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Incursive fractals: a robust mechanism of basin erosion preceding the optimal escape from a potential well
Volume 150, number 8,9
PHYSICS LETTERSA
19 November 1990
Incursive fractals: a robust mechanism of basin erosion preceding the optimal escape from a potential well A.N. L a n s b u r y a n d J.M.T. T h o m p s o n Department of CivilEngineering, UniversityCollegeLondon, GowerStreet, London WCIE 6BT, UK Received 27 March 1990; accepted for publication 10 September 1990 Communicatedby A.P. Fordy
Comparative cell-to-cellmappings of the basins of driven oscillators with cubic and quartic potential wells show remarkable qualitative and quantitative correlations. We conclude that the recently identified erosion by incursive fmctals is a robust phenomenon facilitatingthe optimal escape from a well.
1. Introduction In a recent study of the escape of a driven oscillator from a canonical cubic potential well [ 1 ], attention was focused on the frequency ratio of nonlinear resonance, which gives optimal escape under a minimum forcing magnitude. It was shown that, preceding optimal or near-optimal escape, there is a dramatic erosion of the union of the basins of the constrained attractors [ 1-3]. This occurs a short forcing interval after the homoclinic tangency of the invariant manifolds of the hill-top saddle cycle [4 ], which signals transition from a smooth to a fractal basin boundary [ 5,6 ]. Within this interval the thin fractal zone around the edge of the basin is of little practical concern, but beyond it, thick fractal fingers become rapidly incursive and quickly striate the entire basin. This basin erosion is nicely displayed and quantified by plotting the safe area, within a suitable window, against the forcing magnitude to give an integrity diagram representing the probability of constraint from random starting conditions [3 ]. The steep Dover clifton this diagram defines forcing level above which a physical system in a noisy environment would have little chance of remaining constrained, and might thus be a useful design criterion in engineering and applied science. We here use Hsu's cell-to-cell mapping technique [ 7 ] to make comparative studies between this es-
cape to infinity from a cubic single well and the escape from one-well to cross-well motions in the quartic twin-well Duffing oscillator. For equal values of damping ratio and frequency ratio, we show that there is a striking quantitative and qualitative similarity of behaviour. The incursive fractals have essentially the same form and growth characteristics; and the integrity diagrams, based on normalized forcing magnitudes, can be superimposed to give a remarkably good quantitative fit. It is concluded that basin erosion by incursive fractals is a robust phenomenon that must be expected in other models when tuned to frequencies close to optimal escape.
2. Two archetypal oscillators The first oscillator considered is that introduced by Thompson [ 1 ] to explore the escape from a typical asymmetric well: we refer to it as model F, and write it as
~+ flJc+ x - x 2 = F sin o~t . Here a dot denotes differentiation with respect to time, t, and we write ± - y . Taking fl=O.1, corresponding to a linear damping ratio of ~= 0.05, we focus attention on ~ = 0 . 8 5 , which is numerically equal to the frequency ratio, z,, between the driving and the linear natural frequency of small undamped, undriv-
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
355
Volume 150, number 8,9
PHYSICS LETTERS A
en oscillations about x = 0. This value of coy is close to that giving optimal escape at a minimum value of F. The second is the twin-well Duffing oscillator that has been extensively studied as an archetypal model exhibiting chaotic dynamics [ 8 ]: recent work on its bifurcational structure is particularly relevant [911 ]. We refer to it as model A and write it in the form
This has two equal wells at x = _+ 1 separated by a local potential maximum at x = 0. We are concerned with the escape from the left-hand well, and observe that for small linear vibrations about x = - 1, the two relevant ratios are damping ratio
( = ½fl=k/2x/~,
frequency ratio
v=toF =OJA/X/~.
It is also possible to define a forcing ratio relating the magnitude of the forcing to a measure of the restoring force: one useful measure is tl~e average force between the bottom of the well and the hill-top, which in these equations is numerically equal to the potential barrier. However in the present investigation we normalize F and A to unity at the respective steady state escape values, eliminating the need for a unified forcing ratio.
3. Summary of the steady state responses For each oscillator we study the response at ( = 0.05 and v=0.85 under slowly incremented forcing, and the major relevant bifurcations are summarized in table 1. Each has a regime of resonant hysteresis in the fundamental n = 1 response, associated with a Table 1 Some significant bifurcational values for the two models.
356
jump to resonance at a (saddle-node) fold A and a jump off resonance (under decreasing F, A ) at a fold B. The invariant manifolds, the inset and outset [ 8 ], of the directly-unstable hill-top saddle cycle that evolves continuously from the unstable hill-top equilibrium at F = A =0, exhibit a homoclinic tangency at F M, A M given approximately by FM____ fl sinh(ncoF)
5xco2
~ q - k J c - x + x 3 = A sin 09At.
fold B fold A tangency M escape E
19 November 1990
Singlewell (model F) #=0.1, tot=0.85
Twin well (model A)
F
f=F/F ~
A
a=A/A E
0.047 0.069 0.063 0.109
0.431 0.633 0.578 1.000
0.076 0.105 0.119 0.215
0.353 0.488 0.555 1.000
k= x/~, co^= 0.85x/~
~,0.0633,
AM = 4k cosh(ntOA/2) 3V/~ ~OgA ~0.1193. These results of a Melnikov perturbation analysis [4 ] are in good agreement, at the current (~, v), with numerical observations of the invariant manifolds [1,6,12]. One feature that does not correlate between models is that for the single well we have F S < F M < F A, while for the twin well we have AB
We show finally the escape values, F E, A E, at which a slowly evolving system from F = A = 0 would jump out of the well. This jump is preceded by a period doubling cascade to chaos within the well, and corresponds to a chaotic saddle catastrophe [9]. In model F, it is a blue sky event resulting in escape to infinity, while in model A it is an explosion from a single-well chaotic attractor to a cross-well chaotic attractor. At forcing magnitudes just beyond bifurcation E, all starts within the relevant well lead to escape over the energy barrier. Details of this escape mechanism for model F, given by Thompson and Ueda [ 1,2 ], show that at E the chaotic attractor collides with a directly unstable subharmonic of order six, the remnant of a localized, n = 6, fold-flip-cascade-crisis scenario.
4. Basin sequences by cell-to-ceil mapping Fig. 1 shows for each model a sequence of safe basins at values o f a - A / A E andf=--F/F E of 0.2, 0.4, 0.6, 0.8. White indicates escape over the potential barrier: the criterion of escape adopted for model A was passage beyond the point of maximum energy gradient, x > lx/~; and for model F, which has no comparable maximum, we took the equivalent distance corresponding to, x - 1 > lx//-3. Each sequence
Volume 150, n u m b e r 8,9
PHYSICS LETTERS A
( n o t e nffi3 b a s i n )
19 November 1990
[
Fig. 1. Equally spaced comparative safe basins for single and twin well oscillators with damping ratio ~=0.05, frequency ratio u=0.85. Cell-to-cell mapping with 640 by 350 pixels at phase ~ = 180 °. First column shows model F in the window - O . 8 < x < 1.2, - 1.05 < y < 1.05; second column shows model A in the window - 1.8 < x < 0.2, - 1.05 < y < 1.05.
357
Volume 150, number 8,9
PHYSICS LETTERS A
Fig. 2. Comparative basins in the region of incursive erosion. Data as for fig. 1.
358
19 November 1990
Volume 150, number 8,9
PHYSICS LETTERS A
19 November 1990
::i!~": : ' . : ~ = i ~ '
f=0.92
; ........................
~ .i
: ::'?"
f=0.94
-
[ 096 ,,:
--"'!d:"'z'd "-- . _ . . . . .
~
a=0.92
.
•
I
,...
.,;'.L.~..__ ....:;:L"._--'_~ _______~~ -/' : :"
"i :'(i .........
~
a--0.94 a=0.94
'r'!
a=0.91
'
G,,
-
/~,098 Fig. 3. Comparative basins close to the final steady state escape. Data as for fig. 1.
359
Volume 150, number 8,9
PHYSICS LETTERS A
passes through a hysteresis regime, where the hatched and black regions are the (safe) basins of the two competing n = 1 attractors. In the f = 0.8 picture there is a coexisting n = 3 subharmonic whose basin is indicated by the regularly-dotted (grey) tone. All the black regions have their n = 1 attractors located by white circles at the Poincar6 mapping points. Fig. 2 shows a more finely spaced set in the interval of rapid basin erosion. Despite the relative horizontal elongation of the F basins, due to the shallower gradient of the F potential at negative x, there is a remarkable qualitative and quantitative synchronization of the fractal incursion. Note that despite the granular appearance, due to the coarseness of the cell. grid, all basin boundaries are actually formed by thin whiskers o f a continuous invariant manifold. The single n = 1 attractor throughout this set is marked by a white circle. Fig. 3 shows a final set towards the end of the (a, f ) interval where the black safe basin has very little area. Attractors of the main period-doubling cascade are located by their mapping points: all are n = 2 except for one n = 1 which has not yet period-doubled, and one n = 4 which has period-doubled twice.
5. Comparative integrity curves The erosion of the safe basin areas, within equivalent windows, is illustrated in fig. 4. Here the areas are normalized to unity at A = F = 0, and the horizontal axis represents the normalized forcing magnitudes, a = A / A E,f = F/FE. The integrity curves show remarkable qualitative and quantitative similarities, with a steep cliff and a small raised beach at its foot. The detailed mechanism of basin erosion by incursive fractals involves changes in the Birkhoff signature of the tangled manifolds, and associated attractor-basin bifurcations [ 11 ] giving changes in what Grebogi, Ott and Yorke [ 13 ] call the accessible boundary orbit. Current work on this is to be described in ref. [ 14 ].
6. Concluding remarks Basin erosion by incursive fractals seems to be a robust phenomenon that might have considerable 360
19 November 1990
1.2 Normalized Area of Safe Basin
I
@
@
.~/.~.o~o.F)
Cubic single well
Quartie twin well (Model A)
0.8
• A
0.6 Damping ratio, f=O.05
Q
Frequency ratio, v=0.85
A
0.4
A 0.2
0 0
t
i
i
i
•
0.2
0.4
0.6
0.8
1
1.2
Fig. 4. Integrity curves showing the erosion o f the normalized safe basin areas for the two models.
importance for the integrity of systems in noisy or ill-defined environments [ 15 ]. It could be particulady relevant to the stability of ships in waves, as expounded in ref. [16 ], where the influence of the damping level on the positions and severity of the cliff is fully explored. Here it has particular value in supplying an index ofcapsizability that can be quickly established by a small number of computer or wavetank simulations. Transient capsize, under a short pulse of regular waves, offers a highly valuable design criterion, because our studies show that there is an equally well-defined erosion of the "transient basin" corresponding to escape within as few as 8 forcing cycles.
References [ 1 ] J.M.T. Thompson, Proc. R. Soc. A 421 (1989) 195. [2 ] J.M.T. Thompson and Y. Ueda, Dyn. Stab. Syst. 4 (1989) 285. [3] M.S. Soliman and J.M.T. Thompson, J. Sound Vibr. 135 (1989) 453. [ 4 ] J.M.T. Thompson, S:R. Bishop and L.M. Leung, Phys. Lett. Al21 (1987) ll6.
Volume 150, number 8,9
PHYSICS LETTERS A
[ 5 ] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica D 17 (1985) 125. [ 6 ] F.C. Moon, Chaotic vibrations: an introduction for applied scientists and engineers (Wiley, New York, 1987 ). [ 7 ] C.S Hsu, Cell-to-cell mapping: a method of global analysis for nonlinear systems (Springer, Berlin, 1987 ). [8] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos (Wiley, New York, 1986). [9]H.B. Stewart, A chaotic saddle catastrophe in forced oscillators, in: Dynamical systems approaches to nonlinear problems in systems and circuits, eds. F. Salam and M. Levi (SIAM, Philadelphia, 1987). [10]Y. Ueda, H. Nakajima, T. Hikihara and H.B. Stewart, Forced two-well potential Duffing's oscillator, in: dynamical systems approaches to nonlinear problems in systems and circuits, eds. F. Salam and M. Levi (SIAM, Philadelphia, 1987).
19 November 1990
[ 11 ] Y. Ueda, S. Yoshida, H.B. Stewart and J.M.T. Thompson, Basin explosions and escape phenomena in the twin-well Duff'rag oscillator, compound global bifurcations organizing behaviour, Philos. Trans. R. Soc. A 332 (1990), in press. [ 12 ] N.A. Alexander, J. Sound Vibr. 135 (1989) 63. [13] C. Grcbogi, E. Ott and J.A. Yorke, Physica D 24 (1987) 243. [14] A.N. Lansbury, J.M.T. Thompson and H.B. Stewart, in preparation. [ 15 ] J.M.T. Thompson and M.S. Soliman, Proc. R. Soc. A 428 (1990) 1. [ 16 ] J.M.T. Thompson, R.C.T. Rainey and M.S. Soliman, Ship stability criteria based on chaotic transients from incursive fractals, Philos. Trans. R. Soc. A 332 (1990), in press.
361
PHYSICS LETTERSA
19 November 1990
Incursive fractals: a robust mechanism of basin erosion preceding the optimal escape from a potential well A.N. L a n s b u r y a n d J.M.T. T h o m p s o n Department of CivilEngineering, UniversityCollegeLondon, GowerStreet, London WCIE 6BT, UK Received 27 March 1990; accepted for publication 10 September 1990 Communicatedby A.P. Fordy
Comparative cell-to-cellmappings of the basins of driven oscillators with cubic and quartic potential wells show remarkable qualitative and quantitative correlations. We conclude that the recently identified erosion by incursive fmctals is a robust phenomenon facilitatingthe optimal escape from a well.
1. Introduction In a recent study of the escape of a driven oscillator from a canonical cubic potential well [ 1 ], attention was focused on the frequency ratio of nonlinear resonance, which gives optimal escape under a minimum forcing magnitude. It was shown that, preceding optimal or near-optimal escape, there is a dramatic erosion of the union of the basins of the constrained attractors [ 1-3]. This occurs a short forcing interval after the homoclinic tangency of the invariant manifolds of the hill-top saddle cycle [4 ], which signals transition from a smooth to a fractal basin boundary [ 5,6 ]. Within this interval the thin fractal zone around the edge of the basin is of little practical concern, but beyond it, thick fractal fingers become rapidly incursive and quickly striate the entire basin. This basin erosion is nicely displayed and quantified by plotting the safe area, within a suitable window, against the forcing magnitude to give an integrity diagram representing the probability of constraint from random starting conditions [3 ]. The steep Dover clifton this diagram defines forcing level above which a physical system in a noisy environment would have little chance of remaining constrained, and might thus be a useful design criterion in engineering and applied science. We here use Hsu's cell-to-cell mapping technique [ 7 ] to make comparative studies between this es-
cape to infinity from a cubic single well and the escape from one-well to cross-well motions in the quartic twin-well Duffing oscillator. For equal values of damping ratio and frequency ratio, we show that there is a striking quantitative and qualitative similarity of behaviour. The incursive fractals have essentially the same form and growth characteristics; and the integrity diagrams, based on normalized forcing magnitudes, can be superimposed to give a remarkably good quantitative fit. It is concluded that basin erosion by incursive fractals is a robust phenomenon that must be expected in other models when tuned to frequencies close to optimal escape.
2. Two archetypal oscillators The first oscillator considered is that introduced by Thompson [ 1 ] to explore the escape from a typical asymmetric well: we refer to it as model F, and write it as
~+ flJc+ x - x 2 = F sin o~t . Here a dot denotes differentiation with respect to time, t, and we write ± - y . Taking fl=O.1, corresponding to a linear damping ratio of ~= 0.05, we focus attention on ~ = 0 . 8 5 , which is numerically equal to the frequency ratio, z,, between the driving and the linear natural frequency of small undamped, undriv-
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
355
Volume 150, number 8,9
PHYSICS LETTERS A
en oscillations about x = 0. This value of coy is close to that giving optimal escape at a minimum value of F. The second is the twin-well Duffing oscillator that has been extensively studied as an archetypal model exhibiting chaotic dynamics [ 8 ]: recent work on its bifurcational structure is particularly relevant [911 ]. We refer to it as model A and write it in the form
This has two equal wells at x = _+ 1 separated by a local potential maximum at x = 0. We are concerned with the escape from the left-hand well, and observe that for small linear vibrations about x = - 1, the two relevant ratios are damping ratio
( = ½fl=k/2x/~,
frequency ratio
v=toF =OJA/X/~.
It is also possible to define a forcing ratio relating the magnitude of the forcing to a measure of the restoring force: one useful measure is tl~e average force between the bottom of the well and the hill-top, which in these equations is numerically equal to the potential barrier. However in the present investigation we normalize F and A to unity at the respective steady state escape values, eliminating the need for a unified forcing ratio.
3. Summary of the steady state responses For each oscillator we study the response at ( = 0.05 and v=0.85 under slowly incremented forcing, and the major relevant bifurcations are summarized in table 1. Each has a regime of resonant hysteresis in the fundamental n = 1 response, associated with a Table 1 Some significant bifurcational values for the two models.
356
jump to resonance at a (saddle-node) fold A and a jump off resonance (under decreasing F, A ) at a fold B. The invariant manifolds, the inset and outset [ 8 ], of the directly-unstable hill-top saddle cycle that evolves continuously from the unstable hill-top equilibrium at F = A =0, exhibit a homoclinic tangency at F M, A M given approximately by FM____ fl sinh(ncoF)
5xco2
~ q - k J c - x + x 3 = A sin 09At.
fold B fold A tangency M escape E
19 November 1990
Singlewell (model F) #=0.1, tot=0.85
Twin well (model A)
F
f=F/F ~
A
a=A/A E
0.047 0.069 0.063 0.109
0.431 0.633 0.578 1.000
0.076 0.105 0.119 0.215
0.353 0.488 0.555 1.000
k= x/~, co^= 0.85x/~
~,0.0633,
AM = 4k cosh(ntOA/2) 3V/~ ~OgA ~0.1193. These results of a Melnikov perturbation analysis [4 ] are in good agreement, at the current (~, v), with numerical observations of the invariant manifolds [1,6,12]. One feature that does not correlate between models is that for the single well we have F S < F M < F A, while for the twin well we have AB
We show finally the escape values, F E, A E, at which a slowly evolving system from F = A = 0 would jump out of the well. This jump is preceded by a period doubling cascade to chaos within the well, and corresponds to a chaotic saddle catastrophe [9]. In model F, it is a blue sky event resulting in escape to infinity, while in model A it is an explosion from a single-well chaotic attractor to a cross-well chaotic attractor. At forcing magnitudes just beyond bifurcation E, all starts within the relevant well lead to escape over the energy barrier. Details of this escape mechanism for model F, given by Thompson and Ueda [ 1,2 ], show that at E the chaotic attractor collides with a directly unstable subharmonic of order six, the remnant of a localized, n = 6, fold-flip-cascade-crisis scenario.
4. Basin sequences by cell-to-ceil mapping Fig. 1 shows for each model a sequence of safe basins at values o f a - A / A E andf=--F/F E of 0.2, 0.4, 0.6, 0.8. White indicates escape over the potential barrier: the criterion of escape adopted for model A was passage beyond the point of maximum energy gradient, x > lx/~; and for model F, which has no comparable maximum, we took the equivalent distance corresponding to, x - 1 > lx//-3. Each sequence
Volume 150, n u m b e r 8,9
PHYSICS LETTERS A
( n o t e nffi3 b a s i n )
19 November 1990
[
Fig. 1. Equally spaced comparative safe basins for single and twin well oscillators with damping ratio ~=0.05, frequency ratio u=0.85. Cell-to-cell mapping with 640 by 350 pixels at phase ~ = 180 °. First column shows model F in the window - O . 8 < x < 1.2, - 1.05 < y < 1.05; second column shows model A in the window - 1.8 < x < 0.2, - 1.05 < y < 1.05.
357
Volume 150, number 8,9
PHYSICS LETTERS A
Fig. 2. Comparative basins in the region of incursive erosion. Data as for fig. 1.
358
19 November 1990
Volume 150, number 8,9
PHYSICS LETTERS A
19 November 1990
::i!~": : ' . : ~ = i ~ '
f=0.92
; ........................
~ .i
: ::'?"
f=0.94
-
[ 096 ,,:
--"'!d:"'z'd "-- . _ . . . . .
~
a=0.92
.
•
I
,...
.,;'.L.~..__ ....:;:L"._--'_~ _______~~ -/' : :"
"i :'(i .........
~
a--0.94 a=0.94
'r'!
a=0.91
'
G,,
-
/~,098 Fig. 3. Comparative basins close to the final steady state escape. Data as for fig. 1.
359
Volume 150, number 8,9
PHYSICS LETTERS A
passes through a hysteresis regime, where the hatched and black regions are the (safe) basins of the two competing n = 1 attractors. In the f = 0.8 picture there is a coexisting n = 3 subharmonic whose basin is indicated by the regularly-dotted (grey) tone. All the black regions have their n = 1 attractors located by white circles at the Poincar6 mapping points. Fig. 2 shows a more finely spaced set in the interval of rapid basin erosion. Despite the relative horizontal elongation of the F basins, due to the shallower gradient of the F potential at negative x, there is a remarkable qualitative and quantitative synchronization of the fractal incursion. Note that despite the granular appearance, due to the coarseness of the cell. grid, all basin boundaries are actually formed by thin whiskers o f a continuous invariant manifold. The single n = 1 attractor throughout this set is marked by a white circle. Fig. 3 shows a final set towards the end of the (a, f ) interval where the black safe basin has very little area. Attractors of the main period-doubling cascade are located by their mapping points: all are n = 2 except for one n = 1 which has not yet period-doubled, and one n = 4 which has period-doubled twice.
5. Comparative integrity curves The erosion of the safe basin areas, within equivalent windows, is illustrated in fig. 4. Here the areas are normalized to unity at A = F = 0, and the horizontal axis represents the normalized forcing magnitudes, a = A / A E,f = F/FE. The integrity curves show remarkable qualitative and quantitative similarities, with a steep cliff and a small raised beach at its foot. The detailed mechanism of basin erosion by incursive fractals involves changes in the Birkhoff signature of the tangled manifolds, and associated attractor-basin bifurcations [ 11 ] giving changes in what Grebogi, Ott and Yorke [ 13 ] call the accessible boundary orbit. Current work on this is to be described in ref. [ 14 ].
6. Concluding remarks Basin erosion by incursive fractals seems to be a robust phenomenon that might have considerable 360
19 November 1990
1.2 Normalized Area of Safe Basin
I
@
@
.~/.~.o~o.F)
Cubic single well
Quartie twin well (Model A)
0.8
• A
0.6 Damping ratio, f=O.05
Q
Frequency ratio, v=0.85
A
0.4
A 0.2
0 0
t
i
i
i
•
0.2
0.4
0.6
0.8
1
1.2
Fig. 4. Integrity curves showing the erosion o f the normalized safe basin areas for the two models.
importance for the integrity of systems in noisy or ill-defined environments [ 15 ]. It could be particulady relevant to the stability of ships in waves, as expounded in ref. [16 ], where the influence of the damping level on the positions and severity of the cliff is fully explored. Here it has particular value in supplying an index ofcapsizability that can be quickly established by a small number of computer or wavetank simulations. Transient capsize, under a short pulse of regular waves, offers a highly valuable design criterion, because our studies show that there is an equally well-defined erosion of the "transient basin" corresponding to escape within as few as 8 forcing cycles.
References [ 1 ] J.M.T. Thompson, Proc. R. Soc. A 421 (1989) 195. [2 ] J.M.T. Thompson and Y. Ueda, Dyn. Stab. Syst. 4 (1989) 285. [3] M.S. Soliman and J.M.T. Thompson, J. Sound Vibr. 135 (1989) 453. [ 4 ] J.M.T. Thompson, S:R. Bishop and L.M. Leung, Phys. Lett. Al21 (1987) ll6.
Volume 150, number 8,9
PHYSICS LETTERS A
[ 5 ] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica D 17 (1985) 125. [ 6 ] F.C. Moon, Chaotic vibrations: an introduction for applied scientists and engineers (Wiley, New York, 1987 ). [ 7 ] C.S Hsu, Cell-to-cell mapping: a method of global analysis for nonlinear systems (Springer, Berlin, 1987 ). [8] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos (Wiley, New York, 1986). [9]H.B. Stewart, A chaotic saddle catastrophe in forced oscillators, in: Dynamical systems approaches to nonlinear problems in systems and circuits, eds. F. Salam and M. Levi (SIAM, Philadelphia, 1987). [10]Y. Ueda, H. Nakajima, T. Hikihara and H.B. Stewart, Forced two-well potential Duffing's oscillator, in: dynamical systems approaches to nonlinear problems in systems and circuits, eds. F. Salam and M. Levi (SIAM, Philadelphia, 1987).
19 November 1990
[ 11 ] Y. Ueda, S. Yoshida, H.B. Stewart and J.M.T. Thompson, Basin explosions and escape phenomena in the twin-well Duff'rag oscillator, compound global bifurcations organizing behaviour, Philos. Trans. R. Soc. A 332 (1990), in press. [ 12 ] N.A. Alexander, J. Sound Vibr. 135 (1989) 63. [13] C. Grcbogi, E. Ott and J.A. Yorke, Physica D 24 (1987) 243. [14] A.N. Lansbury, J.M.T. Thompson and H.B. Stewart, in preparation. [ 15 ] J.M.T. Thompson and M.S. Soliman, Proc. R. Soc. A 428 (1990) 1. [ 16 ] J.M.T. Thompson, R.C.T. Rainey and M.S. Soliman, Ship stability criteria based on chaotic transients from incursive fractals, Philos. Trans. R. Soc. A 332 (1990), in press.
361