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Approximate First Order Solutions of Non Linear Dynamical Mappings Many important problems of physics and engineering can be described by equation 1 1 for example in rotor dynamics robots shells and accelerators physics see for example References 1 12 If f 0 t 0 and the eigenvalues of A have all negative real part the asymptotic stability of the origin is preserved in a ball

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1 INTRODUCTION, We consider a dynamical system which is defined by a set of d non linear first order. ordinary differential equations,x Ax f x t 1 1, where x Rd A is a nonsingular matrix and f is a C 1 map Rd 1 with periodic dependence. on t with period T Many important problems of physics and engineering can be described. by equation 1 1 for example in rotor dynamics robots shells and accelerators physics. see for example References 1 12 If f 0 t 0 and the eigenvalues of A have all negative. real part the asymptotic stability of the origin is preserved in a ball whose radius depends. on whereas if they are all imaginary stability is insured by the existence of a Lyapunov. function The dynamics is conveniently described by the Poincare map defined by. xn 1 L xn F xn n 1 2, where x n x nT and L exp T A In general the Poincare map is not known exactly. but there a few relevant exceptions For example a mechanical system with a harmonic. plus an impulsive periodic force defined by the Hamiltonian. X pi 2 qi q,H i V q P t T x R2d 1 3, where P is the periodic Dirac delta function has a Poincare map defined by 1 2 where. L R 1 R d with R a 2 2 matrix rotation matrix of an angle and. F with fi x In this note we are concerned with the construction of. approximate analytical solutions of a class of non linear dynamical mappings as a system. of d non linear first order difference equations of the form given by. D xn F xn n D xn L 1 xn 1 xn 1 4, where D is the discrete analogue of the linear operator which appears in the left hand side.
of 1 1 namely dt A eA t dt e A t and reduces to a first difference D xn xn 1 xn. discretization of dt if A 0 We assume is a small parameter. The study of dynamical mappings has been the subject of many papers see for example. References 13 17 Here we first present some algorithms to construct analytically the. iterates of the map to first order in the small parameter The method is convenient. for investigation of the asymptotic behavior n of the map whenever the first order. solution can be obtained in closed form by a series summation which is the case in several. examples we have considered For small the result is in good agreement with the the. exact iterates of the map This is always the case if d 1 If d 2 then it is well known. that if the fixed point of the map is elliptic and the map is area preserving the problem. of small divisors appears making any perturbative expansion an asymptotic series If the. linear frequency 2 is non resonant we show that for a quadratic map the present. method gives the same result as the Birkhoff normal forms which are optimal At higher. order in or when a resonant frequency is approached approaches a rational number. the method should be modified to avoid the appearance of secular terms and diverging. divisors In that case the simplicity of the method would be lost and the use of normal. forms is suggested, A nice feature of the method is that it applies to non autonomous maps If the map is. periodic in n the result can be usually written in closed form We consider the following. examples a map of R given by xn 1 xn xkn cos n with k 2 3 and the area. preserving maps of R2 given by the He non map and the modulated He non map which. are basic dynamical models in accelerators physics In the last example good agreement. is found for any n if the Hausdorff distance between the exact and approximate orbit is. considered whereas the distance grows linearly with n with a coefficient 2. The plan of the paper is the following in section 2 we prove the basic results In section. 3 we analyze some 1D maps In section 4 we discuss the area preserving maps and the. relation of the proposed methods with Birkhoff normal forms and present the numerical. results for the He non map Finally section 5 is devoted to our concluding remarks. 2 THE APPROXIMATION METHOD, In this section we describe the procedure to obtain first order in approximate solutions. a class of non linear dynamical mappings of the form 1 2 The results are formulated it. in terms of three propositions, Proposition 1 There exists an approximate analytical solution of the mapping 1 2. when L I corresponding to 1 1 with A 0,Dxn xn 1 xn F xn n 2 1. of the form,xn x0 F x0 j O 2 2 2, Proof by introducing the transformation xn x0 yn and inserting it into the right.
hand side of 2 1 we get,F xn n F x0 yn n F x0 n O 2 3. to lowest order in From 2 1 and 2 3 one obtains,Dxn D x0 yn Dyn F x0 n O 2 2 4. because D x0 0 due to x 1 0 As a consequence we finally obtain the linear difference. equation Dyn F x0 n O whose solution leads to 2 2, Proposition 2 There exists an approximate analytical solution of the mapping 1 2 of. xn Ln x0 Ln j F Lj x0 j O 2 2 5,for n 1 2 given the initial value x0. Proof By introducing the transformation,xn Ln x0 yn 2 6.
the new variable yn satisfies the linear recurrence. yn 1 Lyn L F Ln x0 n O 2 7,whose solution is given by 2 5 2 2. Proposition 3 There exists an approximate analytical solution of the map 1 2 with a. non autonomous linear part,xn 1 Rn xn F xn 2 8,where Rn is a d d matrix depending on n given by. xn Ln x0 Ln L 1 2,j F Lj x0 O 2 9,for n 1 2 given the initial value x0 with. Ln 1 Rn Rn 1 R1 R0 L0 I 2 10, Proof Using the sequence of matrices Lj defined above and introducing the transformation. to y variable similar to 2 6 xn Ln x0 yn the map 2 8 becomes a linear map up to. terms of order,yn 1 Rn yn Rn F Ln x0 O y0 0 2 11,whose solution leads precisely to 2 9.
3 A FIRST EXAMPLE, In this section we use the above theorems to solve analytically an example from the class. of difference equations 2 1 This is a map of R defined by. xn 1 xn xkn sin n k 2 3 3 1, where we choose k 2 3 to consider even and odd nonlinearity From 2 2 the solution. to first order in reads,xn x0 xk0 Sn O 2 3 2, for n 0 1 2 where we have defined the following functions. En Cn iSn ei j 3 3,so that a simple calculation shows that. Cn cos j 1,Sn sin j cot, In figure 1 we show a graph of the error defined n 2 xn xnappr xn xnappr which.
grows linearly in n with a coefficient proportional to 2 We notice that for a homogeneous. nonlinearity of degree k such as in the map 3 1 the parameter can be scaled out by. defining X x k 1 The distance xn xnappr grows linearly with n with a coefficient. proportional to 2 x2k 1, 0 so that the error which is basically the distance divided by x0. behaves as n xk 1,0 2 n X 0 n, In the first example we choose a quadratic nonlinearity k 2 and 1 The error is fitted. by n 0 5 2 x20 n 0 5 X02 n as can be seen in figure 1 which refers to x0 0 1 0 2. namely X0 0 02 in the scaled variable, In the second example we choose a cubic nonlinearity k 3 and 1 The error is fitted. by n 0 75 2 x40 n 0 75 X04 n as can be seen in figure 1 which refers to x0 X0. It is worthwhile to remark that the error dependence on n for a map may be different with. respect to the corresponding differential equation For example letting x x2 cos t. the exact solution and the error on the first approximation of order are. x0 sin t 2,x t x0 x20 2 x30 3 5,sin t sin t,0 04 0 015. 0 200 0 200, Figure 1 Plot of the error n for the map 3 1 for k 2 left and k 3 right The straight line is a fit.
with a straight line y c X02 k 1 n, For 0 the distance between the exact and the order solution given by the last term. in 3 4 grows as 2 t2 x30 just as the map 3 1 where the sin n is is replaced by the cos n. since in this case the map is a good integrator of the differential equation For 1 the. error oscillates for the solution of the differential equation whereas for the map which is. no longer a good integrator is it rises linearly with small oscillations. 4 AREA PRESERVING MAPS,The He non map, We consider the important example of a quadratic area preserving map with an elliptic. fixed point at the origin which can be reduced to the standard form. yn 1 yn x2n, where R represents a 2 2 matrix of rotation by an angle In this case it is convenient. to introduce the complex scaled coordinates z x iy so that the map becomes. zn 1 f zn z n ei zn zn z n 2, The small parameter is not present explicitly but is appears after the scaling z z in. front of the quadratic term By applying 2 5 where the vector xn is replaced by zn and. the matrix L by ei we obtain,i X i n j h 2i j 2 i,z0 e e z0 2z0 z 0 e 2i j z 2 4 3.
The sums are performed by using,and the final result is. in i 21 e 1 e 2 1 e,zn e z0 z 2z0 z 0 z 0 4 5,4 0 1 ei 1 e i 1 e 3i. It can be observed that if 2 is irrational the sequence n n mod 2 is dense. on the torus T defined as the interval 0 2 where the end points are identified As a. consequence z z defined by 4 5 where n is replaced by the continuous variable. provides an approximate parameterization of the orbit To any closed orbit one associates. the invariant action J defined as the area divided by 2 Given a Fourier representation. of the orbit z k ck eik one has,1 1 1 dz 1X,J pdq Im z z dz Im z z d k ck 2 4 6. and writing 4 5 as z c1 ei c2 ei2 c 2 e i2 one has J 2 c1 2 c2 2 c 2 2. Normal Forms, The result is the same as the first order application of the normal forms In this case one. looks for a transformation such that the map in the new coordinates. z 1 z z 4 7,is again a rotation,n 1 ei n n n 4 8,The equation which determines and is.
f n n n n ei n n n e i n n n 4 9, whose solution is obtained by a series expansion in At the lowest order corresponding. to a quadratic approximation of and to constant frequency we have. 20 2 11 02 2 1 z z z 20 z 2 11 zz 02 z 2,The computation gives. i 1 i ei i 1,11 02 4 11,4 e 1 2 ei 1 4 e 3i 1, To compare with the previous result we compute zn by observing that in the normalized. variables the map is a constant rotation n ein 0 so that. zn ein 0 e in 0 ein 1 z0 z 0 e in 1 z0 z 0,ein 0 20 e2in 02 11 0 0 02 e 2in 20. ein z0 20 z02 11 z0 z 0 02 z 20 20 e2in z02 11 z0 z 0 02 e 2i z 20. where in the last line the cubic terms have been omitted As a consequence using 4 11. we check that zn given by the normal forms is identical to 4 5 up to O z0 3 terms. namely up to O 2 kx0 k3 in original coordinates, As a consequence the previous recurrence gives in a simple way the same result as the.
normal forms for a quadratic map This is not true if a higher order nonlinearity is used. In this case indeed the perturbation expansions are different even at lowest order For. instance for a cubic nonlinearity the third order normal forms give an amplitude dependent. frequency 2 This contribution is absent from the proposed perturbative. solution which takes into account only the geometric deformation keeping the frequency. constant Its higher order extension would require taking a variable frequency and the. method would become rather involved In such a case the use of normal forms would be. suggested This is also the case if the linear frequency is very close to a resonance in order. to avoid the explosion of the corresponding small divisors. Modulated map, A physically relevant case of a map like 2 6 whose the linear part depends on n is given. by the modulated He non map This is the map 4 1 where R is replaced by R n. this map describes the effect of ripples and a sextupolar nonlinearity in beam dynamics. Rn R n n cos n 4 13, where and 1 We introduce complex coordinates and the sequence of complex. numbers n corresponding to Ln,n ei 0 1 n 1 4 14,where the sum can be evaluated according to. 1 ein sin n 12,j n Re n 1 4 15,1 e i 2 sin 12, The iterate of order n in complex coordinates reads. zn n z0 z Cn 1 2z0 z0 2Cn 1 z0 Cn 3 4 16, where the coefficients Cn m are given analytically in the Appendix in terms of trigono.
metric and Bessel functions,x0 0 1 x0 0 3 x0 0 3 N F. 0 2 0 5 0 5,0 2 0 5 0 5,0 2 0 2 0 5 0 5 0 5 0 5,x0 0 6 x0 0 6 N F Henon map. 1 1 1 1 1 1,Figure 2 Orbits for the He non map with 2 3 2 5. 0 381966 for initial conditions x0 0 1 0 3 0 6 y0 0. and n 2000 iterations The orbits in blu are the exact orbits whereas the orbits in red correspond to the. present approximation or to the normal forms denoted by N F on top of the frame approximation for. x0 0 1 they are indistinguishable, A numerical example for the He non map The Hausdorff distance between the exact. orbit zn and its approximation znappr is of order z0 3 namely 2 kx0 k3 in the original. coordinates We recall that the Hausdorff distance between two sets A and B is. dH A B max max d x B max d x A d x A min kx yk 4 17. x A x B y A, The distance zn znappr grows linearly with n and the coefficient z03 is z0 times the.
error on the frequency z02 Indeed assuming that x and x x are the exact. and the approximated orbit and the exact frequency. kxn xnappr k kxn n x n x n k A n mod 2 max k x k,x 0 0 1 x 0 0 3 Hausdorf. 0 500 0 100 0 0 7, Figure 3Plot of the error n zn znappr zn znappr versus n for initial conditions x0 0 1 0 3 for. the normal form blu and the present approximation red In the right figure the error in the Hausdorff. distance of the orbits for initial conditions in the whole range 0 7 is shown for the present approximation. blu and normal forms red in a Log10 scale, In figure 2 we compare the exact orbits with the orbits obtained with the present approx. imation 4 5 and the normal forms given by the first line of 4 12 whose difference is of. higher order The error defined as n kxn xnappr k kxn xnappr k is shown in figure. 3 where its linear growth is evident Indeed the first term in 4 18 proportional to n. is dominant over the second one which depends on the geometry of the orbits This is. measured by Hausdorff distance between the orbits This error shown in figure 3 for the. whole range of initial conditions 0 x0 0 7 y0 0 varies as x30 as expected. Figure 4 Plot of the tune 2 versus x0 for the He non map with 2 3 2 5. 0 381966 and orbits,with initial conditions x0 y0 0. 5 CONCLUDING REMARKS, In this study we present a simple method formulated by propositions 1 2 3 which can.
be used to construct approximate analytical solutions of a class of non linear dynamical. mappings of the form 1 2 Our study shows interesting similarities between difference. and differential equations The connection of our maps to several branches of mechanics. physics and engineering is discussed e g rotor dynamics robots shells beams with vi. brating support and accelerators physics The analytical results of our examples are in. good agreement with the numerical results obtained by iterating the maps for reasonably. large values of In the case of the classical Henon map the proposed method gives the. same result as normal forms to first order One should notice that the method just as the. normal forms provides an analytic interpolation of the orbits written as a Fourier series. if the map has an elliptic fixed point and the frequency is nonresonant The approximate. analytical solutions of these mappings eqs 2 1 2 5 and 2 8 can be used as a theo. retical guidance for further numerical or analytical studies of these systems for example. stability analysis and control of chaos The advatage of the our results lies in its simplic. ity and ease of application to a wide variety non linear dynamical mappings modulated. autonomous and non autonomous, We give the explicit expression of the coefficients of the modulated map. exp im cot sin j exp im cos j A1, In order to carry out the sum explicitly we need to assume that 1 cot 2 1 In. such a case we can expand the exponentials in series of Bessel functions retaining only. the first two terms Indeed using the generating function expansion exp 21 t t 1. n t Jn z valid for z t and recalling that Jn z 2 z for z 1 we retain. only the terms up to first order in and cot 2 and write. Cn m eim 2 A J0 J 0 2iB J0 J 1 2iD J1 J 0,where we have defined. Jk Jk m J k Jk m cot A eijm,2 2 2 1 eim,B n 1 sin j i in m i in m. D j 0 cos j,1 1 ei m 1 1 ei m, If the modulating frequency is resonant namely 2 p q it is convenient to consider.
the stroboscopic map at n q Then the result simplifies according to. Cn m eim 2 J0 J 0 2iJ1 J 1 A5, and if 2 is irrational it is possible to obtain a parametric representation of the orbit. z z by replacing n with the belonging to the interval 0 2 and to identify the. Fourier coefficients of its expansion,REFERENCES, 1 J Guckenheimer and P Holmes Non Linear Oscillations Dynamical Systems and. Bifurcations of Vector Fields Springer Verlag New York 1983. 2 A H Nayfeh and D T Mook Non Linear Oscillations J Wiley New York 1979. 3 A H Nayfeh Perturbations Methods J Wiley New York 1973. 4 M A Lieberman and A J Lichtenberg Regular and Stochastic Motion Springer. Berlin 1983, 5 R H Helleman Self Generated chaotic behavior in non linear mechanics Fundamental. Problems in Statistical Mechanics Vol 5 Ed E G Cohen North Publ Amsterdam and. New York pp 165 233 1980, 6 G M Mahmoud and T Bountis Synchronized periodic solutions of a class of periodi. cally driven nonlinear oscillators J Appl Mech 55 pp 721 728 1988. 7 G M Mahmoud T Bountis and G Turchetti Synchronization in parametrically. Hamiltonian systems Il Nuovo Cimento Vol 110B No 11 pp 1311 1322 1995. 8 G M Mahmoud A Rauh and A A Mohamed On modulated complex nonlinear. dynamical systems Il Nuovo Cimento Vol 114B pp 31 47 1999. 9 G M Mahmoud and S A Aly Periodic attractors of complex non linear systems Int. J of Non Linear Mech 35 pp 309 323 2000, 10 G M Mahmoud A theorem for n dimensional strongly non linear dynamical systems.
Int J of Non Linear Mech 2001, 11 C M Bender and S A Orszag Advanced Mathematical Methods for Scientists and. Engineers McGraw Hill New York 1978, 12 M Henon Numerical study of quadratic area preserving mappings Quart Appl. Math 27 pp 291 312 1969 and its references, 13 T Bountis Non Linear Models in Hamiltonian Dynamics and Statistical Mechanics.