“Oscillator–wave” model: properties and heuristic instances

Chaos, Solitons and Fractals 17 (2003) 41–60 www.elsevier.com/locate/chaos

‘‘Oscillator–wave’’ model: properties and heuristic instances q Vladimir Damgov *, Plamen Trenchev, Kostadin Sheiretsky Nonlinear Space Dynamics, Space Research Institute at the Bulgarian Academy of Sciences, 6 Moskowska str., P.O. Box 799, 1000 Sofia, Bulgaria Accepted 7 October 2002 Communicated by T. Kapitaniak

Abstract The article considers a generalized model of an oscillator, subjected to the influence of an external wave. It is shown that the systems of diverse physical background, which this model encompasses by their nature, should belong to the broader, proposed in previous works class of ‘‘kick-excited self-adaptive dynamical systems’’. The theoretical treatment includes an analytic approach to the conditions for emergence of small and large amplitudes, i.e. weak and strong nonlinearity of the system. Derived also are generalized conditions for the transition of systems of this ‘‘oscillator–wave’’ type to non-regular and chaotic behaviour. For the purpose of demonstrating the heuristic properties of the generalized oscillator–wave model from this point of view are considered the relevant systems and phenomena of the quantized cyclotron resonance and the megaquantum resonance-wave model of the Solar System. We point to a number of other natural and scientific phenomena, which can be effectively analyzed from the point of view of the developed approach. In particular we stress on the possibility for development and the wide applicability of specific wave influences, for example for the improvement and the speeding up of technological processes. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction The works [1–5] present the class of kick-excited self-adaptive dynamical systems. In these works are provided numeric and analytic proofs for the behaviour and the principal properties of dynamic systems subjected to the influence of an external non-linear exciting force. The considerations are based on a generalized pendulum model the nonlinearity of the external influence being introduced through particularly selected functional dependencies. Here we consider the generalized ‘‘oscillator–wave’’ model and show that the inhomogeneous external influence is realized naturally and does not require any specific conditions. Systems covered by the ‘‘oscillation–wave’’ model immanently belong to the generalized class of kick-excited self-adaptive dynamical systems. Attempting maximal clarity of the sequence of presentation we consider the excitation of oscillations in a non-linear oscillator of the ‘‘pendulum’’ type under the influence of an incoming (fall) wave. We will show that under certain condition in the system will arise non-attenuated oscillations with a frequency close to the systemÕs natural frequency and an amplitude which belongs to a defined discrete spectrum of possible amplitudes. A second important quality also appears––self-adaptive stability of the excited oscillations with given amplitude for a broad range of the incoming waveÕs intensity.

q

An investigation supported by the Bulgarian National Council ‘‘Scientific Research’’ under Contract no. H3-1106/01. Corresponding author. E-mail address: [email protected] (V. Damgov).

*

0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 4 4 6 - 0

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

2. Model of the interaction of an oscillator with an electromagnetic wave: approach applicable for small amplitudes of the systems oscillations Let us consider the interaction of an electromagnetic wave with a weakly dissipative non-linear oscillator. Let the electric charge q having mass m oscillate along the x-axis under the influence of a non-linear returning force around a certain fixed point. The electromagnetic wave also propagates along the x-axis and has a longitudinal electric field component E. The equation of motion for the charge interacting with the wave can be represented as mð€x þ 2dd x_ þ x20 sin xÞ ¼ Eq sinðmt  kxÞ;

ð1Þ

where 2dd is the damping coefficient, x0 is the natural frequency of small oscillations of the charge, m is the wave frequency and k is the wave number. The case considered is: m  x0 . We assume that the excitation of charge oscillations by the influence of the wave does not perturb significantly the symmetry of the chargeÕs motion around its equilibrium position and the coordinate of the charge changes according to the law x ¼ a sin h;

h ¼ xt þ a;

a ¼ aðtÞ;

a ¼ aðtÞ:

ð2Þ

Substituting the solution (2) in the right-hand side of Eq. (1) we obtain Eq sinðmt  ka sin hÞ ¼ Eq

1 X

Jn ðkaÞ sinðmt  nhÞ:

n¼1

Letting x_ ¼ ax cos h, a_ sin h þ a_ a cos h ¼ 0 in accordance with the Krilov–Bogolyubov–Mitropolskii method [6] we obtain to first order:  1 X  da  ¼ 2dd a þ F0 Jn ðkaÞ sinðmt  nhÞ cos h;  dt x n¼1  ð3Þ  1  da X2  x2 F0 X  ¼  J ðkaÞ sinðmt  nhÞ cos h; n  dt 2x x n¼1 where X is the frequency of free oscillations having an amplitude a,   2J1 ðaÞ 2 a2 x0 ffi x20 1  þ Oða4 Þ ; X2 ¼ a 8

ð4Þ

Jn ð Þ are Bessel functions of the first kind, F0 ¼ Eq=m. The excitation of continuous oscillations with a frequency x ¼ xs close to the oscillatorÕs natural frequency is only possible under the condition ð2p=kÞa > 1 where k is the wavelength of the influencing wave. As result of the interaction of the oscillator with the wave a frequency components appears in the force spectrum that is close to its natural oscillation frequency. Then the action of these spectral components becomes dominant and the right-hand side of Eq. (1) attains the form: 1 n h h m  i m  io X 1 Jn ðkaÞ sinðmt  nhÞ ¼ F0 Jðm=xÞ1 ðkaÞ sin xt  Eq  1 a  Jðm=xÞþ1 ðkaÞ sin xt þ þ1 a : ð5Þ m n¼1 x x Under the condition m > x the resonance area of the non-linear oscillator can be entered by several spectral components of the exciting wave each of which could excite the oscillator into stationary oscillations with amplitude belonging to a discrete sequence of possible amplitudes. For fixed parameters of the oscillator and the wave the excitation of oscillations with amplitude from the possible sequence of amplitudes is determined by the initial conditions. In accordance with relation (4) the values of the discrete sequence of stationary amplitudes can be calculated by the formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 m as0 ¼ 8 1  2 2 ffi 4 1  ; s ¼ 1; 2; 3; . . . ð6Þ sx0 s x0 Averaging the right-hand side of Eqs. (3) and taking into account (5), we determine:   das F0   dt ¼ dd a þ 2xs ½Js1 ðkas Þ þ Jsþ1 ðkas Þ sinðpt  cs Þ;   das X2  x2 F0 s   dt ¼ 2x  2x a ½Js1 ðkas Þ  Jsþ1 ðkas Þ cosðpt  cs Þ; s s s

ð7Þ

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

43

where p ¼ m  ðm=xs Þ xs , cs ¼ ðm=xs Þas , xs ¼ v=s. In accordance with the familiar recurrence relations for Bessel functions, Eqs. (7) can be represented in the form:   das mF0   dt ¼ dd as  x2 kas Js ðkas Þ sinðpt  cs Þ; s  ð8Þ  das X2s  x2s F0 0  ¼  Js ðkas Þ cosðpt  cs Þ:  dt 2xs xs as In the case of stationary oscillations (das =dt ¼ 0 and das =dt ¼ 0) from Eqs. (8) we find tg cs ¼

2dd as x2s k Js0 ðkas Þ : ðX2s  x2s Þm Js ðkas Þ

The connection between the intensity of the waveÕs longitudinal component and the amplitude of oscillations has the form:  2 2 2  2 aso xs dd k aso ðX2s  x2s Þ 2 þ : ð9Þ F0 ¼ vJs ðkaso Þ 2Js0 ðkaso Þ For high intensities of the wave equation (9) can be represented as F0 ffi

a2so x2s ðas  aso Þ : 8Js0 ðkas Þ

The first term in formula (9) represents the minimal threshold value F0 of the waveÕs intensity. If the intensity of the wave is smaller than this threshold value only the excitation of forced oscillations with frequency equal to the waveÕs frequency is possible. For wave intensities above the threshold value depending on the initial conditions, the oscillatorÕs motion is realized with one of the amplitudes from the discrete sequence (6). When m > x0 each amplitude is realized for oscillation frequency close to the oscillatorÕs natural frequency Xs . Using the approach, developed in [3], it is not difficult to show that for fixed values of the frequency m and the amplitude F0 of the external force the oscillatorÕs motion with amplitude from the discrete sequence (6) is stable. The performed analysis shows that the continuous wave having a frequency much larger than the frequency of a given oscillator can excite in it oscillations with a frequency close to its natural frequency and an amplitude belonging to a discrete set of possible stable amplitudes. The settling of certain particular amplitude depends on the initial conditions. When the motion becomes stationary the amplitudeÕs value practically does not depend on the waveÕs intensity when the latter changes over a significant range above a certain threshold value. This is reminiscent of EinsteinÕs explanation of the photoelectric effect using PlanckÕs quantization hypothesis. In this case the absorption is also independent of the incoming waveÕs intensity. Besides, the absorbed frequencies can be expressed as integer multiples of a certain basic frequency reminding of resonance phenomena. 3. ‘‘Quantized’’ cyclotron motion As the history of research development shows, the revealing of generation mechanisms in planetary magnetosphere radio sources is connected with the level of understanding of physical phenomena and the development of new concepts in modern radio physics. The most vivid example is the investigation of cyclotron maser processes and the subsequent discovery of similar processes in the nature of all magnetized planets [7]. When an electromagnetic wave interacts with resonators, the effect of ‘‘quantization’’ of all possible stationary stable oscillating amplitudes arises without the requirement of any specifically organized conditions (like the inhomogeneous action of external harmonic force). An electric charge, moving on a circular orbit in a homogeneous permanent magnetic field is here considered. When the charge is irradiated by a flat electromagnetic wave having a length commensurable with the orbitÕs radius, an effect of discretization (quantization) of the possible stable orbital radii (or motion velocities) is observed. A recurrence expression for the possible stable radius values (correspondingly, for the possible rotation speed values) is derived. It is shown that a threshold radius value exists such that for values above it arises a discretization of the possible stable radius values. A general stability investigation is carried out. Let us consider an electric charge q in magnetic field B and electric field E. The equation of motion in three-dimensional Euclidean space is

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

m

d2 r ¼ F ¼ qðE þ V  BÞ  2mbV; dt2

ð10Þ

where m is the mass of the moving charge q, V ¼ dr=dt is the velocity, b is a coefficient of dissipation. Considering Eq. (10) and assuming that the motion is put into effect in the plane z ¼ 0, we can write:   dVx  m dt ¼ qðEx þ Vy  B0 Þ  2mbVx ;   dVy m ¼ qðEy  Vx  B0 Þ  2mbVy : dt

ð11Þ

For constant magnetic field B ¼ ez B0 ¼ const: the cyclotron frequency x0 ¼ 

qB0 : m

ð12Þ

Taking into account Eq. (12), Eq. (11) take the form   dVx ¼  x0 E  x V  2bV ;  dt 0 y x B0 x   dVy x  ¼  B 0 Ey þ x0 Vx  2bVy : dt 0

ð13Þ

A solution, corresponding to rotation plus drift, is sought in the form x ¼ R cos W þ at;

y ¼ R sin W þ bt;

W ¼ xt þ u;

where R, u, a, b are constants in the stationary regime, x ¼ const. The values a and b can be found from the following equations:  x   0 hEx i  x0 b  2ba ¼ 0;  B0  x0   hEy i þ x0 a  2bb ¼ 0; B0 where the sign h i denotes averaging over time t. Integrating (13), we can write  R  Vx ¼  x0 Ex dt  x0 y  2bx þ const1 ;  B 0 R   Vy ¼  x0 Ey dt þ x0 x  2by þ const2 : B0 Neglecting da=dt and db=dt from (14) we obtain    Vx ¼ dx ¼ xR sin W þ dR cos W  du R sin W þ a; dt dt dt    Vy ¼ dy ¼ xR cos W þ dR sin W þ du R cos W þ b: dt dt dt The substitution of (17) into (16) gives   dR du x R  dt cos W  dt R sin W ¼  B 0 Ex dt  a  x0 bt  2bat þ ðx  x0 ÞR sin W  2bR cos W þ const1 ; 0  R  dR du 0  R cos W ¼  x sin W þ B0 Ey dt  b þ x0 at  2bbt  ðx  x0 ÞR cos W  2bR sin W þ const2 : dt dt

ð14Þ

ð15Þ

ð16Þ

ð17Þ

ð18Þ

Considering (18) it becomes clear how const1 and const2 should be determined for the constant part of Eqs. (16) to cancel. Considering also (15) we can write  R  dR du x  dt cos W  dt R sin W ¼  B 0 ðperiodical part of Ex dtÞ þ ðx  x0 ÞR sin W  2bR cos W; 0  ð19Þ R  dR du 0  R cos W ¼  x sin W þ B0 ðperiodical part of Ey dtÞ  ðx  x0 ÞR cos W  2bR sin W: dt dt From Eq. (19) we have   dR ¼ x0   cos Wðperiodical part of R E dtÞ  sin Wðperiodical part of R E dtÞ  2bR;  dt x y B0   R R  du 1 x0   ¼ R B sin Wðperiodical part of Ex dtÞ  cos Wðperiodical part of Ey dtÞ  ðx  x0 Þ: dt 0

ð20Þ

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

45

We consider a plane electromagnetic wave (i.e. Ek ¼ 0, where k is the wave vector, E  cosð~ mt  kx x  ky y  kz z þ aÞ). Let us assume that kx ¼ kz ¼ 0 and ky ¼ k. Then Ey ¼ Ez ¼ 0 and e 0 cos½ð~ Ex ¼ k E m  kbÞt  kR sin W þ a:

ð21Þ

e 0 ¼ E0 , Eq. (21) can be rewritten in the form Assuming that m ¼ m~  kb and k E Ex ¼ E0 cosðmt þ a  kR sin WÞ: We assume m ¼ Nx, N ¼ 1; 2; 3; . . . For the sake of solving Eqs. (20) and considering Eq. (22), we derive the following expansions:   Z cos W periodic part of cosðmt  kR sin W þ aÞdt ( J0 ðkRÞ ¼ finite part f sin ½ðN  1Þxt þ a  u þ sin ½ðN þ 1Þxt þ a þ ug 2N x  1 X sin ½ð2j þ N  1Þxt þ ð2j  1Þu þ a þ sin ½ð2j þ N þ 1Þxt þ ð2j þ 1Þu þ a þ J2j ðkRÞ 2ð2j þ NÞx j¼1  sin ½ð2j  N  1Þxt þ ð2j  1Þu  a þ sin ½ð2j  N þ 1Þxt þ ð2j þ 1Þu  a þ 2ð2j  N Þx  1 X sin ½ð2j  2  NÞxt þ ð2j  2Þu  a þ sin ½ð2j  N Þxt þ 2ju  a þ J2j1 ðkRÞ 2ð2j  1  NÞx j¼1 ) sin ½ð2j  2 þ NÞxt þ ð2j  2Þu þ a þ sin ½ð2j þ NÞxt þ 2ju þ a ;  2ð2j  1 þ N Þx   Z sin W periodical part of cosðmt  kR sin W þ aÞ dt ( J0 ðkRÞ ¼ finite part f cos ½ðN  1Þxt þ a  u  cos ½ðN þ 1Þxt þ a þ ug 2N x  1 X cos ½ð2j þ N  1Þxt þ ð2j  1Þu þ a  cos ½ð2j þ N þ 1Þxt þ ð2j þ 1Þu þ a þ J2j ðkRÞ 2ð2j þ N Þx j¼1  cos ½ð2j  N  1Þxt þ ð2j  1Þu  a  cos ½ð2j  N þ 1Þxt þ ð2j þ 1Þu þ a þ 2ð2j  N Þx  1 X cos ½ð2j  2  N Þxt þ ð2j  2Þu  a  cos ½ð2j  NÞxt þ 2ju  a þ J2j1 ðkRÞ 2ð2j  1  N Þx j¼1 ) cos ½ð2j  2 þ N Þxt þ ð2j  2Þu þ a  cos ½ð2j þ N Þxt þ 2ju þ a ;  2ð2j  1 þ NÞx where J ð Þ are Bessel functions of the first kind. Using (20) we can write the shortened (averaged) equations:   Z     dR x0  E cos W periodic part of cosðmt  kR sin W þ aÞ dt  2bR; ¼   dt B0         Z  du 1 x0 E0  sin W periodic part of cosðmt  kR sin W þ aÞ dt  ðx  x0 Þ:  dt ¼ R B 0 From (23)–(25) we obtain  Z    1  cos W periodic part of cosðmt  kR sin W þ aÞ dt ¼  JN0 ðkRÞ sinðN u  aÞ;  x   Z   N  sin W periodic part of JN ðkRÞ cosðN u  aÞ; cosðmt  kR sin W þ aÞ dt ¼  xkR where JN0 ð Þ is the first derivative of the Bessel function of the first kind.

ð22Þ

ð23Þ

ð24Þ

ð25Þ

ð26Þ

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

Taking into account (26), Eqs. (25) can be rewritten as D E  dR ¼ f ðR; uÞ;  dt D  du E  ¼ gðR; uÞ; dt where

 x0 E 0 0   f ðR; uÞ ¼ J ðkRÞ sinðN u  aÞ  2bR;  x B0 N  x0 E0 N  JN ðkRÞ sinðN u  aÞ  ðx  x0 Þ:  gðR; uÞ ¼ x B0 kR2

The stationary solution corresponds to the conditions     dR du ¼ 0; ¼ 0: dt dt In order to analyze the system stability we produce the following variations:   dR of of d  dt ¼ oR dR þ ou du;   du og og d  dt ¼ oR dR þ ou du:

ð27Þ

ð28Þ

ð29Þ

ð30Þ

Using fR , fu , gR and gu to denote the derivatives of =oR, of =ou, og=oR and og=ou in Eqs. (30) for constant (stationary) values of R and u, corresponding to the steady-state oscillations, the stability condition can be written as Reðk1;2 Þ < 0; where k1;2

fR þ gu  ¼ 2

ð31Þ ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 fR  gu þ fu gR ; 2

ð32Þ

since the time dependence of the small deviation of R and u from their steady-state values is governed by the equations dR ¼ A1 ek1 t þ A2 ek2 t and du ¼ B1 ek1 t þ B2 ek2 t , where A1 , A2 , B1 and B2 are constants. Considering Eq. (32) the condition (31) can be rewritten as fR þ gu < 0; fR gu  fu gR > 0: The partial derivatives can be expressed as  2

   x E 1 N  fR ¼ 0 0 k  JN0 ðkRÞ þ  1 J ðkRÞ ; N  x B0 k 2 R2 kR   x E  fu ¼ 0 0 NJN0 ðkRÞ cosðN u  aÞ;  x B0   2    gR ¼ x0 E0 Nk  2 JN ðkRÞ þ J 0 ðkRÞ cosðNu  aÞ; N  2 2 kR x B0 k R   x0 E0 N 2  k JN ðkRÞ sinðN u  aÞ:  gu ¼  x B0 k 2 R2 Considering Eqs. (29) and (28), Eqs. (34) become  2

  fR ¼ 4b þ x0 E0 k N  1 JN ðkRÞ sinðN u  aÞ;  x B0 k 2 R2   x E 0 0 0  fu ¼ NJ ðkRÞ cosðN u  aÞ;  x B0 N    gR ¼  2 ðx  x0 Þ þ x0 E0 Nk 2 1 J 0 ðkRÞ cosðN u  aÞ;  x B0 R k 2 R2 N  2   g ¼  x0 E0 k N J ðkRÞ sinðN u  aÞ: N  u x B0 k 2 R2

ð33aÞ ð33bÞ

ð34Þ

ð35Þ

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47

Combining, from (35) we can write fR þ gu ¼ 4b  F0 JN ðqÞ sin c;    

2 N2 N2 N 0 2N 2 02 fR gu  fu gR ¼ F0 2 1  2 JN ðqÞ  JN ðqÞ þ F0 4b 2 JN ðqÞ sin c þ 2ðx  x0 Þ JN ðqÞ cos c q q q q þ ðN 2  q2 Þðx  x0 Þ2 þ 4b2 N 2 ;

ð36Þ

ð37Þ

where the following designations are introduced: x0 E0 k ¼ F0 ; x B0

kR ¼ q;

Nu  a ¼ c:

First we consider the case of small amplitudes, i.e. jqj 1. In this case we can use the following assymptotical expressions for the Bessel functions: 1  q N X ð  1Þl  q 2l 1  q N JN ðqÞ ¼ ffi þ ; ð38Þ 2 N! 2 l!ðN þ lÞ! 2 l¼0 JN0 ðqÞ ¼

 q N 1 1 þ 2ðN  1Þ! 2

ð39Þ

From (36), (28) and (29) we find F0 JN0 ðqÞ sin c ¼ 2bq; N F0 2 JN ðqÞ cos c ¼ x  x0 : q Substituting (38) and (39) into (40a) and (40b) we determine    tg c ffi 2b ;  x  x0  i1=2  N 2 2N ðN  1Þ! h  jqj ffi : ð2bÞ2 þ ðx  x0 Þ2  jF0 j

ð40aÞ ð40bÞ

ð41Þ

From (41) it is evident that the spectrum of the possible amplitudes is uninterrupted and in this case there are no conditions for the discretization of the amplitude. When jqj 1 and N  1 from (36) and (37) we find fR þ gu ¼ 4b < 0;

ð42Þ 2

fR gu  fu gR ffi N 2 ðx  x0 Þ þ 4b2 N 2 > 0;

ð43Þ

i.e. the conditions (33a) and (33b) are satisfied and in this case the system motion is stable. Let us now consider the resonance case, which means x  x0 ffi 0;

ð44Þ

or, considering Eq. (40b), this is equivalent to jF0 j

N  jx  x0 j: q2

ð45Þ

Two possibilities follow from Eq. (40b): (i) JN ðqÞ ¼ 0 and cos c 6¼ 0 or (ii) cos c ¼ 0. We show that when the amplitudes are large (q  1) the motion in the case (i) is unstable as long as in the case (ii) it is stable. Case (i) JN ðqÞ ¼ 0

and

cos c 6¼ 0:

ð46Þ

Then JN0 ðqÞ 6¼ 0. In Eq. (37) we neglect (x  x0 ) and JN ðqÞ in correspondence with (44) and (46). We find: fR gu  fu gR ffi ðF02 N 2 =2ÞJN02 ðqÞ þ 4b2 N 2 . However from (40b) it follows

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

F0

i:e: fR gu  fu gR ffi 4b2 N 2 1 

JN0 ðqÞ 2b ¼ ; q sin c

1 sin2 c

<0

ð47Þ

and apparently in this case the motion is unstable. Case (ii) cos c ¼ 0;

ð48Þ

or sin c ¼ ð1Þm ;

m ¼ 0; 1

ð49Þ

(the two cases are possible, e.g. with adding p to c). As here b is not of essential significance, for the sake of simplicity we let b ! 0:

ð50Þ

From Eq. (40a) it follows JN0 ðqÞ ¼ 0:

ð51Þ

The condition (51) determines the possible discrete spectrum of amplitudes q. These amplitudes do not depend on the force E0 (or F0 ). Taking into account Eqs. (44), (48), (50) and (51) from Eq. (37) we find

N2 N2 ð52Þ fR gu  fu gR ffi F02 2 1  2 JN2 ðqÞ: q q When ð53Þ

q>N from (52) it follows that the stability condition (33b) is satisfied: fR gu  fu gR > 0. From (36) and (49) we obtain fR þ gu ffi 4b  ð1Þm F0 JN ðqÞ: Selecting the values for m, it is possible to satisfy also the second stability condition (33a): fR þ gu < 0, or F0 sin cJN ðqÞ > 0:

ð54Þ

4. The wave nature and dynamical quantization of the solar system The commensurability and resonance phenomena of the solar system motion structure––(including planets, asteroids, planetÕs satellites)––have been an object of detailed discussions and experimental examination in the last years [7– 9]. Systematic observations and measurements have been carried out using all available means. A considerable bulk of empiric data for the solar system has been obtained from the ‘‘Voyager 2’’ mission. The understanding of the inevitable resonance character of evolving mature oscillation systems leads to series of interesting concepts about the resonance character of the Solar system motion dynamics. One of the most outstanding among them is MolchanovÕs hypothesis [7] about the complete resonance character of the large planets in-orbit motion. Molchanov has noticed that the mean motion of the nine large planets n1 ; . . . ; n9 are related approximately by nine linear homogeneous equations ðjÞ

ðjÞ

k1 n1 þ þ k9 n9 ¼ 0;

j ¼ 1; . . . ; 9;

ðjÞ ðjÞ k1 ; . . . ; k9 .

with integer coefficients The mean motions of the Jovian, Saturn and Uranian satellites are related to similar equations. If asteroids are considered as planets and excluding Pluto, it is established [9] that the planetary distances obey the following regularity: akþ1 ffi 1:75  0:20; ak i.e. the relation of the semi-major orbital axes of neighbouring planets is almost constant.

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

49

Up to now, whole series of phenomena in the Solar system structure are considered anomalous or obscure. For example, the revolution periods of the celestial bodies vary within comparatively narrow range of values even if their masses differ considerably. Those and other phenomena do not fit properly in the pattern of classical celestial mechanics and are actually ignored due to the absence of traditional explanation. The development of astronautics and modern astrophysics calls for: (i) an understanding of the anomaliesÕ origin; (ii) filling up the gaps in the existing theoretical models; (iii) clarification of certain phenomena, such as the nature of stability of commensurability, periodicity and resonances and their probable preferability, having in mind that the above-mentioned phenomena are observed not only among celestial bodies, but also among artificial satellites, space ferries and orbital stations. Practically all contemporary scientists come to the conclusion that the new picture of the Universe is based on ‘‘closing down the Nature ring’’ that leads to close interrelations between the microcosm physics and astrophysics. The general data on near-the-Earth, deep space and especially Solar system structure cannot be thoroughly interpreted without a general notion of the wave nature of occurring phenomena. Since 1982, a great interest has arisen in the quantization problem––(in the large) and the Universe megasystem wave structure [9–12]. The concept and the corresponding idea of the megaquantum wave structure of astronomic system has been considered. According to the wave Universe concept, the large astronomic systems are considered as wave dynamic systems that are, in a certain sense, analogous to the atomic system [11]. From that point of view the Solar system is considered as a wave dynamical system. Its components––the celestial bodies (the Sun, the planets, the satellites and the small celestial bodies) and the interplanetary continuum (interplanetary plasma, electromagnetic field, etc.) are described within a common dynamic substance––field context. The phenomenologic and dynamic description of such systems is connected with the megaquantum wave dynamics. Assuming that the Solar system is a wave dynamic system and hence, the micro–mega-analogy (MM-analogy) is valid, series of deductions have been drawn on the basis of the dynamic isomorphism of the atomic and the solar system structures [9–12]. Different quantization versions have been used––according to Bohr, De Broglie, Sommerfeld, Scr€ odinger, etc. The essence of all those fundamental works is the substantiation of the existence of megawaves, realizing short-range interactions in a scale, commensurable with the systems scale in any Universal megasystem and, in particular, in the Solar system, representing an analog to De BroglieÕs waves for gigantic astronomic systems. However, the common opinion is that the major problem at present is to make out a more detailed identification, phenomenologic and dynamic description of the solar system megaquantum structure and of the analogous systems, constituting of a central body and satellites [9]. The above-mentioned concepts of the bodiesÕ waves interaction unity in the micro- and macroworld as well as of the basis realization and manifestation behaviour closing down in Nature is the essence and grounds for the presented work. For the purpose of commensurability, resonance and dynamic quantization studies of the Solar system bodies motion structure, different authors use versions of the quantization phenomena within the frames of the Quantum Mechanics. Herewith, a heuristic model is proposed of the discrete distribution of Solar system planets and satellites mean distances from the primaries. That megaquantum resonance-wave model is based on the Solar system structure wave nature concept, as well as on the oscillator amplitude quantization phenomenon, occurring under wave action, discovered on the basis of the classical oscillations theory [10]. The analysis of the latter explains and gives analytic grounds for: (i) the existence of a discrete set of stable (allowed) orbits and a set of resonant, but unstable (forbidden) orbits of the planets and their satellites; (ii) the phase locking of the in-orbit motion; (iii) the trajectory stability only under exceptional initial conditions and in strictly defined areas, and (iv) the independence of the stable stationary orbits from the planetsÕ and satellitesÕ masses. As a rule, the harmonic oscillator is taken as the basis of the field theory interpretations [11] from point of view of the classical theory of oscillations and the quantum-mechanical approach. The classical approach is of greater heuristic value and efficiency when it comes to the action of fully determined (in the classical sense) forces at an oscillator. This fact, noted by other researchers, is confirmed by the analysis, presented below. Herewith, as a general model we take a periodical motion, that is the rotation of a charge at which an electromagnetic wave falls along the X -axis. 1

1 We draw intentionally a simple consideration, so as to outline more vividly the heuristicity and perspectiveness of the presented approach.

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The charge motion equation is €r þ 2b_r þ x20 r ¼ eEsinðmt  krÞ;

ð55Þ

where r is the charge radius-vector; 2b is the dissipation coefficient; e is the charge value; k is the wave vector; E is the wave field intensity and m, t are the frequency and time parameters, respectively. Plotting the equation along the X -axis, we obtain €x þ 2b_x þ x20 x ¼ eEx sinðmt  kxÞ;

ð56Þ

where Ex is the X -axis electrical field component and k is the wave modulus. The solution of Eq. (56), describing a linear oscillator under wave action, is written in the form of a quasiharmonic function xðtÞ ¼ a sinðxt þ aÞ;

ð57Þ

where aðtÞ and aðtÞ are the slowly changing amplitude and phase, x ¼ m=N is the charge periodic motion frequency, and N ¼ 1; 2; 3; . . . is a whole number. By use of the averaging method [6], shortened, these equations are  eE N   a_ ¼ ba þ x JN ðkaÞ sin N a;  kax  x20  x2 eEx 0   J ðkaÞ cos N a;  a_ ¼ 2x ax N

ð58Þ

where JN is the N -order Bessel function of the first kind. In charge stationary periodic motion mode, i.e. when a_ ¼ 0 and a_ ¼ 0, Eqs. (58) can be written as ba ¼

eEx N JN ðkaÞ sin Na; kax

x20  x2 ¼

ð59Þ

2eEx 0 J ðkaÞ cos N a: a N

ð60Þ

According to the amplitude quantization phenomenon theory presented above, the role of the phase a is essential whence the excited oscillation synchronization and adaptive stability maintenance under different perturbing actions are considered. In resonance mode, 2 condition x ¼ x0 is accomplished and Eq. (60) is valid provided that JN0 ðkaÞ ¼ 0 and

cos N a ¼ 0:

In order to examine the stationary amplitude and the phase stability, the set of Eqs. (58) is rewritten, as follows: a_ ¼ f ða; aÞ;

a_ ¼ gða; aÞ:

ð61Þ

The derivatives of =oa, of =oa and og=oa, og=oa obtained for a and a constant values and corresponding to the stationary oscillations are designated by fa , fa and ga , ga . Hence, the stability condition can be rewritten as follows:

Re P1;2 < 0;

where P1;2

fa þ ga  ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s

2 fa  ga þ fa ga ; 2

  fa þ ga b eEx N ¼  sin N a JN ðkaÞ  2kaJN0 ðkaÞ ; 2 2 2 2kxa fa  ga b eEx N ¼  sin N aJN ðkaÞ; 2 2 2kxa2 2   ðeEx N Þ cos3 NaJN ðkaÞ JN0 ðkaÞ  kaJN00 ðkaÞ : fa ga ¼ kx2 a3

2

The resonance case is in conformity with the idea about evolutionary mature systems [9].

ð62Þ

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

51

The P1;2 quantities are derived from the time dependence small deviations of the stationary values of a and a, expressed as Da ¼ A1 eP1 t þ A2 eP2 t ;

Da ¼ B1 eP1 t þ B2 eP2 t ;

where A1 , A2 , B1 and B2 are constants. The examination of the stability solutions (Eqs. (58)) shows, that the stable oscillations amplitude a satisfies condition JN0 ðkaÞ ¼ 0:

ð63Þ

Hence, the stationary charge oscillations can be realized for amplitudes ai , belonging to a strictly defined set of amplitude values. The ai values are determined by the Bessel function extremes, and using Eq. (63) may be presented as kai ¼ jN ;i ;

ð64Þ

where jN ;i is the N -th order Bessel function JN argument value kai at the i-th extremum point. In conformity with the general premises we assume, that the oscillating charge system under wave action may be used as a general model of description of the Solar system. The large Earth orbit semi-axis aE complies with Eq. (63) at N ¼ 8 and i ¼ 5. Those conditions are assumed as basic: kaE ¼ j8;5 . Further, using expression (64), the following semi-major axes relation is obtained: ai kai jN ;i j8;i ¼ ¼ ¼ : ð65Þ aE kaE j8;5 j8;5 At the same time it should be noted that the solar system planets arrangement bears regular character, expressed on the whole by the Titius–Bode law [13], giving the mean planets-to-Sun distances ak by equation ak ¼ 0; 4 þ 0; 3:2k ;

ð66Þ

where ak is expressed in astronomical units, and k ¼ 1 for Mercury, k ¼ 0 for Venus, k ¼ 1 for the Earth, etc. For more than 200 years this law has not found a convincing theoretical explanation. A better expression of the mean distances is proposed by several authors [7,12] in the form as follows rn ¼ r0 d n ;

ð67Þ

where n is the n-th planet distance from the Sun or n-th satellite distance from the primary, r0 is a simple normalizing distance, different for each system, d is a constant, characterizing the geometric progression, valid for all systems. The integer n for Mercury is equal to 1, for Venus is equal to 2, etc. Some of the values do not correspond to an existing celestial body, but to a ‘‘hole’’. Expression (67), compared to the earlier Titius–Bode law (66) is less artificial, as the discount value n ¼ 1 for Mercury is avoided. The Solar system planets mean distances are presented in Table 1. For comparison reasons, the direct astronomic measurements data is given parallel to the result, computed by the classical Titius–Bode law (66) and Eq. (65) according to the oscillator–wave model, described above. All data is expressed in astronomical units (A.U.). Table 1 Mean planet distances in the Solar system Planets in the Solar system

Mercury Venus Earth Mars Asteroids Jupiter Saturn Chiron (CollewllÕs objects) Uranus Neptune Pluto

Titius–Bode Law (66), (Nieto, 1972) A.U.

Computed planet distances using Eq. (65) of the oscillator–wave model

k

ak

i

a=j

0.39 0.72 1.00 1.52 2.78 5.2 9.55 13.71

)8 0 1 2 3 4 5

0.4 0.7 1.0 1.6 2.8 5.2 10.0

1 3 5 9 19 37 71 104

0.392 0.723 1.000 1.530 2.824 5.132 9.474 13.689

19.18 30.03 39.67

6 7 8

19.6 38.8 77.2

147 232 307

19.180 30.035 39.598

Data from direct astronomical measurements of planet distances from the Sun (Allen, 1973) A.U.

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A good correspondence is observed between the computed (Eq. (65)) and astronomically measured radii. Especially significant is the correspondence between the computed and measured radii of Neptune and Pluto. The Titius–Bode law determines the mean distances of those two planets with an error of 23% and 49%, respectively. The computed data of the mean satellite distances from Saturn, Uranus and Jupiter, as well as the mean ring system distances from Saturn, are given in Table 2. The calculations are made on the basis of the oscillator–wave model, using an expression, similar to Eq. (65). For comparison reason, the data obtained from direct astronomical distance measurements [14] are presented as well. Again, a good correspondence is seen between the calculated and measured mean distances. The measurement and computed data normalizing is realized relatively to an arbitrary chosen representative body. Obviously, there are no restriction and any other satellite can be chosen as a representative body for the performance of similar normalizing. It is necessary to note, that according to the oscillator–wave model, there is a certain differentiation between the Solar system (Table 1) and planet–satellite (Table 2) structures. In the first case, the wave action frequency m and Solar system planets revolution frequencies xi relation remains invariable (N ¼ c=xi ¼ 8 ¼ const:). The differences between

Table 2 Mean Saturn, Uranian, Jovian satellites and Saturn-ring-system distances from the primaries in conformity with the proposed oscillator–wave model Satellites

Saturnian satellites Janus Mimas Enceladus Tethys Dione Rhea Titan Hyperion Iapetus Phoebe Uranian satellites Miranda Ariel Umbriel Titania Oberon Jovian satellites Amalthea Io Europa Ganymede Callisto Saturn ring system Crape or C ring: Faint Gap: Dark Main B ring: Very bright Cassini division: Dark Outer A ring: Moderately bright

Data from direct astronomical measurements of satellite mean distances from the primaries [14] 103 A.U.

Normalized data from direct astronomical measurements

N

1.060 1.241 1.592 1.970 2.523 3.524 8.166 9.911 23.718 86.580

0.301 0.352 0.452 0.550 0.716 1.000 2.317 2.812 6.736 24.569

2 3 4 5 7 10 25 31 77 284

jN ;1 =j10;1 0.308 0.357 0.452 0.545 0.729 1.000 2.338 2.838 6.737 24.560

0.872 1.282 1.786 2.930 3.919

0.488 0.718 1.000 1.641 2.104

4 7 10 17 23

jN ;1 =j10;1 0.452 0.728 1.000 1.624 2.152

1.209 2.819 4.489 7.155 12.585

0.096 0.224 0.357 0.568 1.000

1 3 5 9 17

jN ;1 =j17;1 0.096 0.220 0.336 0.563 1.000

Normalized computed mean satellite distances from the primaries in conformity with the oscillator–wave model

jN ;1 =j14;1 0.498 0.595

0.827 0.988

11 13

0.803 0.934

0.602

1.000

14

1.000

0.782

1.299

18

1.261

0.802

1.332

19

1.326

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

53

the planet mean distances from the Sun are determined by the values of the Bessel function arguments j8;i at the extreme i-th points. In the second case, i.e., for planet satellite systems, the correspondence between the calculation and astronomical measurements data is observed different frequency multiplicities (N ¼ m=xi ¼ vary) and the corresponding values of the first Bessel extremes jN ;i . 5. Approach for large amplitudes of the oscillations in a non-linear dynamic system subjected to the influence of a wave Let the non-linear oscillator is an electric charge q with mass m and it is able to oscillate along the X -axis with a small friction force 2d0 X_ . Let an electromagnetic wave propagating along the X -axis acts upon the oscillating charge. Let us assume that the wave has a longitudinal component of the electric field EX . The equation of the charge motion becomes X€ þ 2d0 X_ þ x20 sin X ¼ P0 sinðmtr  kX  uÞ;

ð68Þ

where x0 is the resonant frequency of small amplitude oscillations P0 ¼ EX q=m; m; u and k are the frequency, the initial phase and the wave number respectively, tr is the real time. We will assume m  x0 . Let us introduce the dimensionless time t ¼ x0 tr . In this case, Eq. (68) takes the form 3

m t  kX  u ; ð69Þ X€ þ 2dd x_ þ sin X ¼ F0 sin x0 where 2dd ¼ 2d0 =x0 , F0 ¼ P0 =x20 . In order to integrate Eq. (69) with the methods of the theory of non-linear oscillations, we apply the approach developed in [3,10]. We introduce a new variable y and non-linear time s, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z x x ð70Þ sin x0 dx0 ¼ 2 sin ; y ¼ sign x 2 2 0 dt dX y ¼ ¼ ¼ GðyÞ: ds dy sin½X ðyÞ

ð71Þ

So, the non-linear reactive term sin X in Eq. (69) may be ‘‘excluded’’. The functions X ðyÞ and GðyÞ in (71) are easily expressed by taking into account (70) in the form X ðyÞ ¼ 2 arcsin

y ; 2

1 GðyÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi : 2 1  y4

ð72Þ

Substituting (70) and (71) in (69) we obtain d2 y þy ¼ ds2

  2dd

  dy m tðsÞ  kX ðyÞ  u GðyÞ: þ F0 sin ds x0

ð73Þ

The further consideration will be performed for the following interval of y values: 2 < y < 2. Before the integration of Eq. (73) we will mention, that the solution will be quasiharmonic with nominal frequency xn ¼ m=N , where N  1 is a positive odd number, however xn  x0 . That is why we will write Eq. (73) in the following form:   d2 y dy m 2 þ b y ¼ 2d sin tðsÞ  kX ðyÞ  u GðyÞ þ ðb2  1Þy; ð74Þ þ F d 0 ds2 ds x0 where b  1 corresponds to the difference from the resonant frequency.

3 Similar equations describe the behaviour of cosmic charged particles in certain conditions, the processes in radio-frequency driven, quantum-mechanical Josephson junctions, charge density wave transport and other systems.

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

We assume that in excitation of the charge oscillations by the wave its motion is symmetric with respect to the equilibrium and the charge coordinate changes in agreement with y ¼ R cos bs ¼ R cos w:

ð75Þ

The dependence of the normalized time t on the angle w can be expressed in agreement with (71), (72) and (75) as Z 1 w dW qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : t¼ 2 b 0 1  R4 cos2 W The normalized period of the oscillations is T0 ¼

1 b

Z

2p

0

dW 4 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ K ; b 2 R2 1  4 cos2 W

ð76Þ

where KðR=2Þ is a complete elliptic integral of first kind. By use of (76) the coefficient b is expressed in the form b¼

2mKðR2 Þ : px0 N

Now we can solve Eq. (74). The shortened (reduced) differential equations for the amplitude R and phase u may be written as dR 1 ¼ ds 2pb du 1 ¼ ds 2pb

Z

2p

L½R cos w;  bR sin w; eðw  uÞ sin w dw;

ð77Þ

L½ R cos w;  bR sin w; eðw  uÞ cos w dw;

ð78Þ

0

Z

2p 0

where   F0 pN : L½R cos w; bR sin w; eðw  uÞ ¼ 2dd bR sin w þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2 2KðR=2Þ 1  R4 cos2 w Let us introduce the following designations:     Z KðR=2Þ H1 sn Z dZ; ¼ sinðrZÞ cos ½ DðZÞ cn Z H2 0   Z KðR=2Þ   H3 sn Z ¼ cosðrZÞ sin ½ DðZÞ dZ; H4 cn Z 0     Z KðR=2Þ sn Z H5 dZ; ¼ cosðrZÞ cos ½DðZÞ cn Z H6 0   Z KðR=2Þ   H7 sn Z ¼ sinðrZÞ sin ½ DðZÞ dZ; H8 cn Z 0

ð79Þ ð80Þ ð81Þ ð82Þ

where Z ¼ F ðw; R=2Þ is an incomplete elliptic integral of the first kind, 

 R 2 DðZÞ ¼ 2kE arcsin cn Z ; ; 2 R E½ ;  is an incomplete elliptic integral of second kind, sn Z and cn Z are sine and cosine of the amplitude (the Jacobi elliptic functions), r¼

pN : 2KðR=2Þ

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

55

Taking into account the expressions (79)–(82), the shortened equations (77) and (78) for establishing the amplitude R and phase u take the form dR F0 ½ðH1  H3 Þ cos u  ðH5 þ H7 Þ sin u; ¼ dd R  2pb ds du F0 ¼ ½ðH2  H4 Þ cos u  ðH6 þ H8 Þ sin u  ðb  1Þ: ds 2pbR For the stationary mode (dR=ds ¼ 0, du=ds ¼ 0) we obtain the following expressions for the established values of the amplitude R and the phase u: R¼

F0 ½ðH1  H3 ÞðH6  H8 Þ  ðH2  H4 ÞðH5 þ H7 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pbd ½rðH1  H3 Þ  ðH2  H4 Þ2 þ ½rðH5 þ H7 Þ  ðH6 þ H8 Þ2

u ¼ arctg

rðH1  H3 Þ  ðH2  H4 Þ ; rðH5 þ H7 Þ  ðH6 þ H8 Þ

ð83Þ

ð84Þ

where r ¼ ðb  1Þ=dd . The dependences (83) and (84) have been investigated by numerical methods for the following values of the parameters: dd ¼ 0:01, x0 ¼ 3:14, N ¼ 111, F0 ¼ vary. In Fig. 1 the dependence of the stable roots of Eq. (83) on the amplitude of the external force F0 is presented. It is easy to see the features of the system: (a) in the case of constant external action, stationary oscillations with possible discrete set of stable amplitudes are provided; (b) practical independence of the amplitudes of stationary oscillations on the changes of the values of the external wave amplitude in wide region over some threshold value. Thus a specific effect of quantization of the amplitudes is revealed. The possibility for a given amplitude of the discrete set to be realized is determined by the initial conditions. The mechanism of settling and maintenance of the stationary oscillations is connected with the catch and adaptive automatic tuning of the phase u, determining the interaction and the contribution of the energy from the external high frequency wave source. This mechanism is illustrated in Figs. 2 and 3. In Fig. 2 the values of phase u are represented for a stationary set of stable amplitudes R1 ; R2 ; R3 ; . . ., which are calculated by the formula (84) in case of constant amplitude of the external wave F0 ¼ 20. For each stationary amplitude, which is determined by the initial conditions, corresponds a portion of energy, depending on the initial phase u. Therefore it is possible to find a correspondence between the set of values of the stationary amplitude R1 ; R2 ; R3 ; . . . and the fixed discrete set of stationary phases u1 ; u2 ; u3 ; . . . for each value of the amplitude of the external wave F0 ¼ const: In case of changing of F0 the charge oscillations do not change due to the adaptive tuning of the phase u. This is illustrated in Fig. 3, where the change of the phase u and the corresponding change of the amplitude of the external wave F0 are shown when the value of the settled oscillations amplitude is constant R ¼ 0:756.

Fig. 1. Dependence of possible stable amplitudes on intensity action force (Eq. (83)).

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

Fig. 2. Correspondence between stable oscillation amplitudes and stationary initial phase under unchangeable intensity of external acting wave (Eq. (84)).

Fig. 3. Dependence of stationary initial phase on acting wave intensity alternation under unchangeable stable oscillations amplitude.

6. General conditions for transition to irregular behaviour in an oscillator under wave action The model system under consideration is presented by the following system of equations   x_ 1 ¼ x2 ;   x_ 2 ¼  sin x1 þ l½F0 sinðh  qx1 Þ  dd x2 ;   h_ ¼ m;

ð85Þ

where ½x1 ; x2 hðtÞ 2 R3 , the dot denotes an operation of differentiation by the time t, F0 and m are correspondingly the amplitude and the frequency parameter of the external acting wave, dd reflects the dissipation in the system, 0 l 1 is a small parameter. Proceeding from the non-perturbed system (l ¼ 0),   x_ 1 ¼ x2 ;   x_ 2 ¼  sin x1 ; i.e. from the system with Hamiltonian function H ¼ ðx22 =2Þ þ 1  cos x1 , the phase space divides into two domains and a separating boundary (separatrix), where qualitatively different phenomena occur: I domain, 0 < H < 2; II separatrix, H ¼ 2 and III domain (a regime of rotation), H > 2.

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

57

The solution in the so outlined domains can be presented in the form as follows. I domain, notating e^ ¼ H=2:   x1 ¼ 2 arcsin½j snðt  t0 ; jÞ þ 2pN ; N ¼ 0; 1; 2; 3; . . . ;   x2 ¼ 2j cnðt  t0 ; jÞ; t0 ¼ const: and as well   x1 ¼ 2 arcsin½j snðt  t0 ; jÞ þ 2pN ;   x2 ¼ 2j cnðt  t0 ; jÞ; II separatrix:   x1 ¼ 2 arcsin½thðt  t0 Þ þ 2pN ;   2  x2 ¼ ;  chðt  t0 Þ and as well   x1 ¼ 2 arcsin½thðt  t0 Þ þ 2pN ;   2  x2 ¼  ;  chðt  t0 Þ III domain (a regime with rotation), notating j2 ¼ 2=H :  h t  t i  0  x1 ¼ 2 arcsin sn ; j þ 2pN;   j   x ¼ 2 dn t  t0 ; j ;  2 j j and as well  h t  t i  0  x1 ¼ 2 arcsin sn ; j þ 2pN;  j     x ¼  2 dn t  t0 ; j :  2 j j Let us consider the system dynamics in the domain of the separatrix (the homoclinic trajectory). We work with the solution 0 1 2 arcsin½thðt  t0 Þ x01 A: 2 ¼@ x0 ¼ x02 chðt  t0 Þ According to [16] we can write





d dx1 =dt 0 x2 x ¼ f0 ðxÞ þ lf1 ðx; tÞ  þl ; F0 sinðh  qx1 Þ  dd x2 dx2 =dt  sin x1 dt where in the right-hand side of the equation the initial approximation stands x1 x01 for x ¼ : x0 ¼ x02 x2 Hence



f0  f1  f0 ^ f1 ¼ ¼

x02  sin x01

^

0 F0 sinðh  qx01 Þ  dd x02

2F0 4dd sin fh  2q arcsin½thðt  t0 Þg  2 : chðt  t0 Þ ch ðt  t0 Þ

For the system under consideration the Malnikov distance ([15]) can be expressed: ) Z 1 Z 1( 2F0 4dd sin ½h  2q arcsin ðthðt  t0 ÞÞ þ 2 dt: f0 ^ f1 dt ¼ Dðt0 ; t0 Þ ¼  chðt  t0 Þ ch ðt  t0 Þ 1 1

ð86Þ

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V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

Irregular (chaotic) behaviour occurs for the areas where Dðt0 ; t0 Þ passes through zero. For the practice the case of q ¼ 1 is the most interesting. Substituting h as h ¼ mt þ h0 ¼ mðt  t0 Þ þ ðmt0 þ h0 Þ, Eq. (86) becomes Z 1( 2F0 2F0 Dðt0 ; t0 Þ ¼ sin½mðt  t0 Þ þ ðmt0 þ h0 Þ½1  2th2 ðt  t0 Þ þ cos½mðt  t0 Þ  chðt  t0 Þ chðt  t0 Þ 1 ) 2shðt  t0 Þ 4dd þ dt þ ðmt0 þ h0 Þ 2 ch ðt  t0 Þ ch2 ðt  t0 Þ Z 1 Z 1 Z 1 ð1  sh2 sÞ sh s ds 2 ds  2F ds þ 4d : ¼ 2F0 sinðmt0 þ h0 Þ cos ms sinðmt þ h Þ sin ms 0 0 0 d 2 ch3 s ch3 s 1 1 1 ch s ð87Þ The integrals in (87) are evaluated as follows: 1

Z 1 Z 1 Z 1 ð1  sh2 sÞ sh s sh s  sh s ds ¼ ¼ cos ms 2   m cos ms cos ms d sin ms 2 ds 3 2 s s s s ch ch ch ch 1 1 1 1 1 Z 1 Z 1 2  dch s sin ms  dðsh msÞ mp sin ms 2 ¼  m þm ð88Þ ¼m ¼ mp ; ch s 1 ch s ch 2 ch s 1 1 1

Z Z Z Z 1 sh s 1 1 1 1 sin ms  1 1 dðsin mpÞ m 1 ch ms ds m2 p ¼  ¼ ¼ sin ms 3 ds ¼  sin ms d þ mp ; 2 2  2 2 2 2 2 2 2sh ch s ch s ch s 1 ch s 1 1 1 1 ch s 2 ð89Þ Z 1 ds ¼ 2: ð90Þ 2 s ch 1 According to expressions (88)–(90), Eq. (87) is expressed by

1 1 þ sinðmt0 þ h0 Þ þ 8dd ; Dðt0 ; t0 Þ ¼ 2pm2 F0 ch mp2 sh mp2 or finally Dðt0 ; t0 Þ ¼ 

4pm2 F0 emðp=2Þ sinðmt0 þ h0 Þ þ 8dd : sh mp

Under the condition Dðt0 ; t0 Þ ¼ 0 and taking into account that jsinðmt0 þ h0 Þj < 1 and dd > 0, the general condition for transmission to irregular (chaotic) behaviour in non-linear oscillator under the wave action takes the form 2dd shðjmjpÞ < pm2 jF0 jemðp=2Þ : Obviously this condition is fulfilled in some domains of the space ðdd ; F0 ; mÞ. It is necessary to note that the solution of Eq. (85) strongly depends on the choice of initial conditions. Besides, the homoclinic bifurcation is one of the first bifurcations that occur in the transition from regular to irregular motion in the system under consideration. The latter is related with the condition for overcoming the strong self-adaptive mechanism of the system internal stability [1–3]. It is necessary also to emphasis that the homoclinic tangency implies formation of a very complicated fractal boundary for the basins of attraction [1,4].

7. Conclusion The analysis shows the following two essential features of the system considered. 1. There exists a discrete set of possible stationary stable amplitudes, which can be approximately determined using Eq. (51) under the conditions expressed by Eqs. (45), (48), (53) and (54). 2. There exists a threshold value for the amplitude determined by the condition (53), such that for values above it the discrete states are stable.

V. Damgov et al. / Chaos, Solitons and Fractals 17 (2003) 41–60

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The phenomenon of continuous oscillation excitation with an amplitude belonging to a discrete set of stationary amplitudes has been demonstrated on the basis of a common model––an oscillator under wave influence. It is shown that the conditions necessary for the manifestation of this phenomenon are realized in a natural way in an oscillator system interacting with a continuous electromagnetic wave. Modeling the system of an oscillating charge under wave influence has been considered. It has been shown that the continuous wave with spectral components, considerably higher than the oscillating chargeÕs natural frequency, excites charge oscillations with a quasinatural frequency and an amplitude belonging to a discrete set of the possible stationary amplitudes, depending only on the initial conditions. The considered model may be used for phenomenological investigation of plasma particles with electromagnetic waves interactions and waves in the Earth ionosphere and planetary magnetospheres. A hypothesis of adaptive non-linear-parametric wave generation may be suggested for Solar wind control of Jovian heterometric radiations, Saturn modulated radio emissions and Uranian auroral kilometric radiations. The mechanism is connected with natural interaction inhomogeneity and its type can be defined as cyclotron instability in the generation processes. There is a general agreement between researchers that planetary radiation is emitted in an extraordinary mode by maser cyclotron process and that all celestial bodies with a magnetic field and an energetic electron source are strong radioemitters due to cyclotron maser instability. We hope that the effect presented here may throw new light and enrich the understanding of the generation mechanisms. We have presented a mechanism giving rise to cyclotron processes that might prove fundamental considering the radioemission of the planetary magnetosphere. It can be shown that the mechanism may give rise to radioemission not only in a narrow range of angles almost perpendicular to the magnetic field in the source region, but always when a wave packet falls upon the charged particle oscillator. Here-with is shown the potential for excitation of low-frequency continuous oscillations having a discrete amplitude set under the influence of a wave with incompatibly higher frequency––in that number fall waves from the ultraviolet band, the near and the far IR range and the radioband. Possibly, this mechanism is combined with multiple re-emissions with frequency downward transformation and collision mechanisms which are accompanied by radioemission generation mechanisms due to plasma waves transformation into electromagnetic waves under the ‘‘wave–particle’’ and ‘‘wave–wave’’ interactions. The mechanism may also be combined with maser cyclotron processes, giving initial excitation (initial conditions) in the presence of a magnetic field, whereas subsequently a wave pumping from electromagnetic background is added. The characteristics of the radioemission spectrum might be determined from the properties of the discussed effect–– on one hand, a wave with the same (unchangeable) frequency parameter may excite oscillations in a wide frequency band and for different amplitudes; on the other hand––waves with different frequency parameters may excite oscillations with the same frequency (e.g., in the gyroresonance frequency area and local plasma frequency, due to resonance effects). A heuristic model is proposed of the mean distances between the solar-system planets, their satellites and primaries. The model is based on: (i) the concept of the solar system structure wave nature; (ii) the MM analogy of the micro- and megasystem structures, and (iii) the oscillator amplitude quantization phenomenon, occurring under wave action, discovered on the basis of the classical oscillations theory [1–5]. From the equation, describing the charge rotation under the action of an electromagnetic wave, an expression is obtained for the discrete set of probable stationary motion amplitudes. The discrete amplitudes values––the quantization phenomenon––are defined by the argument values at the extreme points of the N -order Bessel functions. Using this expression, the mean related distances are computed from the solar system planets and the Saturn, Uranian and Jovian satellites to the primaries. The harmonic oscillator obeys the classical laws to a greater extent than other system. A number of problems, related to harmonic oscillators, have the same solution in classical and quantum mechanics. Regardless of its simplicity, the oscillator–wave model obviously reflects a number of processes in the micro- and macroworld. The model is manifested naturally in different material media and harmonizes with the modern ideas about the world into and out of us as a totality of particles and fields. Also, it takes into account the wide extension of the oscillating processes in Nature. In the presence of particle flows and fields of different nature, the model realizes (materializes) widely in Nature in a very natural way. In one way or another, the model has been considered by a number of authors, but the most essential feature of behaviour––the quantization phenomenon, has escaped their attention. The oscillator–wave model confirms the bodiesÕ wave interaction mechanism in the micro- and macroworld, as well as the closing down and convergence of the basic properties manifestation in Nature. The heuristic importance of the model is drawn on the basis of the remarkably fruitful hypothesis about the Solar system wave dynamical structure, propounded recently by a number of researchers. Notwithstanding its simplicity, the proposed model gives a

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phenomenologic and dynamic description of the megaquantum resonance-wave structure of the Solar system and analogous systems, consisting of a central body and satellites. It seems that the model reconciles two opposite approaches––this of the classical celestial mechanics of N bodies and that of the quantum-mechanical wave system theory. Based on the ideas of the Universe wave nature, the model gives an enough detailed identification of the Solar system resonance structure within the frames of the classical theory. The large number of possible resonances are in complete correspondence, e.g., with the fact of existence of numerous resonances in the Jupiter and Saturn satellite systems. The model gives the orbit quantization, independent of the initial dynamic conditions and the planetsÕ and satellitesÕ masses. At the same time, by analogy with the atomic system theory, the model defines the ranges of the initial dynamic conditions, determining the possibility for planet or satellite capture, forming the Solar system planet and satellite structure. Hence, the model may trace the space substance zones, determinating in the course of evolution the present structure of the solar system planets and satellites arrangements. A more detailed study of the oscillator–wave model, proposed herewith, may reveal all possible stable orbits of the Solar system planets and planet–satellite structures, including the fine Saturn ring system structures, etc. Since the system oscillator–wave may have a different physical nature, the observed features have significantly common character. In the case, which has been considered in [1], the interaction non-homogeneity has been especially arranged––by restriction of the external force action over a small part of the trajectory. In this case of the wave action over the oscillator, on the contrary, nothing has been done to provide the inhomogeneity of the interaction. The general character of the oscillator–wave model can be illustrated with most diverse examples from Nature, Science and Technology where such systems are abundant. Everywhere where waves interact with oscillators arise conditions for the excitation of oscillations with a discrete sequence of possible stable amplitudes. It was shown above that the oscillator–wave model can be used as an heuristic model for the distribution of the interplanetary distances and the distances between natural satellites in the Solar System. On the basis of the presented oscillator–wave model it is also possible to create heuristic models of the interaction of electromagnetic waves with plasma particles in the EarthÕs ionosphere and magnetosphere, heuristic models of the generation of powerful low-frequency waves in the space around the Earth when a cosmic electromagnetic background is present, etc. Starting from the considered oscillator–wave model and generalizing for three dimensions it is possible to show analytically for example the possibility of optical pumping of a microwave SHF emitter and the sustaining of a stationary level of the electromagnetic field in the resonator given a sufficiently high level of the initial radiation [16]. High-efficient submillimeter emitter, built on this basis, could be suitable for radiophysical heating of plasma, e.g. in the experiments aimed the achievement of controllable thermonuclear reaction.

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