Tek serbestlik dereceli lineer olmayan salınıcıların yaklaşık simetrileri ve ilk integralleri

Abstract

Bu çalışmada, tek serbestlik dereceli, zorlamalı-sönümlü, aşağıdaki yapıdaki dinamik sistemle modellenebilen salınıcılarm yaklaşık ilk integralleri hesaplanmıştır. dx\ -- = xX=X2 dx2 (1) - - = £2 = -x\ - ex" + e2(7coso;i - 6x2), 0 < e << 1. Burada, 7 ve w sırasıyla kuvvetin genliği (zorlama katsayısı) ve açısal frekansı, S ise sönüm parametresidir. Bu analiz n E Z*,(Z* = Z+/{0, 1}) olduğu durumlar için yapılmıştır. (1) dinamik sistemiyle, n' nin farklı değerleri için doğadaki birçok problemde karşılaşılmaktadır. Örneğin n = 2 değeri için kulak zarı ve Helmholtz sahnıcısı olarak bilinen gemi devrilme problemlerinde kullanılır. (1) denklemi ayrıca n = 3 için, elektronik salınıcılarm modellenmesinde de kullanılır. Bu Duffing salınıcısı olarak da bilinir. Diferansiyel denklemlerin yaklaşık simetri analizi, Baikov, Gazizov ve Ibragimov tarafından geliştirilmiştir. Yazarlar, diferansiyel denklemlerin yaklaşık çözümlerinin yaklaşık simetri analiziyle hesaplanmasında da yeni bir yaklaşım geliştirmişlerdir. Burada kullanılacak olan yöntem temel olarak bu yaklaşımın modifiye bir versiyonudur. Bu nedenle burada kısaca değinilecektir. incelenen dinamik sistem Ai = x* - /<"£<*.<="" style="margin: 0px; padding: 0px; outline: 0px; color: rgb(34, 34, 34); font-family: Verdana, Arial, sans-serif; font-size: 10px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">£ (3) m=0 olarak seçilsin. Bu durumda seçilen vektör alanı X, yaklaşık yüzey kriterini; ?A'Ui=0 = OV) (4) sağlamalıdır. Burada, X' in birinci uzatması; 2 ' emtk (* t) 8xk X = X+X;^d(x,0^?7 (5) m=0 İV olarak tanınılannııştır. Buradan itibaren toplama kuralı tekrarlanan indislere uygulanacak ve parantez içindeki indislere ise uygulanmayacaktır. (2) ile verilen AJ' ye, X' in 1. uzatması uygulanarak ve AJ' nin bu vektör alam altında invaryant kalması koşulları gözönüne alınarak, <* + /oV0jfe - ıtf /£» + - 1 (25) bulunur. Burada r(i + f) "3 2^r(^) olarak tanımlanmıştır, n tek içinse, 1 x1+n 7 / = -(a;? + Xo) + e- - e2- 5 - -(x\ cosujt - X2cosmujt) (26) 2 1 + rı u^ - 1 elde edilir. Bu ilk integrallerin düzey eğrileri Mathematica kullanılarak çizdirilmiştir. Bu düzey eğrilerinin yapısı, n değerinin tek veya çift olmasına göre değişmektedir, n değerinin büyüklüğünün bu değişimde çok fazla bir etkisi yoktur. Eğer n tek ise, (25) denkleminin düzey eğrileri, periyodik çözümlerin eş merkezli çemberlerini kapsayan bir heteroklinik manifoldun varlığını göstermektedir, n 'nin çift tamsayı olma durumunda ise, (26) yaklaşık ilk integralinin düzey eğrilerinde, periyodik çözümlerin eş merkezli çemberlerini kapsayan homoklinik bir manifoldla karşılaşılmaktadır.

In this work, the. approximate first integrals of the one degree of freedom, forced and damped oscillators that can be modelled with the dynamical system below, are inves tigated -- = Xl=x2 dx2 (1) - - - = X2 = -x\ - ex? + e2(jcosu;t - 6x2), 0 < e << 1. at Here, 7 and u>, respectively, are the amplitude of the force (the. force coefficient) and the angular frequency and 6 is the damping parameter. The analysis is done for the cases that n 6 Z*, (Z* = Z+/{0, 1}). Many problems of the nature can be modelled with the dynamical system (1) with different values of n. For instance when n = 2, it models the motion of ear drums and ship capsizing phenomenon and it is called Helmholtz oscillator. It also arises in modelling of the electronic oscillators when n = 3, and it is called Duffing oscillator. Approximate symmetry analysis of the differential equations has been developed by Baikov, Gazizov and Ibragimov. They also introduced a new approach to study the approximate solutions of the differential equations by employing approximate symmetries. The method which will be discussed here is basically a modified version of this approach. Therefore, it deserves to be. mentioned briefly here. We consider the dynamical system in the form below, Ai = & - /i(x, 0 - e//(x, t) - e2fİ(X, t) = 0, (2) where j = 1,...,n, x ? K1 and (' ) stands for the derivative with respect to t and /q(x, t) is a linear function. In order for a vector field, X=f>"^(x,^, (3) to be a second-order approximate symmetry VF admitted by (2). This vector field X must satisfy the surface criteria; XA'-|AJ=0 = CP(e3). (4) Here, X is the first prolongation of X, i.e., X = X+£eTd(x,')^. (5) 171=0 Vİİİ Here, comma stands for the derivative with respect to the coordinate xi. Summation convention applies to the repeated indices and it drops for the indices in the parentheses here after. After acting the first prolongation of X on the frame defined by AJ given in (2) and using approximate surface criterion given by (4) we obtain t + /0fetF2(C1,C2) = eX2tF2(z1e-Xit,z2e-X2t). (14) IX We first substitute (14) into (13)2 to determine the RHS of (12) for m = 1, and then we solve it to get f/1 = eAlt (F3(*ie-Alt,z2e-A2t) + K1(t,zuz2)) if1=ex>t{F4(z1e-x>t,z2e-x*t) + K2(t,zuz2)). Here, Ki= l e-XirHi(r)dr Jo in which H\{t) and H2(t) are the RHSs of (12) for m = 1. Notice that the infinitesimals found hitherto involve arbitrary functions. Proceeding in this manner will lead to the infinitesimals of the second-order approximate symmetries involving arbitrary functions, this, in turn, will yield approximate first integrals with arbitrary functions. For this reason when a solution like Ceatz{zl (16) is suggested, the combination of this type of linear independent solutions can be given as, ^ = Cj«C^' + c;«A(l)*(l)+C{(%+CfW £ j>-^/) ^ t j=h2 (17) Sl+S2>2, Sl,S2>0 of the inverse transformation is applied to the solution (17) and by using equation (3) we obtain generalized symmetries of linear part of (1). From (12) for m = 0, ?7o is obtained. And when this is used in (13)2, we determine Uf. Now, the RHS of the (12) for m = 1 is obtained and we can solve it to determine fj[. But the resonance phenomena arises in these operations. For this reason, some approximate symmetries corresponding to the vector field X are broken. The resonance condi tion is obtained for first and second order approximate symmetry analysis for the DS (1) and the symmetry breaking is defined in the Chapter 2. After applying resonance condition to U\, it can be seen that the symmetries of the linear part corresponding to the parameters CT, Cft1, Cq1 an<^ Q>i s2 are broken when n is odd in (1). Symmetry corresponding to the parameter 6q is not broken. When the solution 7/1 1 obtained from the unbroken terms, is placed in (9) for m = 2, considering also the resonance conditions, by the similar procedures, f]\ also can be obtained. After the second-order approximate symmetry analysis, it can be checked that all second- order approximate symmetries are broken except the one corresponding to the group parameter Cq1. The approximate symmetries are obtained by the application of the inverse transformations; Ctf1 =» X = Xo + eXj + e^X2 (^ +e( _ ^rîsÛ) (X* + t®**1**) + ^^(no^x? +a|)n-1x2 + (1 - n)a2z" (af + x2)^ x\x2) + jX\ - - r-ş- -rw sin cot) I I - - + (-«>+<^j«+*lîl«'-*î) + e2 ( - 2(aa (a:2 + a^)w-1na:i - a2x^{x\ + z2)^((n + \)x\ + 2ns2) + 7^2 + n, J,v cos wi^ ' » 4" 2(w2-l) JJ )dx2 for odd values of n. Here, r(l+n)r(2 + n) ai r/3±»i)422+2n ' r(i + n) (18) «2 = T(3±ü)222+n and Cg1 =» X = Xo + eXi + e2X2 HI X2 + e2 ( 2 ( -x\ - -7-= - -rw sin utf 2(w2-l) /yyöaı (19) î + ^(-2(h + 5ö^i)~"-)))^ for even values of n. Now the approximate first integrals can be found from these approximate second-order symmetries (18-19). Let us now consider a Hamiltonian system with n degrees of freedom 9H. 8H,., Qi = «-, Pi = -74-, (» = l,...>n). (20) opi aqi Here, 7/ = #o(p, q) +... + e^H(k)(p> q) is the Hamiltonian function, and p and q are the canonical momenta and position, respectively. If an approximate symmetry VF of (20) of the form X = 7?f(p,q)A+7/?(P)q)^L (Z = l,...,n) (21) satisfies c^n = 0 (22) is said to be a locally Hamiltonian approximate symmetry VF. In (22), C stands for the Lie derivative, and Q = dpi A dqi where A is the wedge product. Furthermore, if the VF in (21) enjoys the property £x// = 0(efe+1), (23) xi it is called the approximate Noether symmetry VF of order k. For such an approximate symmetry VF there corresponds to an approximate first integral (the approximate version of Noether 's theorem), and it can be found from XJfi = d/+0(efe+1) (24) where / is an approximate first integral of order k, and J is the interior product. This can be extended to the nonautonomous Hamiltonian systems. It can be easily shown that the second-order approximate symmetry VFs given in (18-19) are the approximate Noether symmetry VFs provided that 6 = 0. Approximate first integrals corresponding to these symmetries have been obtained by employing (24). The second- order approximate first integrals corresponding to the symmetry VFs given in (18-19) are;,1+n /=I(*?+*l>+^-^(*? + *!>*) + n l+i + e2( (x-i cosut - xzujsmujt) + ai{x\ + x%)n - 2a2x]+n(xf + xf)^) u> - 1 (25) p/ı i n \ for odd values of n where 03 = " /-w&m and 1 a;1+n 7 / = -(#? + Xo) + e- e2- 5 - -(x\ cosojt - X2Uismu)t) (26) 2 1 + n us* - 1 for even values of n. The contour lines for the approximate first integrals given in (25-26) have been plotted by using Mathematica. The pattern in the contour lines depends on the parity of n given in (1), but not on the value of n. When n is even in (1), the contour lines of (26) point out that there exists a homoclinic manifold surrounding the concentric cylinders of periodic solutions. When n is odd, the contour lines of (25) pinpoint the existence of a heteroclinic manifold which surrounds the concentric cylinders of the periodic solutions.