메뉴 건너뛰기




Volumn 16, Issue 1, 2001, Pages 29-61

Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; DAMPING; DIFFERENTIAL EQUATIONS; PENDULUMS; SYSTEM STABILITY; TWO DIMENSIONAL;

EID: 0035297090     PISSN: 14689367     EISSN: None     Source Type: Journal    
DOI: 10.1080/02681110010001289     Document Type: Article
Times cited : (26)

References (42)
  • 1
    • 0026917425 scopus 로고
    • Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation
    • Ariarathnam, S. T., and Xie, W. C., 1992, Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation. Journal of Applied Mechanics, 59: 664-673.
    • (1992) Journal of Applied Mechanics , vol.59 , pp. 664-673
    • Ariarathnam, S.T.1    Xie, W.C.2
  • 2
    • 0027660418 scopus 로고
    • Lyapunov exponents and stochastic stability of two-dimensional parametrically excited random systems
    • Ariarathnam, S. T., and Xie, W. C., 1993, Lyapunov exponents and stochastic stability of two-dimensional parametrically excited random systems. Journal of Applied Mechanics, 60: 677-682.
    • (1993) Journal of Applied Mechanics , vol.60 , pp. 677-682
    • Ariarathnam, S.T.1    Xie, W.C.2
  • 4
    • 0003540690 scopus 로고
    • Arnold, L. Crauel. H., and Eckmann, J.-P. (eds); (Berlin: Springer-Verlag)
    • Arnold, L. Crauel. H., and Eckmann, J.-P. (eds), 1991, Lyapunov Exponents, Lecture Notes in Mathematics, Vol. 1486 (Berlin: Springer-Verlag).
    • (1991) Lyapunov Exponents, Lecture Notes in Mathematics , vol.1486
  • 7
    • 0022737468 scopus 로고
    • Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications
    • Arnold, L., Papanicolaopu, G., and Wihstutz, V., 1986b, Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications. SIAM Journal of Applied Mathematics. 46: 427-450.
    • (1986) SIAM Journal of Applied Mathematics , vol.46 , pp. 427-450
    • Arnold, L.1    Papanicolaopu, G.2    Wihstutz, V.3
  • 8
    • 0002034361 scopus 로고    scopus 로고
    • Large noise asymptotics of invariant measures with applications to Lyapunov exponent
    • Arnold, L., Eizenberg, A., and Wihstutz, V., 1996a, Large noise asymptotics of invariant measures with applications to Lyapunov exponent. Stochastics and Stochastics Report, 59(1/2): 71-142.
    • (1996) Stochastics and Stochastics Report , vol.59 , Issue.1-2 , pp. 71-142
    • Arnold, L.1    Eizenberg, A.2    Wihstutz, V.3
  • 11
    • 0005500746 scopus 로고
    • Asymptotic expansions of the Lyapunov index for linear stochastic systems with small noise
    • (translation from Prikl. Mat. Mekh., 46: 378-395, 1982)
    • Auslender, E. I., and Mil'stejn, G. N., 1983, Asymptotic expansions of the Lyapunov index for linear stochastic systems with small noise. Journal of Applied Mathematics and Mechanics, 46: 277-283 (translation from Prikl. Mat. Mekh., 46: 378-395, 1982).
    • (1983) Journal of Applied Mathematics and Mechanics , vol.46 , pp. 277-283
    • Auslender, E.I.1    Mil'stejn, G.N.2
  • 13
    • 33646225347 scopus 로고    scopus 로고
    • Stochastic perturbation of a non-linear oscillator
    • Billeke, J., and Bobadilla, G., 1996, Stochastic perturbation of a non-linear oscillator. Rev. Mat. Apl., 17: 1-10.
    • (1996) Rev. Mat. Apl. , vol.17 , pp. 1-10
    • Billeke, J.1    Bobadilla, G.2
  • 15
    • 0033298941 scopus 로고    scopus 로고
    • An explicit description of the Lyapunov exponents of the noisy driven harmonic oscillator
    • Imkeller, P., and Lederer, C., 1999, An explicit description of the Lyapunov exponents of the noisy driven harmonic oscillator. Dynamics and Stability of Systems, 14: 385-405.
    • (1999) Dynamics and Stability of Systems , vol.14 , pp. 385-405
    • Imkeller, P.1    Lederer, C.2
  • 16
    • 0012176603 scopus 로고
    • Stabilization of companion systems by mean zero noise
    • Kao, J., and Wihstutz, V., 1994, Stabilization of companion systems by mean zero noise. Stochastics and Stochastics Reports, 49: 1-25.
    • (1994) Stochastics and Stochastics Reports , vol.49 , pp. 1-25
    • Kao, J.1    Wihstutz, V.2
  • 17
    • 0032003177 scopus 로고    scopus 로고
    • Moment Lyapunov exponent and stability index for linear conservative systems with small random perturbation
    • Khasminskii, R., and Moshchuk, N., 1998, Moment Lyapunov exponent and stability index for linear conservative systems with small random perturbation. SIAM Journal of Applied Mathematics, 58: 245-256.
    • (1998) SIAM Journal of Applied Mathematics , vol.58 , pp. 245-256
    • Khasminskii, R.1    Moshchuk, N.2
  • 20
    • 0001479882 scopus 로고
    • Necessary and sufficient conditions for almost sure sample stability of linear Ito equations
    • Kozin, F., and Prodromou, S., 1971, Necessary and sufficient conditions for almost sure sample stability of linear Ito equations. SIAM Journal of Applied Mathematics, 21: 413-424.
    • (1971) SIAM Journal of Applied Mathematics , vol.21 , pp. 413-424
    • Kozin, F.1    Prodromou, S.2
  • 21
    • 0012144123 scopus 로고
    • Stability of the inverted pendulum subjected to almost periodic and stochastic base motion-an application of the method of averaging
    • Mitchell, R., 1972, Stability of the inverted pendulum subjected to almost periodic and stochastic base motion-an application of the method of averaging. International Journal of Nonlinear Mechanics, 7: 101-123.
    • (1972) International Journal of Nonlinear Mechanics , vol.7 , pp. 101-123
    • Mitchell, R.1
  • 22
    • 0002297290 scopus 로고
    • The structure of eigenfunctions of one dimensional unordered structures
    • Molchanov, S., 1978, The structure of eigenfunctions of one dimensional unordered structures. Math. USSR. Izvestija, 12: 69-101.
    • (1978) Math. USSR. Izvestija , vol.12 , pp. 69-101
    • Molchanov, S.1
  • 24
    • 0028443283 scopus 로고
    • Maximal Lyapunov exponent and almost sure stability for coupled two-degree-of-freedom stochastic systems
    • Namachchivaya, S., Van Roessel, H. J., and Talwar, S., 1994, Maximal Lyapunov exponent and almost sure stability for coupled two-degree-of-freedom stochastic systems. Journal of Applied Mechanics, 61: 446-452.
    • (1994) Journal of Applied Mechanics , vol.61 , pp. 446-452
    • Namachchivaya, S.1    Van Roessel, H.J.2    Talwar, S.3
  • 25
    • 84972500225 scopus 로고
    • On the stability of two-dimensional linear systems
    • Nishioka, K., 1976, On the stability of two-dimensional linear systems. Kodai Math. Sem. Rep., 27: 211-230.
    • (1976) Kodai Math. Sem. Rep. , vol.27 , pp. 211-230
    • Nishioka, K.1
  • 26
    • 0000515929 scopus 로고
    • Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion
    • Pardoux, E., and Wihstutz, V., 1982, Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion. Stochastic Processes and Applications, 40(2): 289-308.
    • (1982) Stochastic Processes and Applications , vol.40 , Issue.2 , pp. 289-308
    • Pardoux, E.1    Wihstutz, V.2
  • 27
    • 0000438813 scopus 로고
    • Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion
    • Pardoux, E., and Wihstutz, V., 1998, Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion. SIAM Journal of Applied Mathematics, 48: 442-457.
    • (1988) SIAM Journal of Applied Mathematics , vol.48 , pp. 442-457
    • Pardoux, E.1    Wihstutz, V.2
  • 28
    • 0022734329 scopus 로고
    • Instability of the harmonic oscillator with small noise
    • Pinsky, M. A., 1986, Instability of the harmonic oscillator with small noise. SIAM Journal of Applied Mathematics, 46: 451-463.
    • (1986) SIAM Journal of Applied Mathematics , vol.46 , pp. 451-463
    • Pinsky, M.A.1
  • 29
    • 33646210491 scopus 로고
    • Lyapunov exponent and rotation number of the stochastic harmonic oscillator
    • Pinsky, M. A., 1987, Lyapunov exponent and rotation number of the stochastic harmonic oscillator. Random Media, IMA Vol. Math. Appl., 7: 229-243.
    • (1987) Random Media, IMA Vol. Math. Appl. , vol.7 , pp. 229-243
    • Pinsky, M.A.1
  • 30
    • 0012186526 scopus 로고
    • Lyapunov exponents of real-noise-driven nilpotent systems and harmonic oscillators
    • Pinsky, M., and Wihstutz, V., 1991a, Lyapunov exponents of real-noise-driven nilpotent systems and harmonic oscillators. Stochastic and Stochastics Report, 35: 93-110.
    • (1991) Stochastic and Stochastics Report , vol.35 , pp. 93-110
    • Pinsky, M.1    Wihstutz, V.2
  • 31
    • 0000063121 scopus 로고
    • Lyapunov exponents for white and real noise driven two-dimensional systems
    • (Providence, RI: Amer. Math. Soc.)
    • Pinsky, M., and Wihstutz, V., 1991b, Lyapunov exponents for white and real noise driven two-dimensional systems. Mathematics of Random Media, Vol. 47 of Lectures in Applied Mathematics (Providence, RI: Amer. Math. Soc.), pp. 201-213.
    • (1991) Mathematics of Random Media, Vol. 47 of Lectures in Applied Mathematics , pp. 201-213
    • Pinsky, M.1    Wihstutz, V.2
  • 33
    • 0012106230 scopus 로고
    • Rotation numbers for stochastic dynamical systems
    • PhD thesis, University of Warwick
    • Ruffino, P., 1995a, Rotation numbers for stochastic dynamical systems. PhD thesis, University of Warwick.
    • (1995)
    • Ruffino, P.1
  • 34
    • 33646210608 scopus 로고
    • 2
    • Preprint, University of Warwick
    • 2. Preprint, University of Warwick.
    • (1995)
    • Ruffino, P.1
  • 35
    • 0030487210 scopus 로고    scopus 로고
    • Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive
    • Schenk-Hoppé, K. R., 1996a, Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive. Zeitschrift für angewandte Mathematik und Physik, 47: 740-759.
    • (1996) Zeitschrift für Angewandte Mathematik und Physik , vol.47 , pp. 740-759
    • Schenk-Hoppé, K.R.1
  • 36
    • 0030290070 scopus 로고    scopus 로고
    • Bifurcation scenarios of the noisy Duffing-van der Pol oscillator
    • Schenk-Hoppé, K. R., 1996b, Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlinear Dynamics, 11: 255-274.
    • (1996) Nonlinear Dynamics , vol.11 , pp. 255-274
    • Schenk-Hoppé, K.R.1
  • 37
    • 33646213750 scopus 로고    scopus 로고
    • Random attractors I/II
    • University of Bielefeld, Department of Economics. Discussion Paper No. 327/328
    • Schenk-Hoppé, K. R., 1998, Random attractors I/II. University of Bielefeld, Department of Economics. Discussion Paper No. 327/328.
    • (1998)
    • Schenk-Hoppé, K.R.1
  • 38
    • 33646208531 scopus 로고
    • Stabilisierung durch kleine Schwingungen
    • van der Pol, B., 1925, Stabilisierung durch kleine Schwingungen. Physica 167.
    • (1925) Physica , vol.167
    • Van Der Pol, B.1
  • 39
    • 33646209486 scopus 로고
    • Lyapunov Exponent und Rotationszahl der stochastischen Eigenwert-theorie
    • Wedig, W., 1989, Lyapunov Exponent und Rotationszahl der stochastischen Eigenwert-theorie. Zeitschrift für angewandte Mathematik und Mechanik, 69: 438-440.
    • (1989) Zeitschrift für Angewandte Mathematik und Mechanik , vol.69 , pp. 438-440
    • Wedig, W.1
  • 40
    • 0012143764 scopus 로고
    • Pitchfork and Hopf bifurcations in stochastic systems-effective methods to calculate Lyapunov exponents
    • edited by P. Kree (Berlin: Springer-Verlag)
    • Wedig, W., 1995, Pitchfork and Hopf bifurcations in stochastic systems-effective methods to calculate Lyapunov exponents. Probabilistic Methods in Applied Physics, Lecture Notes in Physics 451, edited by P. Kree (Berlin: Springer-Verlag), pp. 120-148.
    • (1995) Probabilistic Methods in Applied Physics, Lecture Notes in Physics , vol.451 , pp. 120-148
    • Wedig, W.1
  • 41
    • 0003235070 scopus 로고    scopus 로고
    • Perturbation methods for Lyapunov exponents
    • edited by H. Crauel and M. Gundlach (Berlin: Springer-Verlag)
    • Wihstutz, V., 1999, Perturbation methods for Lyapunov exponents. Stochastic Dynamics, edited by H. Crauel and M. Gundlach (Berlin: Springer-Verlag).
    • (1999) Stochastic Dynamics
    • Wihstutz, V.1
  • 42
    • 0012106696 scopus 로고
    • The stationary diffusion process generated by second order differential operators and their probability currents
    • Wu, Cheng Xun, and Guo, Mao Zheng, 1980, The stationary diffusion process generated by second order differential operators and their probability currents. Chinese Annals of Mathematics, 3-4: 523-540.
    • (1980) Chinese Annals of Mathematics , vol.3-4 , pp. 523-540
    • Wu, C.X.1    Guo, M.Z.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.