SCOPUS 정보 검색 플랫폼

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 67, Issue 2 2, 2003, Pages 262101-2621017

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

(4) Lai, Ying Cheng a Liu, Zonghua a Billings, Lora b Schwartz, Ira B c

a ARIZONA STATE UNIVERSITY (United States)

b MONTCLAIR STATE UNIVERSITY (United States)

c NAVAL RESEARCH LABORATORY (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; LYAPUNOV METHODS; NONLINEAR SYSTEMS; RANDOM PROCESSES;

RANDOM DYNAMICAL SYSTEMS;

CHAOS THEORY;

EID: 0038057925 PISSN: 1063651X EISSN: None Source Type: Journal
DOI: None Document Type: Article

Times cited : (42)

References (86)

1
- 0001623717
- J. P. Crutchfield and B. A. Huberman, Phys. Lett. A 77, 407 (1980); J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, Phys. Rep. 92, 45 (1982).
- (1980) Phys. Lett. A , vol.77 , pp. 407
- Crutchfield, J.P.¹ Huberman, B.A.²

2
- 0002330439
- J. P. Crutchfield and B. A. Huberman, Phys. Lett. A 77, 407 (1980); J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, Phys. Rep. 92, 45 (1982).
- (1982) Phys. Rep. , vol.92 , pp. 45
- Crutchfield, J.P.¹ Farmer, J.D.² Huberman, B.A.³

3
- 33645083962
- J. M. Deutsch, Phys. Rev. Lett. 52, 1230 (1984); J. Phys. A18, 1449 (1985).
- (1984) Phys. Rev. Lett. , vol.52 , pp. 1230
- Deutsch, J.M.¹

4
- 33645083962
- J. M. Deutsch, Phys. Rev. Lett. 52, 1230 (1984); J. Phys. A18, 1449 (1985).
- (1985) J. Phys. A , vol.18 , pp. 1449

5
- 0000716387
- J. Rössler, M. Kiwi, B. Hess, and M. Markus, Phys. Rev. A 39, 5954 (1989).
- (1989) Phys. Rev. A , vol.39 , pp. 5954
- Rössler, J.¹ Kiwi, M.² Hess, B.³ Markus, M.⁴

6
- 0038857272
- J. M. Deutsch and G. Paladin, Phys. Rev. Lett. 62, 695 (1989).
- (1989) Phys. Rev. Lett. , vol.62 , pp. 695
- Deutsch, J.M.¹ Paladin, G.²

7
- 0000740518
- F. Ladrappier and L.-S. Young, Commun. Math. Phys. 117, 529 (1989).
- (1989) Commun. Math. Phys. , vol.117 , pp. 529
- Ladrappier, F.¹ Young, L.-S.²

8
- 0000273155
- L. Yu, E. Ott, and Q. Chen, Physica D 53, 102 (1992).
- (1992) Physica D , vol.53 , pp. 102
- Yu, L.¹ Ott, E.² Chen, Q.³

9
- 0000483903
- J. B. Gao, S. K. Hwang, and J. M. Liu, Phys. Rev. Lett. 82, 1132 (1999); S. K. Hwang, J. B. Gao, and J. M. Liu, Phys. Rev. E 61, 5162 (2000).
- (1999) Phys. Rev. Lett. , vol.82 , pp. 1132
- Gao, J.B.¹ Hwang, S.K.² Liu, J.M.³

10
- 0001280391
- J. B. Gao, S. K. Hwang, and J. M. Liu, Phys. Rev. Lett. 82, 1132 (1999); S. K. Hwang, J. B. Gao, and J. M. Liu, Phys. Rev. E 61, 5162 (2000).
- (2000) Phys. Rev. E , vol.61 , pp. 5162
- Hwang, S.K.¹ Gao, J.B.² Liu, J.M.³

11
- 0034273284
- S. Rim, D.-U. Hwang, I. Kim, and C.-M. Kim, Phys. Rev. Lett. 85, 2304 (2000).
- (2000) Phys. Rev. Lett. , vol.85 , pp. 2304
- Rim, S.¹ Hwang, D.-U.² Kim, I.³ Kim, C.-M.⁴

12
- 0039741988
- T. W. Carr, L. Billings, I. B. Schwartz, and I, Triandaf, Physica D147, 59 (2000); L. Billings and I. B. Schwartz, J. Math. Biol. 44, 33 (2002).
- (2000) Physica D , vol.147 , pp. 59
- Carr, T.W.¹ Billings, L.² Schwartz, I.B.³ Triandaf, I.⁴

13
- 0039741988
- T. W. Carr, L. Billings, I. B. Schwartz, and I, Triandaf, Physica D147, 59 (2000); L. Billings and I. B. Schwartz, J. Math. Biol. 44, 33 (2002).
- (2002) J. Math. Biol. , vol.44 , pp. 33
- Billings, L.¹ Schwartz, I.B.²

14
- 0002053804
- M. V. Jacobson, Commun. Math. Phys. 81, 39 (1981).
- (1981) Commun. Math. Phys. , vol.81 , pp. 39
- Jacobson, M.V.¹

15
- 0001874180
- u(A). The corresponding numerical trajectories lie within δ of the true trajectories given the roundoff error δ. Consider a phase space of N=3 dimensions. In the plane spanned by the two stable directions at A, all computer-generated trajectories lie in a circle of radius 6. The condition that both fixed points are embedded in an ergodic invariant set implies that some time later, trajectories near A visit a neighborhood of B. When the trajectories that started near A approach B, they begin to separate along the new unstable direction. Along this unstable direction, numerical and true trajectories separate from each other exponentially with time at a rate determined by the associated eigenvalue. The numerical trajectories can no longer be shadowed by the true trajectories at B.
- (1970) Proc. Symp. Pure Math , vol.14 , pp. 5
- Abraham, R.¹ Smale, S.²

16
- 0000252317
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1994) Phys. Rev. Lett. , vol.73 , pp. 1927
- Dawson, S.P.¹ Grebogi, C.² Sauer, T.³ Yorke, J.A.⁴

17
- 0001729952
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1996) Phys. Rev. Lett. , vol.76 , pp. 4348
- Dawson, S.P.¹

18
- 0008025028
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1997) Phys. Rev. E , vol.55 , pp. 5350
- Moresco, P.¹ Dawson, S.P.²

19
- 0003153450
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1997) Phys. Rev. Lett. , vol.79 , pp. 59
- Sauer, T.¹ Grebogi, C.² Yorke, J.A.³

20
- 0002623916
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1997) Physica D , vol.109 , pp. 81
- Kostelich, E.J.¹ Kan, I.² Grebogi, C.³ Ott, E.⁴ Yorke, J.A.⁵

21
- 0032614845
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (1999) Phys. Rev. Lett. , vol.82 , pp. 4803
- Lai, Y.-C.¹ Grebogi, C.²

22
- 0034695792
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (2000) Phys. Rev. Lett. , vol.84 , pp. 1689
- Barreto, E.¹ So, P.² Gluckman, B.J.³ Schiff, S.J.⁴

23
- 0034683241
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (2000) Phys. Rev. Lett. , vol.85 , pp. 2490
- Barreto, E.¹ So, P.²

24
- 41349092465
- S. P. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett. 73, 1927 (1994); S. P. Dawson, ibid. 76, 4348 (1996); P. Moresco and S. P. Dawson, Phys. Rev. E 55, 5350 (1997); T. Sauer, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 79, 59 (1997); E. J. Kostelich, I. Kan, C, Grebogi, E. Ott, and J. A. Yorke, Physica D 109, 81 (1997); Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999); E. Barreto, P. So, B. J. Gluckman, and S. J. Schiff, ibid. 84, 1689 (2000); E. Barreto and P. So, ibid. 85, 2490 (2000); H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E 65, 026209 (2002).
- (2002) Phys. Rev. E , vol.65 , pp. 026209
- Kantz, H.¹ Grebogi, C.² Prasad, A.³ Lai, Y.-C.⁴ Sinde, E.⁵

25
- 0000646269
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (1994) Phys. Rev. Lett. , vol.72 , pp. 1451
- Maritan, A.¹ Banavar, J.²

26
- 0000922144
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (1995) Phys. Rev. E , vol.52 , pp. 3238
- Herzel, H.¹ Freund, J.²

27
- 0002941295
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (1996) Phys. Lett. A , vol.217 , pp. 21
- Gade, P.M.¹ Basu, C.²

28
- 0031247067
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (1997) Phys. Rev. E , vol.56 , pp. 4068
- Sanchez, E.¹ Matias, M.A.² Perez-Munuzuri, V.³

29
- 0032529059
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (1998) Europhys. Lett. , vol.43 , pp. 376
- Lai, C.-H.¹ Zhou, C.²

30
- 41349087693
- A. Maritan and J. Banavar, Phys. Rev. Lett. 72, 1451 (1994); H. Herzel and J. Freund, Phys. Rev. E 52, 3238 (1995); P. M. Gade and C. Basu, Phys. Lett. A 217, 21 (1996); E. Sanchez, M. A. Matias, and V. Perez-Munuzuri, Phys. Rev. E 56, 4068 (1997); C.-H. Lai and C. Zhou, Europhys. Lett. 43, 376 (1998); C. Zhou and J. Kurths, Phys. Rev. E 65, 040101 (2002).
- (2002) Phys. Rev. E , vol.65 , pp. 040101
- Zhou, C.¹ Kurths, J.²

31
- 0001703165
- J. C. Sommerer and E. Ott, Science 259, 335 (1993).
- (1993) Science , vol.259 , pp. 335
- Sommerer, J.C.¹ Ott, E.²

32
- 0003145080
- Y.-C. Lai, Phys. Rev. E 53, 57 (1996); Y.-C. Lai, U. Feudel, and C. Grebogi, ibid. 54, 6070 (1996): T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997).
- (1996) Phys. Rev. E , vol.53 , pp. 57
- Lai, Y.-C.¹

33
- 0030420880
- Y.-C. Lai, Phys. Rev. E 53, 57 (1996); Y.-C. Lai, U. Feudel, and C. Grebogi, ibid. 54, 6070 (1996): T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997).
- (1996) Phys. Rev. E , vol.54 , pp. 6070
- Lai, Y.-C.¹ Feudel, U.² Grebogi, C.³

34
- 5344223647
- Y.-C. Lai, Phys. Rev. E 53, 57 (1996); Y.-C. Lai, U. Feudel, and C. Grebogi, ibid. 54, 6070 (1996): T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997).
- (1997) Phys. Rev. Lett. , vol.77 , pp. 5039
- Yalcinkaya, T.¹ Lai, Y.-C.²

35
- 0001277505
- M. Harrison and Y.-C. Lai, Phys. Rev. E 59, R3799 (1999); Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1471 (2000).
- (1999) Phys. Rev. E , vol.59
- Harrison, M.¹ Lai, Y.-C.²

36
- 0039003816
- M. Harrison and Y.-C. Lai, Phys. Rev. E 59, R3799 (1999); Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1471 (2000).
- (2000) Int. J. Bifurcation Chaos Appl. Sci. Eng. , vol.10 , pp. 1471

37
- 33645050744
- note
- The Lyapunov exponents are the time-averaged stretching or contracting rates of infinitesimal vectors along a typical trajectory in the phase space, regardless of whether the system is deterministic or stochastic.

38
- 33645090754
- note
- This consideration does not apply to nonautonomous systems for which there is always a neutral direction along the time axis and therefore always a zero Lyapunov exponent.

39
- 0037171176
- Z. Liu, Y.-C. Lai, L. Billings, and I. B. Schwartz, Phys. Rev. Lett. 88, 124101 (2002).
- (2002) Phys. Rev. Lett. , vol.88 , pp. 124101
- Liu, Z.¹ Lai, Y.-C.² Billings, L.³ Schwartz, I.B.⁴

40
- 33645051281
- note
- In finite times, there can be deviations of the Lyapunov exponents of the noisy attractor from these of the periodic attractor. However, any deviations will vanish asymptotically, due to the averaging effect of the noise.

41
- 0012349493
- C. Grebogi, E. Ott, and J. A. Yorke, Physica D 7, 181 (1983); H. Kantz and P. Grassberger, ibid. 17, 75 (1985).
- (1983) Physica D , vol.7 , pp. 181
- Grebogi, C.¹ Ott, E.² Yorke, J.A.³

42
- 46549094713
- C. Grebogi, E. Ott, and J. A. Yorke, Physica D 7, 181 (1983); H. Kantz and P. Grassberger, ibid. 17, 75 (1985).
- (1985) Physica D , vol.17 , pp. 75
- Kantz, H.¹ Grassberger, P.²

43
- 0004141085
- edited by Bai-lin Hao (World Scientific, Singapore)
- T. Tél, in Directions in Chaos, edited by Bai-lin Hao (World Scientific, Singapore, 1990), Vol. 3; in STATPHYS 19, edited by Bai-lin Hao (World Scientific, Singapore, 1996).
- (1990) Directions in Chaos , vol.3
- Tél, T.¹

44
- 0004210476
- World Scientific, Singapore
- T. Tél, in Directions in Chaos, edited by Bai-lin Hao (World Scientific, Singapore, 1990), Vol. 3; in STATPHYS 19, edited by Bai-lin Hao (World Scientific, Singapore, 1996).
- (1996) STATPHYS 19
- Hao, B.-L.¹

45
- 29444438680
- C, Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. A 37, 1711 (1988).
- (1988) Phys. Rev. A , vol.37 , pp. 1711
- Grebogi, C.¹ Ott, E.² Yorke, J.A.³

46
- 0000644990
- G.-H. Hsu, E. Ott, and C. Grebogi, Phys. Lett. A 127, 199 (1988).
- (1988) Phys. Lett. A , vol.127 , pp. 199
- Hsu, G.-H.¹ Ott, E.² Grebogi, C.³

47
- 45149143640
- H. E. Nusse and J. A. Yorke, Physica D 36, 137 (1989).
- (1989) Physica D , vol.36 , pp. 137
- Nusse, H.E.¹ Yorke, J.A.²

48
- 0002427559
- J. Jacobs, E. Ott, and C. Grebogi, Physica D 108, 1 (1997).
- (1997) Physica D , vol.108 , pp. 1
- Jacobs, J.¹ Ott, E.² Grebogi, C.³

49
- 48749149528
- P. Grassberger, Phys. Lett. A 97, 227 (1983); H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983); P. Grassberger, Phys. Lett. A 107, 101 (1985).
- (1983) Phys. Lett. A , vol.97 , pp. 227
- Grassberger, P.¹

50
- 0346372923
- P. Grassberger, Phys. Lett. A 97, 227 (1983); H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983); P. Grassberger, Phys. Lett. A 107, 101 (1985).
- (1983) Physica D , vol.8 , pp. 435
- Hentschel, H.G.E.¹ Procaccia, I.²

51
- 0000899534
- P. Grassberger, Phys. Lett. A 97, 227 (1983); H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983); P. Grassberger, Phys. Lett. A 107, 101 (1985).
- (1985) Phys. Lett. A , vol.107 , pp. 101
- Grassberger, P.¹

52
- 3342916075
- T. C. Halsey, M. J. Jensen, L. P. Kadanoff, I. Procaccia, and B, I. Shraiman, Phys. Rev. A 33, 1141 (1986).
- (1986) Phys. Rev. A , vol.33 , pp. 1141
- Halsey, T.C.¹ Jensen, M.J.² Kadanoff, L.P.³ Procaccia, I.⁴ Shraiman, B.I.⁵

53
- 0001600430
- B. R. Hunt, E. Ott, and J. A. Yorke, Phys. Rev. E 54, 4819 (1996).
- (1996) Phys. Rev. E , vol.54 , pp. 4819
- Hunt, B.R.¹ Ott, E.² Yorke, J.A.³

54
- 0003451932
- in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Springer, Berlin
- J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979).
- (1979) Lecture Notes in Mathematics , vol.730
- Kaplan, J.L.¹ Yorke, J.A.²

55
- 0003769663
- Princeton University Press, Princeton
- R. M. May, Stability and Complexity in Model Ecosystems (Princeton University Press, Princeton, 1973).
- (1973) Stability and Complexity in Model Ecosystems
- May, R.M.¹

56
- 0000548789
- O. E. Rössler, Phys. Lett. A 71, 155 (1979).
- (1979) Phys. Lett. A , vol.71 , pp. 155
- Rössler, O.E.¹

57
- 0003568337
- Springer-Verlag, Berlin
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer-Verlag, Berlin, 1992).
- (1992) Numerical Solution of Stochastic Differential Equations
- Kloeden, P.E.¹ Platen, E.²

58
- 0003582543
- Cambridge University Press, Cambridge, Chap. 9
- E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993), Chap. 9.
- (1993) Chaos in Dynamical Systems
- Ott, E.¹

59
- 0000241853
- E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
- (1963) J. Atmos. Sci. , vol.20 , pp. 130
- Lorenz, E.N.¹

60
- 33645060816
- note
- 2(p)]. While both the topological rigidity introduced here and the contrast measure characterize the degree of unstable dimension variability, the former is relatively straightforward to compute because it does not involve the computation of unstable periodic orbits.

61
- 0002256723
- [37] For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1983) Prog. Theor. Phys. , vol.69 , pp. 32
- Fujisaka, H.¹ Yamada, T.²

62
- 0000796974
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1984) Prog. Theor. Phys. , vol.72 , pp. 885

63
- 0001778066
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1985) Prog. Theor. Phys. , vol.74 , pp. 919

64
- 0000931523
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1991) J. Phys. A , vol.24 , pp. 4587
- Pikovsky, A.S.¹ Grassberger, P.²

65
- 12044255333
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1993) Phys. Rev. Lett. , vol.70 , pp. 279
- Platt, N.¹ Spiegel, E.A.² Tresser, C.³

66
- 0001475002
- [37] For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1995) Phys. Rev. E , vol.52
- Lai, Y.-C.¹ Grebogi, C.²

67
- 0002881538
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1996) Physica D , vol.96 , pp. 66
- Venkataramani, S.C.¹ Antonsen T.M., Jr.² Ott, E.³ Sommerer, J.C.⁴

68
- 5344223647
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (1997) Phys. Rev. Lett. , vol.77 , pp. 5039
- Yalcinkaya, T.¹ Lai, Y.-C.²

69
- 0035926605
- For example, see H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983); 72, 885 (1984); 74, 919 (1985); A. S. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); S. C. Venkataramani, T. M. Antonsen Jr., E, Ott, and J. C. Sommerer, Physica D 96, 66 (1996); T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 77, 5039 (1997); Z. Liu and Y.-C. Lai, ibid. 86, 4737 (2001).
- (2001) Phys. Rev. Lett. , vol.86 , pp. 4737
- Liu, Z.¹ Lai, Y.-C.²

70
- 0001296111
- J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); N. Platt, S. M. Hammel, and J. F. Heagy, Phys. Rev. Lett. 72, 3498 (1994).
- (1994) Phys. Rev. E , vol.49 , pp. 1140
- Heagy, J.F.¹ Platt, N.² Hammel, S.M.³

71
- 12044249418
- J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); N. Platt, S. M. Hammel, and J. F. Heagy, Phys. Rev. Lett. 72, 3498 (1994).
- (1994) Phys. Rev. Lett. , vol.72 , pp. 3498
- Platt, N.¹ Hammel, S.M.² Heagy, J.F.³

72
- 84983710576
- D. Marthaler, D. Armbruster, Y.-C. Lai, and E. J. Kostelich, Phys. Rev. E 64, 016220 (2001).
- (2001) Phys. Rev. E , vol.64 , pp. 016220
- Marthaler, D.¹ Armbruster, D.² Lai, Y.-C.³ Kostelich, E.J.⁴

73
- 19044376529
- L. Billings, E. Sollt, and I. B. Schwartz, Phys. Rev. Lett. 88, 234101 (2002).
- (2002) Phys. Rev. Lett. , vol.88 , pp. 234101
- Billings, L.¹ Sollt, E.² Schwartz, I.B.³

74
- 0032623990
- I. T. Georgiou and I. B. Schwartz, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 59, 1178 (1999).
- (1999) SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. , vol.59 , pp. 1178
- Georgiou, I.T.¹ Schwartz, I.B.²

75
- 0028003827
- S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, Nature (London) 370, 615 (1994).
- (1994) Nature (London) , vol.370 , pp. 615
- Schiff, S.J.¹ Jerger, K.² Duong, D.H.³ Chang, T.⁴ Spano, M.L.⁵ Ditto, W.L.⁶

76
- 0025078458
- A. L. Goldberger, Ann. Biomed. Eng. 18, 195 (1990); M. A. Woo, W. G. Stevenson, D. K. Moser, R. M. Harper, and R. Trelease, Am. Heart J. 123, 704 (1992).
- (1990) Ann. Biomed. Eng. , vol.18 , pp. 195
- Goldberger, A.L.¹

77
- 0026536071
- A. L. Goldberger, Ann. Biomed. Eng. 18, 195 (1990); M. A. Woo, W. G. Stevenson, D. K. Moser, R. M. Harper, and R. Trelease, Am. Heart J. 123, 704 (1992).
- (1992) Am. Heart J. , vol.123 , pp. 704
- Woo, M.A.¹ Stevenson, W.G.² Moser, D.K.³ Harper, R.M.⁴ Trelease, R.⁵

78
- 84973949757
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (1987) Behav. Brain Sei. , vol.10 , pp. 161
- Skarda, C.A.¹ Freeman, W.J.²

79
- 0028029553
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (1994) J. Theor. Biol. , vol.171 , pp. 13
- Freeman, W.J.¹

80
- 84973949757
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (1995) Int. J. Intell. Syst. , vol.10 , pp. 71

81
- 84973949757
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (2000) Neural Netw. , vol.13 , pp. 11

82
- 0034492939
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (2000) J. Phys. (Paris) , vol.94 , pp. 303

83
- 0035108245
- C. A. Skarda and W. J. Freeman, Behav. Brain Sei. 10, 161 (1987); W. J. Freeman, J. Theor. Biol. 171, 13 (1994); Int. J. Intell. Syst. 10, 71 (1995); Neural Netw. 13, 11 (2000); J. Phys. (Paris) 94, 303 (2000); W. J. Freeman, R. Kozma, and P. J. Werbos, BioSystems 59, 109 (2001).
- (2001) BioSystems , vol.59 , pp. 109
- Freeman, W.J.¹ Kozma, R.² Werbos, P.J.³

84
- 0000009894
- W. Yang, M. Ding, A. J. Mandell, and E. Ott, Phys. Rev. E 51, 102 (1995).
- (1995) Phys. Rev. E , vol.51 , pp. 102
- Yang, W.¹ Ding, M.² Mandell, A.J.³ Ott, E.⁴

85
- 85040901747
- Cambridge University Press, New York
- J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, New York, 1989).
- (1989) The Kinematics of Mixing: Stretching, Chaos, and Transport
- Ottino, J.M.¹

86
- 0001460304
- Y.-C. Lai, Phys. Rev. E 59, R3807 (1999).
- (1999) Phys. Rev. E , vol.59
- Lai, Y.-C.¹

* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.