Euclidean extensions of dynamical systems, Nonlinearity, 14 (2001), 275-300. doi: 10.1088/0951-7715/14/2/306. [5] J. Bellisard, Stability and instability in quantum mechanics, Trends and developments in the eighties (Bielefeld, 1982/1983), World Sci. Publishing, Singapore, 1985, 1-106. [6] L. Bunimovich, H. Jauslin, J. Lebowitz, A. Pellegrinoti and P. Nilaba, Diffusive energy growth in classical and quantum driven oscillators, Journal of Statistical Physics, 62 (1991), 793-817. doi: 10.1007/BF01017984. [7] M. Combescure, Recurrent versus diffusive dynamics for a kicked quantum oscillator, Annales de l'Institute Henri Poincaré (A) Physique Theorique, 57 (1992), 67-87. [8] M. Fields, I. Melbourne and M. Nicol, Symmetric attractors for diffeomorphisms and flows, Proc. London Math. Soc., 72 (1996), 657-696. doi: 10.1112/plms/s3-72.3.657. [9] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34 (1979), 321-336. doi: 10.1007/BF02760611. [10] M. Herman, Construction de diffeomorphismes ergodiques, preprint. [11] R. Johnson and M. Nerurkar, On null Controllability of linear systems with recurrent coefficients and constrained controls, (jointly with R. Johnson), Journal of Dynamics and Differential Equations, 4 (1992), 259-273. doi: 10.1007/BF01049388. [12] H. Keynes and D. Newton, Ergodicity in $(G,\sigma )$ extensions, Springer Verlag Lecture Note in Math., 668 (1978), 173-178. [13] J. Lebowitz and H. Jauslin, Spectral and stability aspects of quantum chaos, Chaos, 1 (1991), 114-121. doi: 10.1063/1.165809. [14] E. Lesigne and D. Volny, Large deviations for generic stationary processes, Colloquium Mathematicum, 84/85 (2000), 75-82. [15] E. Merzbacher, Quantum Mechanics, 5th edition, Wiley, New York, 1965. [16] I. Melbourne, V. Nitica and A. Torok, Transitivity of Euclidean type extensions of hyperbolic systems, Ergodic Theory and Dynamical Systems, 29 (2009), 1582-1602. doi: 10.1017/S0143385708000886. [17] M. Nerurkar, On the construction of smooth ergodic skew products, Ergodic Theory and Dynamical Systems, 8 (1988), 311-326. doi: 10.1017/S0143385700004454. [18] M. Nerurkar, Spectral and stability questions regarding evolution of non-autonomous linear systems, J. of Discrete and Continuous Dynamical Systems, (2004), 114-120. [19] M. Nerurkar and H. Jauslin, Stability of oscillators driven by ergodic processes, J. of Math. physics, 35 (1994), 628-645. doi: 10.1063/1.530657. [20] M. Nerurkar and H. Sussmann, Construction of minimal cocycles arising from specific differential equations, (jointly with H. Sussmann), Israel Journal of Mathematics, 100 (1997), 309-326. doi: 10.1007/BF02773645. [21] M. Nerurkar and H. Sussmann, Construction of ergodic cocycles arising from linear differential equations of special form, Journal of Modern Dynamics, 1 (2007), 205-253. doi: 10.3934/jmd.2007.1.205. [22] V. Nitica and M. Pollicott, Transitivity of Euclidean group extensions of Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 257-269. doi: 10.1017/S0143385704000471. [23] K. Schmidt, Cocycles and Ergodic Transformation Groups, MacMillan of India, 1977.
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