Higher Order Averaging : periodic solutions, linear systems and an application Hartono and A.H.P. van der Burgh Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands Abstract Existence and stability of periodic solutions by using second order averaging when the vector field by first order averaging vanishes, will be studied in this paper as well as its generalization to higher order. Additionally a special averaging algorithm for the computation of higher approximations of the fundamental matrix of linear equations with periodic coefficients is given. As an application the existence and stability of periodic solutions of an inhomogeneous second order equation with time-dependent damping coefficient are studied. Introduction The averaging method is a well-known method for the construction of approximations for solutions of initial value problems for a class of non-linear differential equations, as well as for finding periodic solutions. Usually the respective algorithm is concerned with first or second order approximations. Little attention has been paid to problems where third and higher order approximations have to be considered. Particularly for the construction of stability diagrams of linear equations with time varying coefficients like equations of Hill s type, these higher order approximations are relevant. For the construction of third and higher order approximations we will study two aspects in more detail: the existence of periodic solutions in particular when from first and second order averaging no conclusions about existence of periodic solutions can be drawn and the algorithm for the construction of the approximations. The well-known theorem on the existence of time periodic solutions is based on the existence of critical points of the autonomous system obtained by (first order) averaging. This theorem will be extended to the case that the system obtained by (first order) averaging vanishes identically. The algorithm for higher order averaging is straight forward: however because of the complexity little attention has been paid in applications. As is well-known the averaging method is of an asymptotic nature the respective asymptotic series may not converge. The situation for linear systems with time-varying coefficients is quite different. Consider a linear system of the form: ẋ = ɛa(t, ɛ)x (.) where A(t, ɛ) is T -periodic in t and ɛ a small parameter. The averaging algorithm is concerned with the computation of the fundamental matrix Φ(t, ɛ) which can be represented (Floquet) by: Φ(t, ɛ) = P(t, ɛ)e B(ɛ)t (.2) Lecturer in Jurusan Matematika Universitas Negeri Yogyakarta, Indonesia, on leave as a PhD reseacher at the Delft University of Technology, The Netherlands.
2 where P(t, ɛ) is a T -periodic matrix and B(ɛ) a constant matrix. The algorithm implies the computation of approximations of P(t, ɛ) and B(ɛ) to any order of ɛ. In the case that A(t, ɛ) is an analytic function in ɛ for ɛ < ɛ are may assume that, as P(t, ɛ) and B(ɛ) are also analytic functions in ɛ on the same interval, the power series for P(t, ɛ) and B(ɛ) obtained by the algorithm converge. The organization of this paper is as follows. In section 2 a theorem is discussed on the existence and stability of periodic solutions by using second order averaging when the vector field by first order averaging vanishes, and a generalization of this theorem to higher order is presented. The approximations of the fundamental matrix of linear equations with periodic coefficients by using a special averaging algorithm are given in the section 3. At the end of this paper an example concerning the existence and stability of periodic solutions of an inhomogeneous second order equation with time-dependent damping coefficient is given. 2 Existence and Stability of Periodic Solutions The existence and stability of periodic solutions by using first order averaging, has been studied extensively and can be found for instance in [4, 6]. In this section the existence and stability of periodic solutions (by using the second order averaging ) in the case that the averaged vector field (to first order) vanishes is investigated. A theorem on the validity of approximations for initial value problems in the case that the vector field by first order averaging vanishes is recalled. Theorem.2. Consider the initial value problems { ẋ = ɛf(t, x) + ɛ 2 g(t, x) + ɛ 3 R(t, x, ɛ), (2.3) x() = x o. and f o (y) = T T f(s, y)ds { u = ɛ 2 f o (u) + ɛ2 g o (u) u() = x o. (2.4) with f, g : [, ) D R n, R : [, ) D (, ɛ o ] R n, where D is a bounded domain in R n, f(t, x) t f (t, x) = x u (t, x) and u (t, x) = f(s, x)ds. Suppose then. f, g and R are Lipschitz-continuous in x on D; f, g, R are continuous in t ; 2. f, g and R are T -periodic in t, g o and f o are average of g and f respectively; R is bounded by a constant independent of ɛ for x D; 3. u(t) belongs to an interior subset of D on the time scale ɛ 2 4. the vector fields f, g, R, f/ x, 2 f/ x 2, g/ x, R/ x are defined continuous and bounded by a constant M (independent of ɛ) in [, ) D, ɛ ɛ o ; x(t) = u(t) + O(ɛ) on the time scale ɛ 2. A proof of this theorem can be found in [5]. The following theorem is related to the above one and is concerned with the existence of
3 periodic solutions for the case that the vector field in first order averaging vanishes. Theorem.2.2 Let f = f o + go then equation (2.4) can be written as Suppose p o is a critical point of (2.5) and u = ɛ 2 f (u) (2.5) f (u)/ u u=po, (2.6) then there exists a T -periodic solution ψ(t, ɛ) of equation (2.3) which is close to p o such that lim ɛ ψ(t, ɛ) = p o. Further if f is continuously differentiable in u and the eigenvalues of the matrix f (p o )/ u all have negative real parts, the corresponding periodic solution ψ(t, ɛ) is asymptotically stable for ɛ sufficiently small. If one of the eigenvalues has a positive real part, ψ(t, ɛ) is unstable. To prove this theorem we need the Lipschitz s continuity of f. As is well known the function f(t, x) with f : R n+ R n, t t o a, x D R n ; satisfies the Lipschitz condition with respect to x if in [t o a, t o + a] D if: f(t, x ) f(t, x 2 ) L x x 2 with x, x 2 D and L a constant. Furthermore L is called the Lipschitz constant. Proposition. Suppose that the functions f and g are Lipschitz continuous in x and α, β are real constants. Let f o be the average of f over t i.e. f o (x) = /T T f(t, x)dt where in the general case f o (x) is not identical zero and: U(t, x) = t [f(s, x) f o (x)]ds. Then the functions αf + βg, f g, and U(t, x) are Lipschitz continuous in x. Proof of the theorem. Consider the equations ẋ = ɛf(t, x) + ɛ 2 g(t, x) + ɛ 3 R(t, x, ɛ), (2.7) f, g are T -periodic in t. Introduce a near-identity transformation x = z + ɛu (t, z) + ɛ 2 u 2 (t, z). (2.8) Substituting (2.8) into (2.7) considering f o (x) and choosing u and u 2 as follows: u (t, z) = t f(s, z)ds, u 2 (t, z) = f (z) = T T t [g(s, z) + f z u ]ds, one obtains the transformed equation ( up to order ɛ 3 ) [g(s, z) + f z u f ]ds, ż = ɛ 2 f (z) + ɛ 3 R(t, z, ɛ) (2.9) where R(t, z, ɛ) = f z u 2 + g z u u z f + G + R(t, z, ) + O(ɛ), (2.)
4 in which G is a vector with the k-th component G k as follows: G k = 2 n i= u 2 2 f k i zi 2 + n 2 f k u i u j, z i z j u i is i-th component of u, f k is k-th component of f, and z i is i-th component of z. The periodicity of f and g with respect to t implies the periodicity of u, u 2, and R. Introduce an initial value z() = z o for equation (2.9). As an equivalent integral equation one obtains: t z(t) = z o + ɛ 2 [f (z) + ɛ R]ds. It may be clear that the solution of this equation depends on ɛ as well as on z o i.e. z(t) = z(t, ɛ, z o ) Further, one can calculate z(t + T ) as follows: i j z(t + T ) = z o + ɛ 2 t+t [f (z) + ɛ R]ds = z o + ɛ 2 T [f (z) + ɛ R]ds + ɛ 2 t+t [f (z) + ɛ R]ds T (2.) To have a time periodic solution one should have z(t) = z(t + T ) from which it follows that: h(z o, ɛ) = T [f (z) + ɛ R]ds =. As z = z(t, z o, ɛ) one obtains for t = : z(, z o, ɛ) = z o and when ɛ equal zero one finds that z(t, z o, ) = z o. So evidently h(p o, ) =, and h(z o, ) = T f (z(s, z o, ))ds = T f (z o )ds (2.2) From (2.6) it follows that = T f (z o ). h(z o, )/ z o zo=p o. (2.3) Finally according to the Implicit Function Theorem there exist a unique function p : ( ɛ o, ɛ o ) R n with p() = p o and h(p(ɛ), ɛ) = for ɛ ( ɛ o, ɛ o ). So h(z o, ɛ) = has unique solution z o (ɛ) and z o (ɛ) p o when ɛ. Thus the transformed equation (2.9) has a T -periodic solution with initial value z o (ɛ). Suppose the solution is ψ (z o (ɛ), t). Because of the periodicity of u and u 2 then the original equation (2.3) has a T -periodic solution, that is ψ(z o (ɛ), t) = ψ + ɛu (t, ψ ) + ɛ 2 u 2 (t, ψ ), (2.4) and satisfies ψ p o when ɛ. To study the stability of this periodic solution one can show that its stability depends on the stability of the periodic solution of the transformed system. First, set z = ψ + w. Differentiating this term and substituting into (2.9) gives : ẇ + ψ = ɛ 2 f (ψ + w) + ɛ 3 R(t, ψ + w, ɛ) = ɛ 2 f (ψ ) + ɛ 2 f (ψ ) w w+ (2.5) ɛ 3 R(t, ψ, ɛ) + ɛ 3 R(ψ ) w w + O(w2 ).
5 As is known ψ (t, ɛ) is a periodic solution of the transformed equation (2.9), so and it follows that ψ = ɛ 2 f (ψ ) + ɛ 3 R(t, ψ, ɛ), (2.6) ẇ = ɛ 2 f (ψ ) w w + ɛ3 R(ψ ) w w + O(w2 ) = ɛ 2 f (p o) w w + ɛ2 [ f (ψ ) w f (p o) w ]w + ɛ3 R(ψ ) w w + O(w2 ). (2.7) Assume that f w and R w continuous, and define the continuous function K(t, ɛ) by: K(t, ɛ) = f (ψ ) w f (p o ) w. As is known ψ (t, ɛ) p o when ɛ, so K(t, ɛ) when ɛ. Secondly, consider the linear part of equation (2.7) w = ɛ 2 [ f (p o ) w + K(t, ɛ) + ɛ R(ψ ) ] w. (2.8) w Suppose that α j, j =, 2, n are the eigenvalues of matrix f (p o) w. Then the characteristic exponents of equation (2.8) λ j (ɛ), j =, 2, n can be considered as single-valued continuous functions of ɛ with λ j () = α j. So if Re(α j ) < (respectively Re(α j ) > ) then there exists a positive ɛ o such that Re(λ j (ɛ)) < (respectively Re(λ j (ɛ) > )) for all ɛ ɛ o. In other words, the sign of the real part of the characteristic exponent is equal to the sign of the real parts of the eigenvalues of the matrix f (p o) w for ɛ sufficiently small. We now apply theorem 7.2 in [6] page 86, saying that if Re(λ j ) < then the trivial solution w = of equation (2.7) is asymptotically stable. But the trivial solution w = corresponds with z = ψ, so one can deduce that ψ is asymptotically stable. According the Floquet theorem, every fundamental matrix Φ(t, ɛ) of equation (2.8) can be written as Φ(t, ɛ) = P(t, ɛ)e B(ɛ)t, and the eigenvalues of matrix B(ɛ) are characteristic exponent of equation (2.8). If one transforms the variable w to a new variable v, according w = P(t, ɛ)v, then the equation (2.7) becomes: v = ɛ 2 B(ɛ)v + O(v 2 ). (2.9) Now theorem 7.3 in [6] page 88 can be applied, yielding that if at least one of the Re(λ j ) is positive then the solution v = of equation (2.9) is unstable. The trivial solution v = corresponds with the trivial solution w =, and the trivial solution w = corresponds with the solution z = ψ. Thus in other words one can conclude that ψ is unstable. Now it will be shown that if ψ is a stable periodic solution of system (2.9) then ψ is also a stable periodic solution of the original system (2.3). Suppose η(t) is a solution of (2.3), then η(t) can be written as η(t) = η (t) + ɛu (t, η ) + ɛ 2 u 2 (t, η ), (2.2) where η (t) is some solution in (2.9). According the fourth assumption of the theorem and the proposition above it can be concluded that u and u 2 satisfy the Lipschitz condition. Thus from (2.4) and (2.2) it follows that, ψ(t) η(t) ψ (t) η (t) + ɛ u (t, ψ ) u (t, η ) + ɛ 2 u 2 (t, ψ ) u 2 (t, η ) (2.2) N(ɛ) ψ (t) η (t).
6 Hence the stability of ψ follows from the stability of ψ. The result obtained above can be extended to more general cases. As has been shown when the vector field f(t, x) vanishes by first order averaging one has to consider second order averaging. In a similar way when higher order averaging, say n-th order averaging yields the trivial vector field one has to consider (n + )-th order averaging and has to determine critical points of the (n + )-th order non-trivial vector field. Consider the initial value problem for the system ẋ = ɛf (t, x) + + ɛ k f k (t, x) + ɛ k+ˆf(t, x, ɛ), x() = xo. (2.22) where f,, f k, ˆf are periodic in t. By substituting the near identity transformation x = y + ɛu (t, y) + + ɛ k u k (t, y) (2.23) into (2.22) one obtains the following transformed system : ẏ = ɛg (y) + + ɛ k g k (y) + ɛ k+ ĝ(t, y, ɛ). (2.24) By neglecting the last term of (2.24) one finds the averaged system : ẇ = ɛg (w) + + ɛ k g k (w). (2.25) The term g in the averaged equation is the average of f in equation (2.22), the term g 2 depends not only on f 2 but also on f and u. The higher the index i of the term g i (w) the more complicated this term becomes. Theorem.2.3. Assume that the vector field in (2.22) is smooth and periodic in t. Let K be a compact subset of R n and let W be a larger compact subset containing K in its interior. Let ɛ o be such that the near identity transformation (2.23) is valid (invertible) for y in W and ɛ < ɛ o. Suppose that g,, g k in the averaged system (2.25) are identically zero. The solution of the dz/dτ = g k (z), z() = x o, τ = ɛ k t (2.26) remains in K in τ C. Then there exist constants c and ɛ such that x(t, x o, ɛ) z(t, x o, ɛ) < cɛ for t C/ɛ k, < ɛ ɛ, (2.27) for all x o in K. Furthermore if p is a critical point of and ż = ɛ k g k (z) (2.28) g k (z)/ z z=p. (2.29) then there exist a periodic solution of (2.22) in the ɛ-neighbourhood of p. Besides that if ĝ k+ (z)/ z is continuous, then this periodic solution is asymptotically stable if all of the eigenvalues of the matrix g k (p)/ z have negative real part and unstable if there exist at least one eigenvalue of that matrix with positive real part. Remarks. The first part of this theorem is generalization of Theorem.2., resulting in approximations on longer time scales. For more general results on higher order averaging one can consult [2], where however approximations are studied on a /ɛ time scale. The second part of this theorem seems not to be known that is this theorem gives conditions for the existence and the stability of periodic solution of the original equation depending on a higher order term of which the determinant of the matrix obtained by linearization in the neighbourhood of the critical point does not vanish. To prove the first part of this theorem one can use the method used in the proof of Theorem.2.. The proof of the second part of Theorem.2.3 can be given on the basis of the principles given in the proof of Theorem.2.2.
7 3 Higher Order Averaging for Linear Equations In general solutions of systems of linear differential equation with time-periodic coefficients are not always periodic. The Floquet theorem shows that the fundamental matrix of this system can be written as a product of a periodic matrix with an exponential matrix. As is known there are no general methods to calculate this fundamental matrix. In this section an example how to approximate solutions of systems of linear differential equations with time-periodic coefficients by using higher order averaging will be given. Consider the equation ẋ = (ɛa (t) + ɛ 2 A 2 (t) + ɛ n A n (t))x, (3.3) where A i (t), i =, 2,, n are T -periodic n n-matrices in t and x is a n column vector. According to the Floquet theorem the fundamental matrix of equation (3.3) can be written as follows: P(t, ɛ)e B(ɛ)t (3.3) where P(t, ɛ) is n n-matrix, T -periodic in t and B(ɛ) is a n n constant matrix depending on ɛ. As the right hand side of equation (3.3) is linear, the near identity transformation can be chosen in linear form as follows: x = (I + ɛv (t) + ɛ 2 V 2 (t) + + ɛ n V n (t))y. (3.32) Substituting (3.32) into (3.3) one obtain the transformed system where A, F and F are ẏ = F (AF Ḟ)y (3.33) A = ɛa (t) + ɛ 2 A 2 (t) + + ɛ n A n (t) F = I + ɛv (t) + ɛ 2 V 2 (t) + + ɛ n V n (t) (3.34) If one choose V (t) = F = I + j= ( )j [ n i= ɛi V i ] j. t [A (s) A () ]ds, A () = T T A (t)dt, V 2 (t) =. V n (t) = t [A (s)v (s) + A 2 (s) V (s)a () A () ]ds, A () = T t n j= i+j=n T [A (t)v (t) + A 2 (t) V (t)a () ]dt, n A i (s)v j (s) V j (s)a (n j ) ds, j= V = I A (n ) = T T n j= i+j=n n A i (t)v j (t) V j (t)a (n j ) dt, then one obtains the transformed equation (3.33) up to order ɛ n+ : ẏ = (ɛa () + ɛ 2 A () + + ɛ n A (n ) )y + O(ɛ n+ ) (3.35) = n ɛ j+ A (j) y + O(ɛ n+ ). j= j=
8 Truncating the order ɛ n+ terms yields the averaged equation n ż = ɛ j+ A (j) z, (3.36) j= and its solution (with initial condition x o ) is n z = exp [ ɛ j+ A (j) ] t x o. (3.37) j= By substituting (3.37) into (3.32) the solution of equation (3.3) with initial condition x o can be approximated by x app i.e. : ( n ) x app = z + ɛ i V i z (3.38) i= ( n n = [I + ɛ i V i )] exp [ ɛ j+ A (j) ] t x o. i= In other words, the fundamental matrix of (3.3), P(t, ɛ)e B(ɛ)t, can be approximated by ( n n [I + ɛ i V i )] exp [ ɛ j+ A (j) ] t. i= Now it follows that if (3.3) and (3.36) have the same initial value then x x app = O(ɛ n ) on time scale /ɛ. This result is a special case of the n-th order averaging as given in [2], where the system ẋ = ɛf(t, x, ɛ) (3.39) is considered. As this system is non-linear the near-identity transformation as well as the resulting n-th order averaged system are much more complicated. As will be shown in the following section the algorithm for linear systems as presented in this paper can be applied straightforwardly to special examples yielding interesting results. 4 Application In this section the theory of the previous sections is illustrated with an example. A special equation is studied by first and higher order averaging. It will be shown that higher order averaging is essential for obtaining interesting results. The periodic solutions of an inhomogeneous second order equation with time-dependent damping coefficient: j= j= ẍ + (c + ɛ cos 2t)ẋ + (m 2 + α)x + A cos ωt = (4.4) are studied, where c, α, ɛ, A are small parameters and m, ω positive integers. A rather special property of equation (4.4) is that the coefficient of ẋ is time dependent and it seems that only little attention has been paid in the literature to an equation of type (4.4). For m = and A = some results especially related to the stability of the trivial solution can be found in []. As will be shown in a separate paper equation (4.4) may be used as a model equation for the study of rain-wind induced vibrations of a special oscillator. In a more general context the homogeneous equation (4.4) may be considered as a variational equation for a corresponding non-linear equation with a periodic solution. The study of periodic solutions of equation (4.4) involves the existence of the periodic solutions as well as the construction of approximations. Also the stability of these periodic solutions will
9 be studied. Because of the presence of a number of small parameters in equation (4.4) the averaging method for the construction of approximations for the periodic solutions will be used. The parameters c, α and A are considered to be small implying that they are expressed in the characteristic small parameter ɛ of the problem: c = ɛc + ɛ 2 c 2 + ɛ 3 c 3, α = ɛα + ɛ 2 α 2 + ɛ 3 α 3, A = ɛa + ɛ 2 A 2 + ɛ 3 A 3. (4.4) where c i, α i and A i, i =, 2, 3 are of O(). Note that throughout the analysis the parametric excitation ɛ cos 2t remains of O(ɛ). For m, ω {, 2, 3}, it will be shown that an O()-periodic solution exists if m = ω and if m ω the periodic solution is of order ɛ. Further, if c = O(ɛ), α = O(ɛ), and A = O(ɛ), for m = ω = both stable and unstable periodic solutions exist but for m = ω = 2, 3 only stable periodic solutions are found. For the case that c = O(ɛ 2 ), α = O(ɛ 2 ), and A = O(ɛ 2 ), for m = ω = 2, 3 only stable periodic solutions are found. But for m = 3 and α = 9 64 ɛ2 + O(ɛ 3 ), c = O(ɛ 3 ), A = O(ɛ 3 ) both stable and unstable periodic solutions exist. The stability of the periodic solutions follows from a new stability diagram related to equation (4.4) with A. According to the Floquet theorem the homogeneous equation has unbounded solutions when c is negative, so in this section we only consider the cases c = and c positive. 4. THE CASE m = ω 4.. Application of the averaging method: first order approximation. For the cases c, α, A are O(ɛ), the averaging method can be used to analyze the stability diagram of equation (4.4). To obtain the standard form for the application of the averaging method one can put c = c ɛ, α = α ɛ, A = A ɛ, (4.42) and transform x and ẋ to the new variables y and y 2 by: x = y cos mt + m y 2 sin mt, ẋ = my sin mt + y 2 cos mt. (4.43) The standard form is: y 2 = ɛk(t) y y 2 + ɛ A m sin mt cos ωt A cos mt cos ωt, (4.44) where K(t) = m sin2 mt(c + cos 2t) + α 2m sin 2mt 2m sin 2mt(c + cos 2t) + α m sin 2 mt 2 m 2 sin 2mt(c + cos 2t) + α cos 2 mt cos 2 mt(c + cos 2t) + α 2m sin 2mt For m =, 2, and 3 the averaged equation of (4.44) and its critical points are presented in table. The critical points of the averaged equation in table correspond with a O() time periodic solution of equation (4.44). The stability of these solutions follows from the eigenvalues of the coefficient matrix as can easily be verified. For m = and given c positive, after rescaling the parameters, the eigenvalues of the coefficient matrix become: ( λ,2 = c ɛ 2 ± ) 2 4 ɛ2 α 2..
* The first order averaged equation and its critical points 4 m= ż = ɛ 2 c 2 α z + ɛ, 2 α 4 2 c 2 A ( αa α 2 +c2 4, ) ( 2 c)a α 2 +c2 4 m=2 ż = ɛ 2 c 8 α z + ɛ, 2 α 2 c 2 A ( 4 Aα c 2 + 4 α2, A c c 2 + 4 α2 ) m=3 ż = ɛ 2 c 9 α z + ɛ 2 α 2 c 2 A ( 2α A 2α 2 +9c2, ) 9A c 2α 2 +9c2, Table : The first order averaged equation for (4.4) and its critical points for the case m = ω, z is a 2 column vector According to the character of the eigenvalues the α ɛ plane can be divided into two regions (see Figure b ) by the curves α = ± 4 ɛ2 c 2. On this curve the determinant of the coefficient matrix is equal to zero, implying that the averaged equation does not have an isolated critical point. In region I the real part of the eigenvalues are negative, thus in this region the periodic solutions are stable. In the region II the periodic solutions are unstable because in this region the eigenvalues are real-valued, one positive and one negative. In case c = the α ɛ plane divided into two regions (see Figure a) by the lines α = ± 2ɛ. In the region II the periodic solutions are unstable because the eigenvalues are real-valued, one positive and one negative. In region I the eigenvalues are purely imaginary. For m = 2 and m = 3, and given c positive the determinant of the coefficient matrix is not equal to zero. In these cases there exists one critical point and the eigenvalues of the coefficient matrix are complex-valued with negative real part implying that equation (4.4) has always stable periodic solutions. 4..2 Application of the averaging method to second order. By applying first order averaging for m = 2 and m = 3 one finds a critical point and hence a periodic solution depending on the parameters c (damping), α (detuning) and A (forcing). In the α ɛ plane one does not find a stability diagram similar to the ones in Figure. i.e. for c > the critical point is locally but also globally stable. Higher order averaging will not affect this qualitative picture because of the dominant O(ɛ) terms involving damping (c > ) in the averaged equations. By reducing the order of magnitude of the damping and forcing as well as the detuning up to O(ɛ 2 ) but keeping the parametric excitation at O(ɛ) one may find a region of instability. This can be achieved by considering
ε ε II II II I I 2c I α α a. c = b. c > Figure : Stability diagrams of periodic solution of equation (4.4) for m =. In the shaded regions the periodic solutions are unstable the expansion c = ɛc + ɛ 2 c 2 + ɛ 3 c 3 +, α = ɛα + ɛ 2 α 2 + ɛ 3 α 3 +, A = ɛa + ɛ 2 A 2 + ɛ 3 A 3 +. (4.45) and setting c = α = A =. As the second order averaging are applied, one can truncate the expansion : c = ɛ 2 c 2, α = ɛ 2 α 2, A = ɛ 2 A 2. (4.46) For m = 2, substitution of (4.46) and (4.43) into (4.4) yields after second order averaging: ż = ɛ 2 2 c 2 8 (α 2 6 ) z + ɛ 2. (4.47) 2 (α 2 6 ) 2 c 2 2 A 2 The critical point of (4.47) is ( 6 A 2(α 2 6 ) 4 c2 2 + 6 (α 2, 6 )2 4 A 2c 2 4 c2 2 + 6 (α 2 ). 6 )2 The determinant of the coefficient matrix is B 2 = 4 c2 2 + 6 (α 2 6 )2, and its eigenvalues are λ,2 = c 2 2 ± 2 6 (α 2 6 )2. By rescaling the parameters, the determinant and the eigenvalues become ɛ ( 4 4 c2 + 6 (α 6 ɛ2 ) 2 ) and ( ) λ,2 = ɛ 2 c 2 ± 6 2 (α 6 ɛ2 ) 2, respectively. For given c 2 positive, B 2 is never zero and its eigenvalues are complex-valued with negative real part for α 2 6. Thus (4.4) has stable periodic solutions. But for the case c 2 = the eigenvalues are purely imaginary and equal zero when α 2 = 6.
2 ε α= /6 ε 2 ε α=9/64 ε 2 α α 2a. m = 2 2b. m = 3 Figure 2: Curves on which the eigenvalues of (4.47) and (4.48) are zero and on both sides of curves the eigenvalues are pure imaginary. In this case the α ɛ plane can be divided in two regions separated by the curve α = 6 ɛ2 : on this curve the averaged equation does not have a critical point (see Figure 2a) implying that no periodic solutions are found in second order approximation. In a similar way for m = 3, by using second order averaging one obtains: ż = ɛ 2 2 c 2 8 (α 2 9 64 ) z + ɛ 2. (4.48) 2 (α 2 9 64 ) 2 c 2 2 A 2 The critical point of (4.48) is ( 36 A 2(α 2 9 64 ) 4 c2 2 + 36 (α 2 9, 64 )2 4 c ) 2A 2 4 c2 2 + 36 (α 2 9. 64 )2 The determinant of the coefficient matrix of (4.48) is 4 c2 2 + 36 (α 2 9 64 )2, and its eigenvalues are λ,2 = c 2 2 ± 2 9 (α 2 9 64 )2. By rescaling the parameters, the eigenvalues and the determinant become ɛ ( 4 4 c2 + 36 (α 9 64 ɛ2 ) 2 ) and ( λ,2 = ɛ 2 c 2 ± ) 2 9 (α 9 64 ɛ2 ) 2, respectively. The situation for m = 3 is qualitatively the same as the situation for m = 2. The curve on which the equation (4.48) does not have a critical point is however slightly different i.e. α = 9 64 ɛ2 (see Figure 2b ). 4..3 Application of the averaging method to third order. In this subsection the case m = 2, 3 by using third order averaging are investigated. When the averaging method to second order are used to investigate the cases m = 2 and m = 3,
3 two curves are obtained in the α ɛ plane that are α = 6 ɛ2 and α = 9 64 ɛ2 respectively, on which curves the averaged equation does not have critical points. However for m = 3 one can obtain an interesting result when one reduces the order of magnitude of the parameters c (damping) and A (forcing) up to O(ɛ 3 ) i.e. c = ɛ 3 c 3, A = ɛ 3 A 3. It turns out that for the detuning one should consider α = 9 64 ɛ2 + ɛ 3 α 3. As will be shown the curve α = 9 64 ɛ2 will split in two curves α = 9 64 ɛ2 ± 3 52 ɛ3 for c =, defining a domain of instability which has not found by second order averaging and scaling of the parameters. In case m = 2, to eliminate the order ɛ and ɛ 2 effects one sets c = α = c 2 = A = A 2 = and α 2 = 6. Thus the expansion in the series (4.45) up to order ɛ3 are: c = ɛ 3 c 3, α = 6 ɛ2 + ɛ 3 α 3, A = ɛ 3 A 3. (4.49) By substituting (4.49) and (4.43) into (4.4) one obtains after third order averaging: ż = ɛ 3 2 c 3 8 α 3 z + ɛ 3. (4.5) 2 α 3 2 c 3 2 A 3 The equation (4.5) does not have a critical point if and only if c 3 = α 3 =. Thus for m = 2 the curve on which the equation (4.5) does not have a critical point is α = 6 ɛ2 (the same result was obtained by using the averaging method to second order). In order to eliminate order ɛ and ɛ 2 effects in case m = 3 one sets c = α = c 2 = A = A 2 = and α 2 = 9 64. Thus the expansions in the series (4.45) up to order ɛ3 are: c = ɛ 3 c 3, α = 9 64 ɛ2 + ɛ 3 α 3, A = ɛ 3 A 3. (4.5) By substituting (4.5) and (4.43) into (4.4) one obtains after third order averaging: ż = ɛ 3 2 c 3 + 24 8 α 3 z + ɛ 3. (4.52) 2 α 3 2 c 3 24 2 A 3 After rescaling the parameters the determinant of the coefficient matrix B 3 of equation (4.52) becomes: and its eigenvalues are B 3 = 36 (α 9 64 ɛ2 ) 2 4 ( ɛ6 52 2 c2 ), λ,2 = ɛ 3 ( c 2 ± 2 ) ɛ 6 52 2 9 (α 9 64 ɛ2 ) 2. The existence of an isolated critical point of equation (4.52) correspond with the existence of a periodic solution of equation (4.4), and the stability of this periodic solution depend on the eigenvalues of matrix B 3. The equation (4.52) has an isolated critical point if B 3, and does not have a critical point if B 3 =. In the α ɛ plane, B 3 = corresponds with the curve: α = 9 64 ɛ2 ± 3 ɛ6 (52c) 52 2. (4.53) Given c positive, this curve divides the α ɛ plane into two regions, that are region I and II (see Figure 3b ). In region I the real part of the eigenvalues are negative, and in the region II the eigenvalues are real-valued, one positive and one negative. So the periodic solution of (4.4) is stable in region I but unstable in region II. In the case c =, the α ɛ plane is divided into two regions by the curves α = 9 64 ɛ2 ± 3 52 ɛ3 ( see Figure 3a ).
4 ε ε II II I I I α α 3a. c = 3b. c > Figure 3: Stability diagrams of periodic solutions of equation (4.4) for m = 3. In the shaded regions the periodic solutions are unstable. 4.2 THE CASE m ω In case m = ω, the general form of the averaged equation can be written as ż = Cz + b, where b is 2 column vector which depends on the parameter A. However for case m ω the parameter A does not occur in the averaged equation, and the general form of the averaged equation is ż = Cz. Thus the only isolated critical point of this system is origin. This implies that the periodic solution of (4.44) is in the ɛ-neighbourhood of the origin. In other words the amplitude of the periodic solution of equation (4.4) is of order ɛ. The stability diagrams in Fig. - Fig 3. depend on the coefficient matrix C, and on the curves, which separates region I and II, the determinant of the coefficient matrix C is equal to zero. Because in both cases the same coefficient matrix is obtained the stability diagram also applies to the stability of the periodic solution of the inhomogeneous equation. The difference is only the order of magnitude of the amplitude of the periodic solution; in case m = ω the amplitude is O() but in case m ω the amplitude is O(ɛ). 5 Conclusion and Remarks (n + )-th order averaging may be used to find periodic solutions when form n-th order averaging no conclusions about existence of periodic solutions can be drawn i.e. when the first n terms of the averaged equation vanish (identically zero). For linear systems one can put the near identity transformation also in linear form resulting in a relatively simple algorithm. As an example higher order averaging is applied to find the stability diagram of an inhomogeneous second order equation with time-dependent damping coefficient. References [] Batchelor, D. B., Parametric resonance of systems with time-varying dissipation, Appl. Phys. Lett., Vol. 29 (976) pp. 28-28. [2] Ellison, J. A., Saenz, A. W. and Dumas, H. S., Improved N th Order Averaging Theory for Periodic Systems, J. Differential Equation 84 (99) pp. 383-43.
5 [3] Hartono and van der Burgh, A. H. P., Periodic Solutions of an Inhomogeneous Second Order Equation with Time-Dependent Damping Coefficient, in Proceedings of the 8th Biennial ASME Conference, September 9-2, 2 Pittsburgh USA, Symposium on Dynamics and Control of Time-Varying Systems and Structures. [4] Murdock, J. A., Perturbations: theory and method, John Wiley & Sons, Inc., New York, 99. [5] Sanders, J. A. and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag New York Inc., 985. [6] Verhulst, F., Nonlinear Differential Equations and Dynamical Systems, Second Edition, Springer-Verlag New York Inc., 996.