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Cascade mode locking: a possible route to chaos in the two-wave Hamiltonian system
Volume 152, number 3,4
PHYSICS LETTERS A
14 January 1991
Cascade mode locking: a possible route to chaos in the two-wave Hamiltonian system Y. Gell and R. Nakach Association Euratom CEA sur Ia Fusion Contrôlée, CEN Cadarache, 13108 St. Paul Lez Durance, Cedex, France Received 10 January 1990; accepted for publication 9 November 1990 Communicated by A.R. Bishop
We present a numerical study of the two-wave Hamiltonian system which reveals the route to chaos as a process involving the mode-locking phenomenon. Using a Fourier analysis, we found that final chaos is reached after a cascade of lockings allowing for the generation of large amplitude low frequency oscillations. Due to the nonlinearity of the system, these oscillations combine with all the existing peaks in the spectrum. The consequence of this interaction might be the building up of a raised spectrum consisting of broad diffuse patterns, which is the signature of chaotic motion.
The routes to chaos taken by nonlinear physical systems have been of major interest during the last few years. A rather large number of realistic physical systems both conservative and dissipative have been found to behave in a chaotic manner when some parameters of the system reach threshold values [1]. Among the conservative systems in which chaotic behaviour is revealed, of considerable interest is the well-known “two-wave system” consisting of two electrostatic waves propagating with different phase velocities in the same direction, and affecting the motion of a charged particle. The equation of motion governing this system is a paradigm equation which adequately modelizes a variety of physical processes [2], and the associated Hamiltonian might serve as a basic model for investigating some mathematical aspects of the theory of intrinsic stochasticity. Previous experience in searching the route to chaos indicates that the Fourier power spectrum analysis is an efficient and convenient technique for analyzing the evolution to chaos of dynamical systems. In applying this technique to the “two-wave system” we realized [3] that finding the route to chaos amounts to finding the route to the generation of low frequency modes of sizeable amplitudes. Once such low frequency modes have been generated, they and their harmonics (appearing as sizeable peaks in the power 0375-9601/91/$ 03.50 © 1991
—
spectra) are translated into the full spectrum via the nonlinear coupling between modes inherent to the system. When these modes couple with the already existing sizeable peaks in the power spectrum, a raised spectrum consisting of broad diffuse patterns might result. This structure is the signature of chaotic motion. In this work we propose a mechanism, based on the mode-locking phenomenon ~‘ for generating low frequency oscillations consequently leading to chaos. The presentation of this mechanism is best done by following a numerical study of the “two-wave system” revealing a process of cascade lockings which leads to the generation of low frequency modes in the system. The first step in this cascade process is detected by following the change in the frequency of a pronounced mode in the power spectrum, when changing the strength of the perturbation, observing the locking of this mode to a fixed frequency inherent to the system. This locking sets in when the perturbation reaches a particular value, and persists even when increasing the perturbation further. Simultaneously with the locking of this mode, one observes the appearance of a new sizeable mode which, in its
The literature ofthissubject is rather extensive. A few instructive examples are given in ref. (4].
Elsevier Science Publishers B.V. (North-Holland)
165
Volume 152, number 3,4
PHYSICS LETTERS A
14 January 1991
(a) 4
“C,.)
I,
C/’)
0
-4
0••
‘~
(b)
~ 0
i iT~ThJi
-4
o
2
4
6
8
Fig. 1. (a)— (d) Power spectrum S( w) of the solution v( t) = dx/dt of eq. (1), using a log
10 scale, as a function of the angular frequency owith initialconditionsx0=0, v0=5.8 andparametersa= 16, u=85, (a) ~=0.12, (b) =0.169, (C) ~=0.23, (d) =0.236.
turn, has its frequency changing with the change of the strength of the perturbation, beginning thus the second step of this cascade process. The locking of this mode is found to be at a rational frequency ratio of the previous locked frequency and is accompanied by a new sizeable mode and so on. To see how 166
this cascading locking process is realized in a specific physical system, we study numerically the power spectrum associated with the nonlinear differential equation describing the motion of a particle affected by two electrostatic waves propagating in the same direction but having different phase velocities. As is
Volume 152, number 3,4
PHYSICS LETTERS A
(c).
14 January 1991
,,
4 ,/,;C,1;
/‘
~“
(d) 4,1,
~_....“
“
//~‘
I
~
i
I
I
/ 0
-.4
0••
.
Fig. 1. Continued.
well known [21 this equation of motion can be written in a normalized form as d2x +sin x= sin(ax— Vt), (1)
frequency parameters of the system. To obtain the power spectrum associated with eq. (1), we first solve it numerically and then apply a standard procedure for obtaining an accurate power spectrum. To this end, we employ a fast-Fourier-
where ~ is considered as a small parameter and a and v are respectively the normalized wave number and
transform algorithm to process 4096-point time Series ofthe velocity of the particle which was initially
—~
167
Volume 152, number 3,4
PHYSICS LETTERS A
shaped by a cosine bell window of Hanning type [5] used to eliminate spurious frequency components associated with sharp edges in the time series. In order to exhibit locking of modes as a scenario to a possible route to chaos, we find it convenient to consider motion of particles far away from the separatrix (the separatrix being determined by the unperturbed motion). A typical example of the velocity power spectrum associated with such a motion corresponding to the parameter set a=l6, v=85, ~=0.l2 with initial values x0=0, v0= 5.8, is given in fig. 1. In fig. 1, major well-defined and separated peaks are clearly distinguished signifying a regular motion. The most prominent peak seen in fig. 1 and marked by an arrow is associated with the free oscillation at frequency w0= 5.585 and corresponding to the unperturbed motion of the particle, shifted, however, somewhat from this value due to the finiteness of the perturbation. Recognized is also an other major independent peak to the left ofthe “w0” peak at a frequency a~= 4.162 and marked by an arrow accordingly. Now, due to the nonlinearity of the system, all the other major spectral lines appearing in fig. 1 can be attributed to a linear combination of these two independent frequencies co~and w1, in the form w1=mw0+nW1 where m and n take appropriate positive or negative integer values. The most important line resulting from this combination is clearly distinguished at w0, = = 1.423. The free frequency depends only slightly on while w1 depends on it rather significantly. In order to see that, we show in fig. 2 the shift in position ofthe “w0” and “w1” peaks with the changes of the perturbation As is seen from fig. 2, the “w0” peak is essentially fixed while w1 changes significantly with linearly in the lower parameter range and acquiring a nonlinear dependence when approaching ~ values for which a chaotic type of motion sets in. The finite range of ~ for which the motion becomes chaotic is represented by a dotted line in fig. 2. The significance of this stochastic region was considered by us in a previous work [61 and will not be elaborated here further. When increasing ~beyond this region, the motion becomesagain regular, however, the “w1” peak does not depend on ~ anymore, and remains constant fixed to a rational, locking value of w0, w1 / = ~for this set of parameters. Simultaneously with the setting in of this “w1” mode-locking, a new sig—
—
~,
~.
~,
168
14 January 1991
chaotic motion Wi
4
° ~
2
W3
4 w37
1 ~1 1
—
w2
4 W
2
_________________________________________ 0.06 0.08 0.10
0.12
0.14
0.16
0.18
0.20 0.22
.
Fig. 2. The position in frequency of the peaks “w0” “~“ “wa” and “w3” which are defined in the text, as a function of the perturbation strength ~. The points of this graph are deduced from power spectra similar to the one shown in fig. l,and initial values and parameters are as in fig. 1.
nificant peak appears at frequency w2 as can be seen in fig. lb which corresponds to the motion of the particle for the same set of parameters and initial values except that ~ has been increased to the value ~= 0.169. This “W2” mode depends on ~ and apparently follows the same basic evolution scheme as “W1”, namely, having a linear dependence which is followed by a nonlinear one. Similarly to “wi”, the “w2” mode in its turn gets also locked to an intrinsic fixed frequency in the system which is this time w1 we recall that the fixed frequency was w0 for the locking of the “w1” mode. This dependence of”w2” is shown in fig. 2 in which the locking of the frequency is clearly exhibited at a value which is a rational value of w1, co2/w1 = With the locking ofthis “W2” mode, a new independent “w3” mode appears, as is seen in the power spectrum presented in fig. 1c. ~.
Volume 152, number 3,4
PHYSICS LETFERSA
The ~ dependence of this mode has essentially the same basic features as those of “Wi” and “a2”, and this mode is locked in its turn to a rational value of W2, the ratio being in this case W3/W3 =4. The dependence of the “W3” mode and its locking is presented in fig. 2. By increasing ~ up to ~= 0.236, the amplitude of this “W3” mode increases and when combined with the “W2” mode which is close to it, results in a sizeable low frequency mode W3...2 =a3—w2, seen and indicated in fig. ld. Notice that the existence of several large locked peaks in the spectrum, including especially a low frequency peak, will lead due to their linear combinations to a power spectrum filled up with sizeable peaks, which is, as mentioned in ref. [31,a significant stage in the route to chaos. As seen in fig. 2, the locking range of the “W3” mode is rather narrow and when increasing beyond this range, up to ~= 0.2368, one can observe final stochasticity setting in, which is signified by a raised power spectrum. This cascading locking process is not limited to this particular set of parameters and initial values. Other examples exhibiting this process have been found as well. Let us mention an interesting case corresponding to a relatively strong perturbation (f of order 1) with initial values associated with an islet of stability, x0=4.7l, v0=4.5. In trying to get an overall view on the cascade locking process just presented, we notice that except for the intrinsic free running frequency w~and its harmonics and the external frequency V, sizeable peaks in the power spectrum have generally their origin in some resonance process taking place in the system. Indeed, if one considers the terms of the Fourier decomposition of the perturbation, one realizes that the terms which are definitely nonresonating will average out to zero. Only the peaks associated with terms corresponding to a resonance process which are thus varying slowly enough in time, can be pronouncedly detected in a numerical study. Another consideration to take into account when studying the size of the peaks in the spectrum is the ratio p/q (p and q integers) between the resonating frequencies. As is well known, when p and q are high value integers, one usually cannot expect a conesponding sizeable peak in the power spectrum. Before the locking process sets in, the different modes in the system considered here have only one fixed frequency (v) to resonate with. A mode having
14 January 1991
a frequency with favourable p/q ratio to the fixed frequency will be represented by a sizeable peak in the power spectrum while all other modes will be represented by much smaller signals. However, the generation of a new fixed frequency due to the locking process creates new opportunities for other modes to resonate with a fixed frequency at a favourable p/ q ratio. (This was our working assumption which helped us to decide to what a fixed frequency mode a mode with changing frequency in the system gets locked to.) Once a mode characterized by a frequency, say, W2, gets locked to a fixed frequency mode, (Os, with which it has a favourable p/q ratio, the nonlinear coupling between modes in the system will result in the appearance of a sizeable peak at a fixed frequency Wi ...2=WI—W2. The frequency of this peak will be smaller than Wi. Indeed, if W~>W2 then w~, 2 is of course smaller than Wi, if W3 is larger than w~,it is still smaller than 2w~(since 2w~will have by assumption a less favourable p/q ratio to (02 than wi). Thus, also in such a case the frequency W~_3 will be again smaller than w~.The peak having a frequency W~3 will represent a fixed frequency mode with which other modes in the system can resonate with. If a mode “(03” will resonate with this WI,..2 mode and gets locked to it, the mode having the frequency ~i,23=~I,.2~~3 will be again a fixed frequency mode having a frequencylower than Wi,.2 and in its turn will be a new mode to which other modes can resonate with and get locked to. In this manner one can expect the generation of lower and frequency modes. The sizeable peaks corresponding to these modes combine with all the peaks of the spectrum to fill it up. When the frequency of these locked modes or their difference is low enough the power spectrum will be dense enough with sizeable peaks to be recognized as a chaotic type power spectrum. In the example we have studied in this work we believe to have seen the first steps in this process, we suppose the convergence of the following steps in the process to proceed rather fast with the increase of ~ so it is difficult to detect the generation of the very low frequency modes which ultimately causes stochasticity in the system. We would like to mention that such difficulties are encountered also in following the period doubling route to chaos, where only 169
Volume 152, number 3,4
PHYSICS LETTERS A
the first few steps in the period doubling process are detected. A detailed analysis of the cascade locking process for the case considered here is presently in preparation. One ofus (Y.G.) would like to thank Dr. J. Tachon for his hospitality at the Centre d’Etudes Nucléaires de Cadarache during the winter 1988—1989 when this work started. We would also like to thank Dr. F. Geniet for many helpful discussions.
170
14 January 1991
References [1] F.C. Moon, Chaotic vibrations (Wiley, New York, 1987). [2] A.B. Rechester and T. Stix, Phys. Rev. A 19 (1979) 1656, and references therein. [3] Y. Gell and R. Nakach, Phys. Rev. A 34 (1986) 4276. [4] V.1 Arnold, Izv. Akad. Nauk Ser. Mat. 25 (1961) 21; Usp. Mat. Nauk 38 (1983) 189 [Russ. Math. Surv. 38 (1983) 215];
M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960, 1970; RE. Ecke, J.D. Farmer and D.K. Umberger, Nonlinearity 2 (1989) 175. [51P.1. Richards, IEEE Spectrum 4 (1967) 83; G.D. Bergland, IEEE Spectrum 42 (1969) 42. [61 Y. Gell and R. Nakach, Euratom CEA Reports, EUR-CEAFC-l367 (1989).
PHYSICS LETTERS A
14 January 1991
Cascade mode locking: a possible route to chaos in the two-wave Hamiltonian system Y. Gell and R. Nakach Association Euratom CEA sur Ia Fusion Contrôlée, CEN Cadarache, 13108 St. Paul Lez Durance, Cedex, France Received 10 January 1990; accepted for publication 9 November 1990 Communicated by A.R. Bishop
We present a numerical study of the two-wave Hamiltonian system which reveals the route to chaos as a process involving the mode-locking phenomenon. Using a Fourier analysis, we found that final chaos is reached after a cascade of lockings allowing for the generation of large amplitude low frequency oscillations. Due to the nonlinearity of the system, these oscillations combine with all the existing peaks in the spectrum. The consequence of this interaction might be the building up of a raised spectrum consisting of broad diffuse patterns, which is the signature of chaotic motion.
The routes to chaos taken by nonlinear physical systems have been of major interest during the last few years. A rather large number of realistic physical systems both conservative and dissipative have been found to behave in a chaotic manner when some parameters of the system reach threshold values [1]. Among the conservative systems in which chaotic behaviour is revealed, of considerable interest is the well-known “two-wave system” consisting of two electrostatic waves propagating with different phase velocities in the same direction, and affecting the motion of a charged particle. The equation of motion governing this system is a paradigm equation which adequately modelizes a variety of physical processes [2], and the associated Hamiltonian might serve as a basic model for investigating some mathematical aspects of the theory of intrinsic stochasticity. Previous experience in searching the route to chaos indicates that the Fourier power spectrum analysis is an efficient and convenient technique for analyzing the evolution to chaos of dynamical systems. In applying this technique to the “two-wave system” we realized [3] that finding the route to chaos amounts to finding the route to the generation of low frequency modes of sizeable amplitudes. Once such low frequency modes have been generated, they and their harmonics (appearing as sizeable peaks in the power 0375-9601/91/$ 03.50 © 1991
—
spectra) are translated into the full spectrum via the nonlinear coupling between modes inherent to the system. When these modes couple with the already existing sizeable peaks in the power spectrum, a raised spectrum consisting of broad diffuse patterns might result. This structure is the signature of chaotic motion. In this work we propose a mechanism, based on the mode-locking phenomenon ~‘ for generating low frequency oscillations consequently leading to chaos. The presentation of this mechanism is best done by following a numerical study of the “two-wave system” revealing a process of cascade lockings which leads to the generation of low frequency modes in the system. The first step in this cascade process is detected by following the change in the frequency of a pronounced mode in the power spectrum, when changing the strength of the perturbation, observing the locking of this mode to a fixed frequency inherent to the system. This locking sets in when the perturbation reaches a particular value, and persists even when increasing the perturbation further. Simultaneously with the locking of this mode, one observes the appearance of a new sizeable mode which, in its
The literature ofthissubject is rather extensive. A few instructive examples are given in ref. (4].
Elsevier Science Publishers B.V. (North-Holland)
165
Volume 152, number 3,4
PHYSICS LETTERS A
14 January 1991
(a) 4
“C,.)
I,
C/’)
0
-4
0••
‘~
(b)
~ 0
i iT~ThJi
-4
o
2
4
6
8
Fig. 1. (a)— (d) Power spectrum S( w) of the solution v( t) = dx/dt of eq. (1), using a log
10 scale, as a function of the angular frequency owith initialconditionsx0=0, v0=5.8 andparametersa= 16, u=85, (a) ~=0.12, (b) =0.169, (C) ~=0.23, (d) =0.236.
turn, has its frequency changing with the change of the strength of the perturbation, beginning thus the second step of this cascade process. The locking of this mode is found to be at a rational frequency ratio of the previous locked frequency and is accompanied by a new sizeable mode and so on. To see how 166
this cascading locking process is realized in a specific physical system, we study numerically the power spectrum associated with the nonlinear differential equation describing the motion of a particle affected by two electrostatic waves propagating in the same direction but having different phase velocities. As is
Volume 152, number 3,4
PHYSICS LETTERS A
(c).
14 January 1991
,,
4 ,/,;C,1;
/‘
~“
(d) 4,1,
~_....“
“
//~‘
I
~
i
I
I
/ 0
-.4
0••
.
Fig. 1. Continued.
well known [21 this equation of motion can be written in a normalized form as d2x +sin x= sin(ax— Vt), (1)
frequency parameters of the system. To obtain the power spectrum associated with eq. (1), we first solve it numerically and then apply a standard procedure for obtaining an accurate power spectrum. To this end, we employ a fast-Fourier-
where ~ is considered as a small parameter and a and v are respectively the normalized wave number and
transform algorithm to process 4096-point time Series ofthe velocity of the particle which was initially
—~
167
Volume 152, number 3,4
PHYSICS LETTERS A
shaped by a cosine bell window of Hanning type [5] used to eliminate spurious frequency components associated with sharp edges in the time series. In order to exhibit locking of modes as a scenario to a possible route to chaos, we find it convenient to consider motion of particles far away from the separatrix (the separatrix being determined by the unperturbed motion). A typical example of the velocity power spectrum associated with such a motion corresponding to the parameter set a=l6, v=85, ~=0.l2 with initial values x0=0, v0= 5.8, is given in fig. 1. In fig. 1, major well-defined and separated peaks are clearly distinguished signifying a regular motion. The most prominent peak seen in fig. 1 and marked by an arrow is associated with the free oscillation at frequency w0= 5.585 and corresponding to the unperturbed motion of the particle, shifted, however, somewhat from this value due to the finiteness of the perturbation. Recognized is also an other major independent peak to the left ofthe “w0” peak at a frequency a~= 4.162 and marked by an arrow accordingly. Now, due to the nonlinearity of the system, all the other major spectral lines appearing in fig. 1 can be attributed to a linear combination of these two independent frequencies co~and w1, in the form w1=mw0+nW1 where m and n take appropriate positive or negative integer values. The most important line resulting from this combination is clearly distinguished at w0, = = 1.423. The free frequency depends only slightly on while w1 depends on it rather significantly. In order to see that, we show in fig. 2 the shift in position ofthe “w0” and “w1” peaks with the changes of the perturbation As is seen from fig. 2, the “w0” peak is essentially fixed while w1 changes significantly with linearly in the lower parameter range and acquiring a nonlinear dependence when approaching ~ values for which a chaotic type of motion sets in. The finite range of ~ for which the motion becomes chaotic is represented by a dotted line in fig. 2. The significance of this stochastic region was considered by us in a previous work [61 and will not be elaborated here further. When increasing ~beyond this region, the motion becomesagain regular, however, the “w1” peak does not depend on ~ anymore, and remains constant fixed to a rational, locking value of w0, w1 / = ~for this set of parameters. Simultaneously with the setting in of this “w1” mode-locking, a new sig—
—
~,
~.
~,
168
14 January 1991
chaotic motion Wi
4
° ~
2
W3
4 w37
1 ~1 1
—
w2
4 W
2
_________________________________________ 0.06 0.08 0.10
0.12
0.14
0.16
0.18
0.20 0.22
.
Fig. 2. The position in frequency of the peaks “w0” “~“ “wa” and “w3” which are defined in the text, as a function of the perturbation strength ~. The points of this graph are deduced from power spectra similar to the one shown in fig. l,and initial values and parameters are as in fig. 1.
nificant peak appears at frequency w2 as can be seen in fig. lb which corresponds to the motion of the particle for the same set of parameters and initial values except that ~ has been increased to the value ~= 0.169. This “W2” mode depends on ~ and apparently follows the same basic evolution scheme as “W1”, namely, having a linear dependence which is followed by a nonlinear one. Similarly to “wi”, the “w2” mode in its turn gets also locked to an intrinsic fixed frequency in the system which is this time w1 we recall that the fixed frequency was w0 for the locking of the “w1” mode. This dependence of”w2” is shown in fig. 2 in which the locking of the frequency is clearly exhibited at a value which is a rational value of w1, co2/w1 = With the locking ofthis “W2” mode, a new independent “w3” mode appears, as is seen in the power spectrum presented in fig. 1c. ~.
Volume 152, number 3,4
PHYSICS LETFERSA
The ~ dependence of this mode has essentially the same basic features as those of “Wi” and “a2”, and this mode is locked in its turn to a rational value of W2, the ratio being in this case W3/W3 =4. The dependence of the “W3” mode and its locking is presented in fig. 2. By increasing ~ up to ~= 0.236, the amplitude of this “W3” mode increases and when combined with the “W2” mode which is close to it, results in a sizeable low frequency mode W3...2 =a3—w2, seen and indicated in fig. ld. Notice that the existence of several large locked peaks in the spectrum, including especially a low frequency peak, will lead due to their linear combinations to a power spectrum filled up with sizeable peaks, which is, as mentioned in ref. [31,a significant stage in the route to chaos. As seen in fig. 2, the locking range of the “W3” mode is rather narrow and when increasing beyond this range, up to ~= 0.2368, one can observe final stochasticity setting in, which is signified by a raised power spectrum. This cascading locking process is not limited to this particular set of parameters and initial values. Other examples exhibiting this process have been found as well. Let us mention an interesting case corresponding to a relatively strong perturbation (f of order 1) with initial values associated with an islet of stability, x0=4.7l, v0=4.5. In trying to get an overall view on the cascade locking process just presented, we notice that except for the intrinsic free running frequency w~and its harmonics and the external frequency V, sizeable peaks in the power spectrum have generally their origin in some resonance process taking place in the system. Indeed, if one considers the terms of the Fourier decomposition of the perturbation, one realizes that the terms which are definitely nonresonating will average out to zero. Only the peaks associated with terms corresponding to a resonance process which are thus varying slowly enough in time, can be pronouncedly detected in a numerical study. Another consideration to take into account when studying the size of the peaks in the spectrum is the ratio p/q (p and q integers) between the resonating frequencies. As is well known, when p and q are high value integers, one usually cannot expect a conesponding sizeable peak in the power spectrum. Before the locking process sets in, the different modes in the system considered here have only one fixed frequency (v) to resonate with. A mode having
14 January 1991
a frequency with favourable p/q ratio to the fixed frequency will be represented by a sizeable peak in the power spectrum while all other modes will be represented by much smaller signals. However, the generation of a new fixed frequency due to the locking process creates new opportunities for other modes to resonate with a fixed frequency at a favourable p/ q ratio. (This was our working assumption which helped us to decide to what a fixed frequency mode a mode with changing frequency in the system gets locked to.) Once a mode characterized by a frequency, say, W2, gets locked to a fixed frequency mode, (Os, with which it has a favourable p/q ratio, the nonlinear coupling between modes in the system will result in the appearance of a sizeable peak at a fixed frequency Wi ...2=WI—W2. The frequency of this peak will be smaller than Wi. Indeed, if W~>W2 then w~, 2 is of course smaller than Wi, if W3 is larger than w~,it is still smaller than 2w~(since 2w~will have by assumption a less favourable p/q ratio to (02 than wi). Thus, also in such a case the frequency W~_3 will be again smaller than w~.The peak having a frequency W~3 will represent a fixed frequency mode with which other modes in the system can resonate with. If a mode “(03” will resonate with this WI,..2 mode and gets locked to it, the mode having the frequency ~i,23=~I,.2~~3 will be again a fixed frequency mode having a frequencylower than Wi,.2 and in its turn will be a new mode to which other modes can resonate with and get locked to. In this manner one can expect the generation of lower and frequency modes. The sizeable peaks corresponding to these modes combine with all the peaks of the spectrum to fill it up. When the frequency of these locked modes or their difference is low enough the power spectrum will be dense enough with sizeable peaks to be recognized as a chaotic type power spectrum. In the example we have studied in this work we believe to have seen the first steps in this process, we suppose the convergence of the following steps in the process to proceed rather fast with the increase of ~ so it is difficult to detect the generation of the very low frequency modes which ultimately causes stochasticity in the system. We would like to mention that such difficulties are encountered also in following the period doubling route to chaos, where only 169
Volume 152, number 3,4
PHYSICS LETTERS A
the first few steps in the period doubling process are detected. A detailed analysis of the cascade locking process for the case considered here is presently in preparation. One ofus (Y.G.) would like to thank Dr. J. Tachon for his hospitality at the Centre d’Etudes Nucléaires de Cadarache during the winter 1988—1989 when this work started. We would also like to thank Dr. F. Geniet for many helpful discussions.
170
14 January 1991
References [1] F.C. Moon, Chaotic vibrations (Wiley, New York, 1987). [2] A.B. Rechester and T. Stix, Phys. Rev. A 19 (1979) 1656, and references therein. [3] Y. Gell and R. Nakach, Phys. Rev. A 34 (1986) 4276. [4] V.1 Arnold, Izv. Akad. Nauk Ser. Mat. 25 (1961) 21; Usp. Mat. Nauk 38 (1983) 189 [Russ. Math. Surv. 38 (1983) 215];
M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960, 1970; RE. Ecke, J.D. Farmer and D.K. Umberger, Nonlinearity 2 (1989) 175. [51P.1. Richards, IEEE Spectrum 4 (1967) 83; G.D. Bergland, IEEE Spectrum 42 (1969) 42. [61 Y. Gell and R. Nakach, Euratom CEA Reports, EUR-CEAFC-l367 (1989).