Geometrical Decomposition of Multi-scale Systems

Transcription

1 Geometrical Decomposition of Multi-scale Systems V.A. Sobolev Department of Differential Equations and Control Theory Samara State University Russia Abstract This paper is devoted to the slow/fast decomposition of multi-scale system by using the integral manifolds method Keywords: decomposition, integral manifolds, multi-scale systems, singular perturbation AMS subject classification: Primary 34D15, Secondary 34C45, 34E15, 70K70. This work was written while the author was visitor at the Boole Centre for Research in Informatics, University College Cork. 1

2 Contents 1 Introduction. Elements of the Geometric Theory of Singularly Perturbed Systems Slow integral manifolds Asymptotic representation of integral manifolds Stability of slow integral manifolds Critical cases Singular Singularly Perturbed Systems Existence of slow integral manifold Explicit and Implicit Slow Integral Manifolds Parametric representation of integral manifolds Example High-Gain Control Asymptotics with fractional exponents of small parameter Systems with Slow Dissipation Gyroscopic Systems Precessional and Nutational Motions Vertical gyro with radial corrections Heavy gyroscope A one rigid-link flexible-joint manipulator Branching of slow integral manifolds Formulation of the problem. Preliminaries Quasi-homogeneous degenerate equations Homogeneous systems Quasi-polynomial degenerate equations Cheap control

3 1 Introduction. Elements of the Geometric Theory of Singularly Perturbed Systems 1.1 Slow integral manifolds It is common knowledge that a wide range of processes in various nature is characterized by extreme differences in the rates of variables and the singularly perturbed ordinary differential systems are used as models of such processes [13, 16, 32, 33, 35, 36]). Consider the ordinary differential system dx ε dy = f(x, y, t, ε), = g(x, y, t, ε) (1) with vector variables and and small positive parameter. A usual approach in the qualitative study of (1) is to consider first the degenerate system dx = f(x, y, t, 0), 0 = g(x, y, t, 0) (2) and then to draw conclusions for the qualitative behavior of the full system (1) for sufficiently small ε. A special case of this approach is the quasi steady state assumption. A mathematical justification of that method can be given by means of the theory of integral manifolds for singularly perturbed systems (1) (see e.g. [1, 2, 5, 9, 10, 12, 25, 32, 38, 39]). Integral manifolds method was applied with advantage to a wide range of problems [4, 6, 7, 10, 15, 24, 26, 27, 28, 30, 31, 34]. In order to recall a basic result of the geometric theory of singularly perturbed systems we introduce the following notation and assumptions. The system of equations dx = f(x, y, t, ε) (3) represents the slow subsystem, and the system of equations ε dy = g(x, y, t, ε) (4) the fast subsystem, so it is natural to call (3) the slow subsystem and (4) the fast subsystem of system (1). In the present paper we use a method for qualitative asymptotic analysis of differential equations with singular perturbations. The method relies on the theory of integral manifolds, which essentially 3

4 replaces the original system by another system, on an integral manifold, of dimension equal to that of the slow subsystem. In the zero-epsilon approximation (ɛ = 0) this method leads to a modification of the quasi-steady-state approximation. Recall, that a smooth surface S in R m R n R is called an integral manifold of system (1) if any trajectory of the system that has at least one point in common with S lies entirely in S. Formally, if (x(t 0 ), y(t 0 ), t 0 ) S, then the trajectory (x(t, ε), y(t, ε), t) lies entirely in S. An integral manifold of an autonomous system ẋ = f(x, y, ε), εẏ = g(x, y, ε) (5) has the form S 1 (, ), where S 1 is a surface in the phase space R m R n. The only integral manifolds of system (1) of relevance here are those of dimension m (the dimension of the slow variables) that can be represented as the graphs of vector-valued functions y = h(x, t, ε). We also stipulate that h(x, t, 0) = h (0) (x, t), where h (0) (x, t) is a function whose graph is a sheet of the slow surface, and we assume that h(x, t, ε) is a sufficiently smooth function of ε. In autonomous systems the integral manifolds will be graphs of functions y = h(x, ε). Integral manifolds of the above form are called manifolds of slow motions the origin of this term lies in nonlinear mechanics. An integral manifold may be regarded as a surface on which the phase velocity has a local minimum, that is, a surface characterized by the most persistent phase changes (motions). Integral manifolds of slow motions constitute a refinement of the sheets of the slow surface, obtained by taking the small parameter ε into consideration. The motion along an integral manifold is governed by the equation ẋ = f(x, h(x, t, ε), t, ε). (6) If x(t, ε) is a solution of this equation, then the pair x(t, ε), y(t, ε), where y(t, ε) = h(x(t, ε), t, ε), is a solution of the original system (1), since it defines a trajectory on the integral manifold. Consider the associated subsystem, that is, dy dτ = g(x, y, t, 0), τ = t/ε, treating x, y as parameters. We shall assume that some of the steady states y 0 = y 0 (x, t) of this subsystem are asymptotically stable and that a trajectory 4

5 starting at any point of the domain approaches one of these states as closely as desired as t. This assumption will hold, for example, if the matrix ( g/ y)(x, y 0 (x, t), t, 0) is stable for part of the stationary states and the domain can be represented as the union of the domains of attraction of the asymptotically stable steady states. Let I i be the interval I i := {ε R : 0 < ε < ε i }, where 0 ε i 1, i = 0, 1,.... (A 1 ). f : R m R n R I 0 R m, g : R m R n R I 0 R n are sufficiently smooth and uniformly bounded together with their derivatives. (A 2 ). There are some region G R m and a map h : G R R m of the same smoothness as g such that g(x, h(x, t), t, 0) 0 (x, t) G R. (A 3 ). The spectrum of the Jacobian matrix g y (x, h(x, t), t, 0) is uniformly separated from the imaginary axis for all (x, t) G R. Then the following result is valid (see e.g. [32, 39]): Proposition 1. Under the assumptions (A 1 ) (A 3 ) there is a sufficiently small positive ε 1, ε 1 ε 0, such that for ε I 1 system (1) has a smooth integral manifold M ε with the representation M ε := {(x, y, t) R n+m+1 : y = ψ(x, t, ε), (x, t) G R.} Remark. The global boundedness assumption in (A 1 ) with respect to (x, y) can be relaxed by modifying f and g outside some bounded region of R n R m. 1.2 Asymptotic representation of integral manifolds When the method of integral manifolds is being used to solve a specific problem, a central question is the calculation of the function h(x, t, ε) in terms of which manifold is described. Exact calculation is generally impossible, and various approximations are necessary. One possibility is asymptotic expansion of h(x, t, ε) in powers of the small parameter: h(x, t, ε) = h 0 (x, t) + εh 1 (x, t) ε k h k (x, t) (7) Substituting this formal expansion in equation ε h t + ε h f(x, h(x, t, ε), t, ε) = g(x, h, ε). (8) x 5

6 we obtain the relationship ε ε k h k k 0 t + ε ε k h k k 0 x f(x, ε k h k, t, ε) = k 0 Use the formal asymptotic representations = g(x, k 0 ε k h k, t, ε). (9) f(x, ε k h k, t, ε), g(x, ε k h k, t, ε), k 0 k 0 f(x, ε k h k, t, ε) = ε k f (k) (x, h 0,..., h k 1, t), k 0 k 0 g(x, ε k h k, t, ε) = B(x, t) ε k h k + ε k g (k) (x, h 0,..., h k 1, t). k 0 k 1 k 1 It was taken into account for g that g(x, h (0) (x, t), t, 0) = 0. The notion B(x, t) here is used for t he matrix ( g/ y)(x, h 0, t, 0). Substituting this formal expansions in (9) and equating equal powers of ε, we obtain Since B is invertible h k 1 t + 0 p k 1 h k = B 1 [g (k) h k 1 t h p x f (k 1 p) = Bh k + g (k). 0 p k 1 h p x f (k 1 p) ]. (10) Note, that at first asymptotic expansions of slow integral manifolds were used in [27, 30, 31] and for systems with several small parameters were obtained in [22]. 1.3 Stability of slow integral manifolds In applications it is often assumed that the spectrum of the Jacobian matrix g y (x, h(x, t), t, 0) is located in the left half plane. Under this additional hypothesis the manifold M ε is exponentially attracting for ε I 1. In this case, the solution x = x(t, ε), y = y(t, ε) of the original system satisfied the initial condition x(t 0, ε) = x 0, y(t 0, ε) = y 0 can be represented as x(t, ε) = v(t, ε) + εϕ 1 (t, ε), y(t, ε) = ȳ(t, ε) + ϕ 2 (t, ε). (11) 6

7 There exists a point v 0 which is the initial data for a solution v(t, ε) of the equation v = f(v, h(v, t, ε), t, ε); the functions ϕ 1 (t, ε), ϕ 2 (t, ε) are small corrections that determine the degree to which trajectories passing near the manifold tend asymptotically to the corresponding trajectories on the manifold as t increases. They satisfy the following inequalities: ϕ i (t, ε) N y 0 h(x 0, t 0, ε) exp[ β(t t 0 )/ε], i = 1, 2, (12) for t t 0. From (11) and (12) we obtain the following reduction principle for a stable integral manifold defined by a function y = h(x, t, ε): a solution x = x(t, ε), y = h(x(t, ε), t, ε) of the original system (1) is stable (asymptotically stable, unstable) if and only if the corresponding solution of the system of equations v = F (v, t, ε) on the integral manifold is stable (asymptotically stable, unstable). Lyapunov reduction principle was extended on ordinary differential systems with Lipschitz right hand sides by Pliss [18], and on singularly perturbed systems in [29, 32]. Thanks to the reduction principle and the representation (11), the qualitative behavior of trajectories of the original system near the integral manifold may be investigated by analyzing the equation on the manifold. 1.4 Critical cases The case that assumption (A 3 ) is violated is called critical. We distinguish following subcases: 1. The Jacobian matrix g y (x, y, t, 0) is singular on some subspace of R m R n R. In that case, system (1) is referred to as a singular singularly perturbed system. This subcase has been treated in [8, 11, 20, 35]. In this case it is possible to introduce new variables in such a way that the transformed differential system has the structurally hyperbolic fast subsystem of lower dimension. It means that the original differential system has the slow integral manifold of higher dimension. This case and an application to a high-gain control problem [26] is investigated in the next Section. 2. The Jacobian matrix g y (x, y, t, 0) has eigenvalues on the imaginary axis with nonvanishing imaginary parts. A similar case has been investigated in [21, 23, 27, 32]. If the part of eigenvalues are pure imaginary but under taking into account the perturbations of order they move to the left complex half-plane, then the system under consideration has the stable slow integral manifolds. Some problems of mechanics of gyroscopes and manipulators with high-frequency and weakly damped transient regimes are discussed in the Section 3. 7

8 3. The Jacobian matrix g y (x, y, t, 0) is singular on the set M 0 := {(x, y, t) R m R n R : y = h(x, t), (x, t) G R}. In that case, y = h(x, t) is generically an isolated root of g = 0 but not a simple one. This case will be studied in the Section 4. 2 Singular Singularly Perturbed Systems Consider the system εż = Z(z, t, ε), z R m+n, t R, (13) where 0 ε 1, the vector-function Z is sufficiently smooth. Suppose that the limit system Z(z, t, 0) = 0 (ε = 0) has a family of solutions z = ψ(v, t), v R m, t R, (14) with sufficiently smooth vector=function ψ. We try to find a slow integral manifolds z = P (v, t, ε), (15) a flow on which is described by the equation v = Q(v, t, ε). (16) We shall restrict our consideration to smooth integral surfaces situated in ε-neighborhood of the slow surface z = ψ(v, t), i. e. P (v, t, 0) = ψ(v, t), a motions of which is described by differential equations of form (16) with smooth r.h.s. The equation (13) describes motions with speeds of order O(ε 1 ), and (16) describes motions with speeds of order O(1). Thus, the integral manifold (15) is a manifold of slow motions or a slow integral manifold. Consider the singularly perturbed system ẋ = X(x, y, t, ε), x R m, t R, (17) εẏ = Y (x, y, t, ε), y R n. (18) If the equation Y (x, y, t, 0) = 0 has the isolated solution y = ϕ(x, t) (19) 8

9 and det B(x, t) 0, B(x, t) = Y y (x, ϕ, t, 0), then under some additional conditions on eigenvalues of B(x, t) the system (17), (18) has in ε neighborhood of the slow surface (19) the slow integral surface the motion on which is described by the equation y = h(x, t, ε), (20) ẋ = X(x, h(x, t, ε), t, ε). (21) For the system (17), (18) the role of v plays the vector x, the integral manifold (15) describes by the equation (20), and the equation (16) takes the form (21). 2.1 Existence of slow integral manifold Suppose the following hypotheses hold: the rank of matrix ψ v (v, t) is equal to m; the rank of matrix A(v, t) = Z z (ψ(v, t), t, 0) is equal to n; the matrix A(v, t) has m fold eigenvalue and n other eigenvalues λ i (v, t) satisfy the inequality Reλ i (v, t) 2α < 0, t R, v R m. (22) Differentiation with respect to v of Z(ψ(v, t), t, 0) = 0 gives Z z (ψ(v, t), t, 0)ψ v (v, t) = 0, or A(v, t)ψ v (v, t) = 0. The last equality means that (m + n) (m + n) matrix A(v, t) possesses m linear independent eigenvectors, which are columns of ψ v (v, t))), corresponded to multiple zero eigenvalue. Let D1 T be (m + n) n matrix, the columns of which give the base of A kern, and D2 T is such (m+n) m matrix, that (D1 T, D2 T ) is the regular matrix. Then or DA = A T (D T 1 D T 2 ) = (0 B T ), ( ) 0, D = B ( ) D1 Thus, the left multiplying of A on the regular matrix D intercepts the zero m (m + n) block and the regular n (m + n) block of B. The rank of B is equal to n. Consequently, without the loss of generality the system (13) may be considered of form D 2 εẋ = f 1 (x, y 2, t, ε), x R m, (23). 9

10 εẏ 2 = f 2 (x, y 2, t, ε), y 2 R n, (24) and the following conditions take place 1. The equation f 2 (x, y 2, t, 0) = 0 has a smooth isolated root y 2 = ϕ(x, t) with x R m, t R, and f 2 (x, ϕ(x, t), t, 0) = The Jacobian matrix ( ) f1x f A(x, t) = 1y2 f 2x f 2y2 y2 =ϕ(x,t),ε=0 on the surface y 2 = ϕ(x, t) has m fold zero eigenvalue and m dimensional kern, and the matrix B(x, t) = f 2y2 (x, ϕ(x, t), t, 0) has n eigenvalues satisfying (22)under v = x. 3. In the domain Ω = {(x, y 2, t, ε) x R m, y 2 ϕ(x, t) p, t R, 0 ε ε 0 } the functions f 1, f 2 and the matrix A continuously differentiable (k + 2) times (k 0). Using the change of variable y 2 = y 1 + ϕ(x, t) in (47), (48), we obtain the following equations for x and y 1 εẋ = C(x, t)y 1 + F 1 (x, y 1, t) + εx(x, y 1, t, ε), (25) where εẏ 1 = B(x, t)y 1 + F 2 (x, y 1, t) + εy (x, y 1, t, ε), (26) C(x, t) = f 1y2 (x, ϕ(x, t), t, 0), B(x, t) = f 2y2 (x, ϕ(x, t), t, 0), F 1 (x, y 1, y) = f 1 (x, y 1 + ϕ(x, t), t, 0) C(x, t), F 2 (x, y 1, t) = f 2 (x, y 1 + ϕ(x, t), t, 0) B(x, t), εx(x, y 1, t, ε) = f 1 (x, y 1 + ϕ(x, t), t, ε) f 1 (x, y 1 + ϕ(x, t), t, 0), εy (x, y 1, t.ε) = f 2 (x, y 1 + ϕ(x, t), t, ε) f 2 (x, y 1 + ϕ(x, t), t, 0). Note that the vector-functions F i (i = 1, 2) satisfy relations F i (x, y 1, t) = O( y 1 2 ). Thus, ε 1 F i (x, εy, t) are continuous in Ω. We now have the following theorem. Theorem 2.1 Let the assumptions 1-3are true, then there exists ε 1, 0 < ε 1 < ε 0, such that for any ε (0, ε 1 ) the system (25), (26) possesses a unique 10

11 slow integral manifold y 1 = εp(x, t, ε). On this manifold the flow of the system is governed by the equation ẋ = X 1 (x, t, ε), where X 1 (x, t, ε) = C(x, t)p(x, t, ε) + X(x, εp, t, ε) + ε 1 F 1 (x, εp, t), and the function p(x, t, ε) is k times continuously differentiable with respect to x and t. Note that the change of variable y 1 = εy leeds the system (25), (26) to the form (17), (18)and the role of variable v plays x. 2.2 Explicit and Implicit Slow Integral Manifolds To describe slow integral manifolds for systems like (17), (18) the explicit representation (20) is usual. In this case the approximation of h(x, t, ε) may be obtained as asymptotical expansion in power of ε. In general, it is impossible to solve for the function h(x, t, 0) from the equation Y (x, y, t, 0) = 0 exactly. In this case the slow integral manifold can be obtained in an implicit form. The flow on the slow integral manifold as a zero approximation is governed by the following differential-algebraic system: ẋ = X(x, y, t, 0), (27) 0 = Y (x, y, t, 0). (28) To obtain the first approximation, it is necessary to differentiate Y (x, y, t, ε) by virtue of (17), (18) ε d Y = Y yy + εy t + εy x X. As a first approximation, the flow on the slow integral manifolds is governed by the differential-algebraic system ẋ = X(x, y, t, ε), (29) Y y Y + εy t + εy x X = 0, (30) where the terms of order o(ε) can be neglected. To obtain the second order approximation, it is necessary to differentiate Y (x, y, t, ε) twice by virtue of (17), (18). Unfortunately, the corresponding relationships are too cumbersome. Because of this, we consider the case of 11

12 autonomous systems. Then the second order approximation take the form of (29) and Y + ε(y y ) 1 Y x X+ +ε 2 Yy 2 {Y x X x + Y xx X Y xy (Y y ) 1 Y x X Y x X y (Y y ) 1 Y x Y yx X(Y y ) 1 Y x + Y yy (Y y ) 1 Y x X(Y y ) 1 Y x X} = 0. (31) In (29), (30) all terms multiplying by the second power of ε can be neglected. To obtain the k th order approximation, it is necessary t6o differentiate Y (x, y, t, ε) k times with respect to t by virtue of (17), (18). To ensure these formulae it is sufficient to note that under the calculation h(x, t, ε) as asymptotical expansions h = h 0 + O(ε), h = h 0 + εh 1 + O(ε 2 ), h = h 0 + εh 1 + ε 2 h 2 + O(ε 3 ) the using (27) (31) gives the same result as the using of (10). To illustrate this, consider as example the following system ẋ = y, εẏ = x 2 + y 2 a, a > 0. The first approximation of slow integral manifold is y 2 + x 2 a + εx = 0. It is easy to check that the second order approximation y 2 + (x + ε/2) 2 = a ε 2 /4 gives the exact equation for this manifold. 2.3 Parametric representation of integral manifolds The implicit form of integral manifolds has evident disadvantages, but on numerous problems it is impossible to find a solution of Y (x, y, t, 0) = 0 in the explicit form y = ϕ(x, t), but sometimes the solution of Y (x, y, t, 0) = 0 can be found as parametric function x = χ 0 (v, t), y = ϕ 0 (v, t), (32) where v R m, and the following identity is true Y (χ 0 (v, t), ϕ 0 (v, t), t, 0) 0, t R, v R m. (33) In this case the slow integral manifold may be found in parametric form x = χ(v, t, ε), y = ϕ(v, t, ε), (34) 12

13 where t R, v R m, χ(v, t, 0) = χ 0, ϕ(v, t, 0) = ϕ 0. The flow on the manifold is governed by the equation v = F (v, t, ε), (35) the function F (v, t, ε) will be determined below. The functions χ, ϕ, F can be found as asymptotical expansions χ(v, t, ε) = χ 0 (v, t) + εχ 1 (v, t) ε k χ k (v, t) +..., ϕ(v, t, ε) = ϕ 0 (v, t) + εϕ 1 (v, t) ε k ϕ k (v, t) +..., F (v, t, ε) = F 0 (v, t) + εf 1 (v, t) ε k F k (v, t) +... (36) in line with (35) from the equations χ t + χ F = X(χ, ϕ, t, ε), (37) v ε ϕ t + ε ϕ = Y (χ, ϕ, t, ε). (38) v Equating the coefficients under the same powers of the small parameter we obtain χ 0 t + χ 0 v F 0 = X(χ 0, ϕ 0, t, 0), Y (χ 0, ϕ 0, t, 0) = 0, χ 1 t + χ 1 v F 0 + χ 0 v F 1 = X x (χ 0, ϕ 0, t, 0)χ 1 + +X y (χ 0, ϕ 0, t, 0)ϕ 1 + X 1, ϕ 0 t + ϕ 0 v F 0 = Y x (χ 0, ϕ 0, t, 0)χ 1 + Y y (χ 0, ϕ 0, t, 0)ϕ 1 + Y 1, X 1 = X ε (χ 0, ϕ 0, t, 0), Y 1 = Y ε (χ 0, ϕ 0, t, 0). The relationships (37), (38) contain unknown functions χ, ϕ, F. According to concrete problem it is possible to consider one of these functions or any m scalar components of χ, ϕ and F as known functions, and all others may be found from (37), (38). Moreover, it is possible at any step of calculations of coefficients in (36) to choose any m components of these coefficients as given functions. In the case that F is given function, the equations (36), (37) are used to calculate the coefficients of asymptotical expansions of χ and ϕ. If it is possible to predeterminate the function χ, then these equations give the feasibility to calculate F and ϕ. Note that in the case of explicit form y = h(x, t, ε) v = x, χ = v, ϕ = h(v, t, ε), F = X(v, h(v, t, ε), t, ε), 13

14 and (36) takes the form ε h t + ε h X(v, h, t, ε) = Y (v, h, t, ε), v h = h(v, t, ε). If dim x = dim y and the role of v plays y, then ϕ = v and χ t + χ F = X(χ, v, t, ε), εf = Y (χ, v, t, ε). (39) v It immediately follows the equation for χ ε χ t + χ Y (χ, v, t, ε) = εx(χ, v, t, ε), (40) v whence, under the assumption det( χ 0 ) 0 it is possible to calculate χ as v asymptotic expansion. Note that Y (χ 0, ϕ 0, t, 0) = 0 means that the equation (35) is regularly perturbed. Turning back to the system (13), we use the p-0arametric form to describe the slow integral manifolds and the reduced differential equation z = P (v, t, ε), v = Q(v, t, ε). (41) The functions P and Q will be found as asymptotical expansions P (v, t, ε) = P 0 (v, t) + εp 1 (v, t) ε k P k (v, t) +..., Q(v, t, ε) = Q 0 (v, t) + εq 1 (v, t) ε k Q k (v, t) (42) The differentiation of P with respect to t by virtue of (13), (16) gives ε P t + ε P Q = Z(P, t, ε). (43) v Write the Taylor series expansion of Z(P, t, ε) about ε = 0 Z(P, t, ε) = Z(P 0, t, 0) + εz 1 (P 0, P 1, t) and represent Z k (K 1) of form +ε k Z k (P 0, P 1,..., P k, t)..., Z k (P 0,..., P k, t) = Z 2 (P 0, t, 0)P k + R k (P 0, P 1,..., P k 1, t). In particular, Z 1 (P 0, P 1, t) = Z 2 (P 0, t, 0)P 1 + Z ε (P 0, t, 0). Equating equal powers of ε, we obtain from (43) under ε = 0 Z(P 0, t, 0) = 0. 14

15 In keeping with (14), let P 0 (v, t) = ψ(v, t). Using the following notion we obtain under first power of ε A(v, t) = Z z (ψ(v, t), t, 0), ψ t + ψ v Q 0 = AP 1 + R 1. (44) The equation (44) contains two unknown functions P 1 and Q 0. With respect to P 1 the equation (44) may be considered as nonhomogeneous linear algebraic system with singular matrix deta(v, t) 0, v R m, t R. Thereby, Q 0 is required to provide the solvability of the system. It is apparent that we have some a freedom in choosing in the design of Q 0 and P 1. To determinate this functions uniquely, multiply left the equation (44) by the matrix D, introduced in subsection 2.1, and obtain ψ D 1 t + D ψ 1 v Q 0 = D 1 R 1, (45) ψ D 2 t + D ψ 2 v Q 0 = BP 1 + D 2 R 1. (46) It it is granted additionally that the matrix is invertible, then (45) gives D 1 = ψ/ v Q 0 = (D 1 ψ v ) 1 D 1 (R 1 ψ t ), and this permits to determinate uniquely P 1 from (46): P 1 = B 1 D 2 (ψ t + ψ v Q 0 R 1 ). The design of following pairs of coefficients P k, Q k 1 is carry out by the same way. Equating coefficients under ε k, we obtain the equation P k 1 t + ϕ k 1 v Q P i k 1 + i=1 v Q k i 1 = AP k + R k, which is decomposed into two equations 15

16 ψ (D 1 v )Q k 1 + D 1 ( P k 1 k 1 P i + t i 1 v Q k i 1) = D 1 R k, D 2 [ P k 1 t + ψ k 1 v Q P i k 1 + i=1 v Q k i 1] = BP k + D 2 R k, by multiplying left on D. The last equations give Q k 1 = (D 1 ψ v ) 1 D 1 [R k k 1 i=1 P i v Q k i 1 P k 1 ]; t P k = B 1 D 2 [ P k 1 t + ψ v Q k 1 + k 1 i=1 P i v Q k i 1 R k ] Example As a very simple example consider the system εẋ = f(x, y 2, t, ε) + εf 1 (x, y 2, t, ε), (47) Introducing a new variable εẏ = kf(x, y 2, t, ε) + εf 2 (x, y 2, t, ε), (48) x 1 = y kx, we obtain the following differential equation for a slow variable ẋ 1 = f 2 (x, y 2, t, ε) kf 1 (x, y 2, t, ε). To obtain the full system, it is possible to use either of two equations for x and y as a fast equation. 2.4 High-Gain Control Consider the control system ẋ = f(x) + B(x)u, x(0) = x 0, where x R n, u R r, t 0. The vector function f and the matrix function B are taken to be sufficiently smooth and bounded. The control vector u is to be selected in such a way as to transfer the vector x from x = x 0 to a sufficiently 16

17 small neighborhood of a smooth m dimensional surface S(x) = 0. Commonly employed feedback control is u = 1 ε KS(x), where K is a constant r m matrix and ε is a small positive parameter, see [37]. Suppose that we can choose the matrix K in such a way that the matrix N(x, t) = GBK is stable and invertible and introduce the new variable y = S(x), then x and y satisfy the system εẋ = εf(x) B(x)Ky, x(0) = x 0 εẏ = εg(x)f(x) G(x)B(x)Ky, y(0) = y 0 = S(x 0 ), where G(x) = S/ x. The reduced (ε = 0) algebraic problem possesses n parametric family of solutions x = v, y = 0. The role of A plays the singular matrix ( ) 0 B1 K. 0 N The last singular singularly perturbed differential system possesses n dimensional slow integral manifold x = v, y = εn 1 (v, t)g(v)f(v, t) + O(ε 2 ). The flow on the manifold is governed by Introduce the new variables v = [I B 1 (v, t)kn 1 (v, t)g(v)]f(v, t) + O(ε 2 ). x = v + B 1 (v, t)kn 1 (v, t)z; y = z + εn 1(x, t)g(x)f(x, t). Then for v, z we obtain the equations v = (I B 1 KN 1 G)f + O(ε), εż = (N + O(ε))z. It is clear now that the representations x = v + O(ε vt/ε ), y = εϕ(v, t, ε) + O(ε vt/ε ), ϕ = N 1 Gf + O(ε) take place under v > 0 for all t > 0. Thus, under the given control low u = 1 ε KS(x), 17

18 the trajectory very quickly attains the ε neighborhood of S(x) = 0. It is easy to see that the modified control u = 1 ε K [ S(x) εn 1 (x, t)g(x)f(x, t) ], with the stable matrix N(x, t) = GBK, is more preferable because it guides the trajectory of x in a time t to e ν tε 1 neighborhood of S(x) = 0. Under this control for the variable x we obtain the equation εẋ = ε [ I B(x, t)k(gbk) 1 G(x) ] f(x, t) B 1 (x, t)ks(x), and for the variable y = S(x) we obtain the equation εẏ = N(x, t)y, y = O ( e νε 1 t ), ν > 0, t > 0, ε Asymptotics with fractional exponents of small parameter Firstly consider cases when some assumptions of (A1 A3) break down. Thus, for the system εẋ 1 = εf 1 (x 1, x 2, x 3, t, ε), εẋ 2 = εf 2 (x 1, x 2, x 3, t, ε), with A = A(x 1, t) εẋ 3 = D(x 1, t)x 2 + εf 3 (x 1, x 2, x 3, t, ε), (49) x i R n i, i = 1, 2, 3, A = D 0 the multiplicity n 1 + n 2 + n 3 = n of zero eigenvalue of A(v, t) is grater the the number of corresponding eigenvectors n k (k > 0) under the condition D 0. Using of new variable x 2 = ε 1/2 x 2, we obtain the system ẋ = f 1 (x 1, εx 2, x 3, t, ε), εẋ2 = f 2 (x 1, εx 2, x 3, t, ε), εx3 = D(x 1, t)x 2 + εf 3 (x 1, εx 2, x 3, t, ε). (50) In this case the slow integral manifold can be found using the asymptotic expansions in power of ε, i. e. we need to use the fractional power of ε. Let x 3 = ϕ(x 1, t) be the isolated root of f 2 (x 1, 0, x 3, t, 0) = 0. 18

19 The last differential system has the form (17), (18) with a small parameter ε. The role of B plays the matrix B = ( O C(x 1, t) D(x 1, t) O ), where C(x 1, t) = f 2x2 (x 1, 0, ϕ(x 1, t), t, 0). If detb 0, then the following situations are most typical : The eigenvalues of B have nonzero real parts and (50) possesses a conditionally stable slow integral manifold (the usual situations for optimal control problems [10, 24, 25, 26]). The eigenvalues of B are pure imaginary, such systems appear under the modelling of gyroscopic systems and double spin satellites [27, 30, 31, 32]. Asymptotical expansions with fractional exponents of small parameter will be used in Section 4. 3 Systems with Slow Dissipation Nonsingular matrix with eigenvalues situated on the imaginary axis and problems of mechanics. Let the part of eigenvalues are pure imaginary but under taking into account the perturbations of order they move to the left complex half-plane. In this case the considered system has the stable slow integral manifolds. Some important problems of mechanics of gyroscopes, satellites and manipulators with high-frequency and weakly damped transient regimes were studied in [21, 23, 27, 32]. 3.1 Gyroscopic Systems dx = y, ε d (Ay) = (G + εb)y + εr + εq, R = 1 2 [ ] T (Ay) y. (51) Here x n, A(t, x) is a symmetric positive definite matrix, G(x, t) is a skewsymmetric matrix of gyroscopic forces, B(x, t) is a symmetric positive definite matrix of dissipative forces, Q(x, t) is a vector of generalized forces, ε = H 1 is a small positive parameter. The precessional equations take the form (G + εb) dx x = εq. (52) 19

20 The equations (52) are obtained from (119) by neglecting only part of terms multiplying by the small parameter. All roots of characteristic equation det(g λi) = 0 are situated on the imaginary axis and the main assumption of Tikhonov theorem [33] is violated. To justify a permissibility of precessional equations we use the integral manifolds method. 3.2 Precessional and Nutational Motions Return to the system (119). The initial variables x and y are connected with new slow variable v and new fast variable z by following relations x = v + εh(v, z, t, ε), (53) y = z + εh(x, t, ε) = z + εh(v + εh(v, z, t, ε), t, ε). (54) Differential equations for v and z are dv = εh(v, t, ε), (55) ε d (Az) = (G + εb)z + εr(v, z, t) + ε2 R 1 (v, z, t, ε). (56) A = A(v + εh, t), B = B(v + εh, t), G = G(v + εh, t), h = h(v + εh, t, ε), H = H(v, z, t, ε), R 1 = P (v + εh, z, t, ε), P (x, y, t, ε) = 1 2 ( ) T A x y + 1 ( ) T A 2 x h y (Ah) x y. The following asymptotic formulae are true h(x, t, ε) = h 1 (x, t) + εh 2 (x, t) + O(ε 2 ), H(v, z, t, ε) = H 1 (v, z, t) + εh 2 (v, z, t) + O(ε 2 ), h 1 = G 1 Q, h 2 = G [ ] 1 Bh 1 + (Ah 1) t, H 1 = G 1 Az, H 2 = [ G 1 A G 1 B ] G 1 Az + O( z 2 ). t In expressions for h 1, h 2 matrices A, B, G and function Q depend on x and t, and in expressions for H 1, H 2 this functions and matrices depend on v and t. The equation (55) describes slow precessional motions of gyroscopic system, and the equation (56) describes fast nutational motions. The formula (53) shows that the vector of generalized coordinates x is a superposition of precessional and nutational motions. By the way, the original equations are split into two subsystems (55), (56) by the use of (53), (54). As this takes place, the approximate presentations for h and H were obtained. 20

21 Let we consider an initial value problem for (119) with initial conditions x(t 0 ) = x 0, y(t 0 ) = εy 0, then z(t 0 ) = εz 0 = ε(y 0 h(x 0, t 0, ε)), and an initial value v(t 0 ) = v 0 for the equation of precessional motions (55) is found from the equation x 0 = v 0 + εh(v 0, εz 0, t 0, ε) as an asymptotic expansion v 0 = x 0 + ε 2 G 1 (x 0, t 0 )[y 0 G 1 (x 0, t 0 )Q(x 0, t 0 )] + O(ε 3 ). Turning back to problem of precessional theory justification it is possible to see that the r.h.s. of precessional equations (52) represented of form v = ε[g(v, t) + εb(v, t)] 1 Q(v, t), coincides with the r.h.s. of (55) with an accuracy of order O(ε) inclusive in the case of nonautonomous system, and with an accuracy of order O(ε 2 ) in the case of autonomous one. Taken into account that for the system under consideration, nutation oscillations are quench, it may be interred that the using of precession equations is justified. Present without proof some results from [21] devoting to investigations of integral manifolds in gyroscopic type systems. Consider the following ordinary differential equations system da = Aa + P 1y + P 2 z + Q 1 (t, a, x, y, z, ɛ); (57) dx = Bx + P 3y + P 4 z + Q 2 (t, a, x, y, z, ɛ); (58) ɛ dy = C(ɛ)y + ɛδ 1P 5 z + ɛq 3 (t, a, x, y, z, ɛ); (59) ɛ dz = Dz + δ 2P 6 y + Q 4 (t, a, x, y, z, ɛ), (60) where Q 1, a R k ; Q 2, x R l ; Q 3, y R m ; Q 4, z R n ; A, B, C(ɛ), D, P i (i = 1..6) are the matrices of corresponding dimensions; ɛ is a small positive parameter, C(ɛ) = C 0 +ɛc 1, δ 1 δ 2 = 0. Let denotes a norm in finite-dimensional space. Suppose the following assumptions hold. 1. Functions Q i (i = 1..4) in Ω = { < t <, a R k, x r 1, y r 2, z r 3, 0 ɛ ɛ 0 }, are continuous and satisfy Q i (t, a, 0, 0, 0, ɛ) < M and the Lipschitz condition with the constant λ with respect a, x, y, z, where M and λ are sufficiently small positive numbers; 2. Eigenvalues λ i of A, B, C(ɛ), D satisfy: 21

22 (a) A Reλ i = 0, (i = 1..k); (b) B Reλ i < α 1 < 0, (i = 1..l); (c) C(ɛ) Reλ i (ɛ) < ɛα 2 (α 2 > 0), (i = 1..m); (d) D Reλ i < α 3 < 0, (i = 1..n); during which e At Ke α t /2 ; e Bt Ke αt ; e C(ɛ)t/ε Ke αt ; e D/(ɛ)t Ke αt/ɛ with some K, α > 0 and any t > 0. As usual, by an integral manifold of (57) (60) is meant some set S in R 1 R k R l R m R n, consisting of trajectories of this system. The integral manifold x = f(t, a, ɛ), y = g(t, a, ɛ), z = h(t, a, ɛ) is called (L, ) - manifold if norms of functions f, g, h under all t, a, ɛ from Ω are bounded by L and these functions satisfy the Lipschitz condition with respect to a with a constant. Theorem 3.1 Let the conditions 1 and 2 are true. There exist such L 0 > 0, 0 > 0, that for all L < L 0, < 0 with sufficiently small M, λ, ɛ the system (57) (60) possesses a unique (L, ) - integral manifold, given by equations x = f(t, a, ɛ), y = g(t, a, ɛ), z = h(t, a, ɛ). The behaviour of (57) (60) on this manifold is described by the equation da = Aa + P 1g(t, a, ɛ) + P 2 h(t, a, ɛ) + Q 1 (t, a, f(t, a, ɛ), g(t, a, ɛ), h(t, a, ɛ), ɛ) (61) Let a(t), x(t), y(t), z(t) are solutions of (57) (60), satisfying the conditions a(t 0 ) = a 0, x(t 0 ) = t 0, y(t 0 ) = y 0, z(t 0 ) = z 0. The following statement characterises a stability of an integral manifold S. Theorem 3.2 If the system (57) (60) satisfy 1 2, then there exist such positive numbers N, γ, that under t > t 0 the following inequality are true x(t) f(t, a(t), ɛ) Nη 0 e γ(t t 0), y(t) g(t, a(t), ɛ) Nη 0 e γ(t t 0), z(t) h(t, a(t), ɛ) Nη 0 e γ ɛ (t t 0), where η 0 = x 0 f(t 0, a 0, ɛ) + y 0 g(t 0, a 0, ɛ) + z 0 h(t 0, a 0, ɛ), and f, g, h are the functions defined in 3.1. Establish asymptotical properties of f, g, h, by assuming that, C 0 is regular matrix, A is similar to a diagonal matrix and functions Q i (i = 1..4) have bounded partial derivatives with respect to all variables until to (r + 1) order. 22

23 Lemma 1 If conditions of the theorem 3.1 and the inequalities hold Q i (t, a, 0, 0, 0, ɛ) ɛ r+1 M, (i = 2..4), then norms of functions f, g, h are bounded by the value ɛ r+1 L. Introduce new variables in (57) (60): x = x 1 + f 0 (t, a) + ɛf 1 (t, a) ɛ r f r (t, a), (62) y = y 1 + g 0 (t, a) + ɛg 1 (t, a) ɛ r g r (t, a), (63) z = z 1 + h 0 (t, a) + ɛh 1 (t, a) ɛ r h r (t, a) (64) Substituting expressions (62) (64) into (57) (60) and equating x 1, y 1, z 1 to zero, we obtain the formal equalities for f i, g i, h i, (i = 1..r). These equalities with an accuracy of terms multiplying by ɛ to the power r + 1 may be represented of form r ɛ i f i r i=0 t + ɛ i f ( i r r Aa + P1 ɛ j g j + P 2 ɛ j h j + i=0 a j=0 j=0 r r r +Q 1 (t, a, ɛ j f j, ɛ j g j, ɛ j h j, ɛ) ) r B ɛ i f i + j=0 j=0 j=0 i=0 r r r r r +P 3 ɛ i g i + P 4 ɛ i h i + Q 2 (t, a, ɛ i f i, ɛ i g i, ɛ i h i, ɛ), (65) i=0 i=0 i=0 i=0 i=0 ɛ ( r ɛ i g i r ɛ i g ( i r r Aa + P1 ɛ j g j + P 2 ɛ j h j + i=0 t i=0 a j=0 j=0 r r r +Q 1 (t, a, ɛ j f j, ɛ j g j, ɛ j h j, ɛ) )) r r C(ɛ) ɛ i g i + ɛδ 1 P 5 ɛ j h j + j=0 j=0 j=0 i=0 j=0 r r r +ɛq 3 (t, a, ɛ j f j, ɛ j g j, ɛ j h j, ɛ) )) (66) j=0 j=0 j=0 ɛ ( r ɛ i h i r ɛ i h ( i r r Aa + P1 ɛ j g j + P 2 ɛ j h j + i=0 t i=0 a j=0 j=0 r r r +Q 1 (t, a, ɛ j f j, ɛ j g j, ɛ j h j, ɛ) )) r r D ɛ i h i + δ 2 P 6 ɛ j g j + j=0 j=0 j=0 i=0 j=0 r r r +Q 4 (t, a, ɛ j f j, ɛ j g j, ɛ j h j, ɛ) )) (67) j=0 j=0 j=0 Expanding all functions in (65) (67) into formal asymptotic series in powers of a small parameter and equating coefficients under the same powers of ɛ, we obtain equations which allow to find f i, g i, h i, (i = 1..r). 23

24 The differential system for new variables a, x 1, y 1, z 1, introduced by (62) (64), possesses an integral manifold describing by functions f r+1 (t, a, ɛ), g r+1 (t, a, ɛ), h r+1 (t, a, ɛ) and norm of these functions are bounded by the value ɛ r+1 L. Turning back to variables a, x, y, z we see that the system (57) (60) possesses the integral manifold x = f(t, a, ɛ), y = g(t, a, ɛ), z = h(t, a, ɛ), representing of form: f(t, a, ɛ) = f 0 (t, a) + ɛf 1 (t, a) ɛ r f r (t, a) + ɛ r+1 fr+1 (t, a, ɛ), (68) g(t, a, ɛ) = ɛg 1 (t, a) ɛ r g r (t, a) + ɛ r+1 g r+1 (t, a, ɛ), (69) h(t, a, ɛ) = h 0 (t, a) + ɛh 1 (t, a) ɛ r h r (t, a) + ɛ r+1 hr+1 (t, a, ɛ), (70) where ɛ r+1 fr+1 = f r+1 (t, a, ɛ), ɛ r+1 g r+1 = g r+1 (t, a, ɛ), ɛ r+1 hr+1 = h r+1 (t, a, ɛ). Thus, the following statement takes place Theorem 3.3 If (1) (2) hold and moreover, functions Q i (i = 1..4) have bounded partial derivatives with respect to all variables to r + 1 order, A is similar to a diagonal matrix and det C 0 0,, then there exist such numbers M 1 (M 1 < M 0 ), λ 1 (λ 1 < λ 0 ), ɛ 2 (ɛ 2 < ɛ 1 ), that for all M < M 1, λ < λ 1, 0 < ɛ < ɛ 2 the functions f, g, h, in Theorem 3.1, have bounded partial derivatives with respect to a until rth order and can be represented in the form (68) (70). Coefficients of expansions (68) (70) are uniquely determined by (65) (67). More general results concerning with gyroscopic systems may be found in [4, 32]. 3.3 Vertical gyro with radial corrections The equations of small oscillations of gyroscopic system about the equilibrium position have the form Aẍ + (HG + B)ẋ + Cx = 0. (71) where A is a symmetric positive-definite matrix of inertia, G is a skew-symmetric positive-definite matrix of gyroscopic forces, B is a symmetric positive-definite matrix of dissipative forces, C is the matrix of potential and nonconservative forces, H the gyroscope angular momentum, thus we let ε = H 1 is a small positive parameter. If G is non-singular matrix, the characteristic roots of this linear autonomous system break down into groups of roots of order O(ε) and order O(1/ε). In those cases in which the roots of order O(1/ε) lie in the left half-plane, we can set up a slow invariant manifold for which the reduction principle valid. The equations of motion over this manifold describe only precessional oscillations. The corresponding precessional equations which can be considered as an approximate equations on the integral manifolds are 24

25 (HG + D)ẋ + Cx = 0. (72) Investigation of a vertical gyroscope with radial correction leads to the equations J α H β + d α kβ = 0, J β + H α + d β + kα = 0. In these equations J is the equatorial mass moment of inertia of the gyroscope, H is its angular momentum, d is the coefficient of friction. Forces H β and H α are gyroscopic, while kβ and kα are nonconservative forces, in the theory of gyroscopic systems they are referred to as forces of radial corrections [32]. For this system the precessional equations take the form H β + kβ d α = 0, H α + kα + d β = 0 It is easy to see that even in the case d = 0 the zero solution of the precessional equations is asymptotically stable, whereas the zero solution of original equations is unstable. It means that in the case d = 0 the using of precessional equations is inappropriate. More detailed analysis shows that the value of the damping factor for which asymptotic stability will prevail d > kj/h. As it was noted, the angular momentum H of the gyroscope is very large compared to kj. Therefore, the lower limit for the damping factor is very small. 3.4 Heavy gyroscope Using either general equations of motions or Lagrange equations, differential equations governing the motions of the axis of the heavy gyroscope in Cardano suspension can be derived [32] A(β) α + H cos β β + E sin 2β α β = m 1 α, B 0 β H cos β α 1 2 E sin 2β α2 = m 2 β P l cos β, A(β) = (A + A 1 ) cos 2 β + C 1 sin 2 β + A 2, B 0 = A + B 1, E = C 1 A A 1. Introduce the new time variable t = T τ and the dimensionless parameters A(β) B 0 = a(β), E B 0 = e, P lt B 0 = ν, 25

26 m 1 T B 0 = b 1, m 2 T B 0 = b 2, B 0 T/H = ε. and note that T can be chosen in such a way that a(β), e, ν, b 1, b 2 are values of order O(1), ε 1. Using the new variables α 1 = αt, β 1 = βt leads to equations of form 119 with x = ( α β ), y = ( α1 β 1 ), A(x, t) = ( a(β) ( ) ( ) 0 cos β b1 0 G(x, t) =, B(x, t) =, cos β 0 0 b 2 ( ) 0 Q(x, t) =. ν cos β We derive the representations for slow variables α = v 1 + ε z ( 2 + O(ε 2 v1 z ), v = cos v 2 v 2 β = v 2 ε a(v ( 2)z 1 + O(ε 2 z1 z ), z = cos v 2 z 2 where v 1, v 2, z 1, z 2 satisfy the equations ( v 1 = εν ε 3 ν 2 e sin v 2 + νb ) 1b 2 + O(ε 4 ), cos 2 v 2 v 2 = ε νb 1 cos 2 v 2 + O(ε 4 ), εż 1 = ε b 1 a(v 2 ) z 1 cos v 2 a(v 2 ) z 2 εz 1 z 2 tgv 2 + O(ε 2 z ), εż 2 = cos v 2 z 1 εb 2 z 2 + εγ 0 z1tgv O(ε 2 z ), (73) γ 0 = a(v 2 ) e cos2 v 2 = C 1 + A 2 B 0. These relationships show that the motion of the heavy gyroscope is very close to the regular precession on a bounded time intervals, but in the time of order O(1/ε 2 ) the angle β has attained the value π/2, which is to say that the frames of gyroscopic suspension tend to add up. The influence of a random perturbations on a gyroscopic systems will is studied in Chapter Integral Manifolds and Optimal Estimation in Gyroscopic Systems. ), ), ), 26

27 3.5 A one rigid-link flexible-joint manipulator Consider a simple model of rigid-link flexible joint manipulator [6, 28]. Let J m is the motor inertia, J 1 is the link inertia, M is the link mass, k is the stiffness. The model is described by the following equations: J 1 q 1 + Mg sin q 1 + k(q 1 q m ) = 0 J m q m k(q 1 q m ) = u In the equations, q 1 is the link angle, q m is the rotor angle, u is the torque input. The differential system may be considered as singularly perturbed one under ε = µ 2 = 1/k. In state form last system is expressed as µw 1 = ε 2 Mg µ q 1 = w 1 J 1 sin q 1 q 1 q m J 1 µ q m = w m µẇ m = (q 1 q m + µ 2 u)/j m It is clear that we obtain the singular singularly perturbed system. Using the simple algebra it is possible to rewrite the original system in the form J 1 q 1 + J m 1 q m + Mg sin q 1 = u q 1 q m + Mg sin q 1 + k(1/j 1 + 1/J m )(q 1 q m ) = u and the use of new variables gives the system x 1 = (J 1 q 1 + J m q m )/(J 1 + J m ), x 2 = ẋ 1 y = q 1 q m, ẋ 1 = x 2, ẋ 2 = Mg sin(x 1 + J m u y) + J 1 + J m J 1 + J m J 1 + J m εÿ = (1/J 1 + 1/J m )y ε Mg J 1 sin(x 1 + J m J 1 + J m y) ε u J m Note that to investigation of robotic-like systems with a weak energy dissipation the following results from [23] are useful. Consider the differential system ẋ = Ax + Bẏ + Qy + X(x, y, ẏ, ε), εÿ = εcẏ + P y + εrx + εy (x, y, ẏ, ε) (74) 27

28 with a small positive parameter ε. t; Here x, X are m-vectors; y, Y are - n- vectors; A, B, Q, C, P, R are constant matrices of corresponding dimensions. Suppose the following assumptions hold: 1. Function X and Y are defined and continuous in Ω = {x R m, y < ρ, 0 ε ε 0 }, bounded by M and satisfy the Lipschitz condition with a constant λ with respect to x, y, ẏ, where M andλ are sufficiently small positive numbers. 2. There exist such numbers K, α, β(k > 0, α > β 0), that the following inequalities take place (I is the identity matrix). Setting z = (y, ẏ) T, rewrite (74) as e At Ke β t ( < t < ) e D(ε)t Ke αεt (0 t < ) ( ) 0 ε I D(ε) = p ε C ẋ = Ax + P 1 z + X 1 (x, z, ε), P 1 = (QB), X 1 (x, z, ε) = X(x, y, ẏ, ε), εż = D(ε)z + εz(x, z, ε), (75) Z(x, z, ε) = ( 0 Y (x, y, ẏ, ε) + R(x) The system (75) possesses a local integral manifold z = H(x, ε). The flow on this manifold is described by ẋ = Ax + P 1 H(x, ε) + X 1 (x, H(x, ε), ε). (76) If P is regular matrix, then functions describing the integral manifold of (74) can be found as asymptotic expansions y = F (x, ε) = εf 1 (x) ε k f k (x) + ε k+1 f k+1 (x, ε) ẏ = G(x, ε) = εg 1 (x) ε k g k (x) + ε k+1 g k+1 (x, ε) where functions f i, g i are calculated using following identities: 0 = P f 1 + Rx + Y (x, 0, 0, 0), 28 )

29 g 1 = f 1 (Ax + X(x, 0, 0, 0)), x g 1 x (Ax + X(x, 0, 0, 0)) = Cg 1 + P f 2 + Y Y (x, 0, 0, 0) + (x, 0, 0, 0), y ẏ g 2 = f 2 x (Ax+X(x, 0, 0, 0))+ f 1 x {Bg 1+Qf 1 +[ x y (x, 0, 0, 0)]f 1+[ x ẏ (x, 0, 0, 0)]g 1}, and so on. The flow on this manifold is given by the equation ẋ = Ax + BG(x, ε) + QF (x, ε) + X(x, F (x, ε), G(x, ε), ε). (77) Note that the reducibility principle is true for this integral manifold. In [23] these results were used to obtain the sufficient condition of orientation stability of satellites with double spin in the case of a small friction and large stiffness of dampers. These results may be applied to the analysis of manipulator model under consideration. Following [10, 28], the control function u is adopted as a sum u = u f + u s, where u f = u f (y, ẏ), u s = u s (x 1, x 2 ). Setting u f = 2bẏ, b > 0; u s = Mg sin x 1 Mg + ε cos x 1 J m ( Mg sin x 1 2bx 2 ) + ε(j 1 + J m )v s, J 1 + J m J 1 + J m we obtain with an accuracy of order O(ε 2 ) the equation ẋ 1 = x 2, ẋ 2 = εv s on the slow integral manifold with a new control function v s. The corresponding fast variable z which describes the behaviour of the system under consideration near this integral manifold, with an accuracy of the same order satisfies the following fast equation ε z = (1/J 1 + 1/J m )z ε 2bż J m. It means, that transition regimes present a slowly extinguishing oscillations of high-frequency. 29

30 4 Branching of slow integral manifolds In this Section we formulate the problem and recall the method of gauge functions by considering a degenerate two-dimensional autonomous singularly perturbed differential system. The case of a quasi-homogeneous degenerate system, the case of an autonomous homogeneous singularly perturbed system are treated and the case of a quasi-homogeneous degenerate system. Several examples are given, one example describes a partly cheap optimal control problem. We consider singularly perturbed differential systems whose degenerate equations have an isolated but not simple solution. In that case, the standard theory to establish a slow integral manifold near this solution does not work. Applying scaling transformations and using the technique of gauge functions we reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate our method by several examples. 4.1 Formulation of the problem. Preliminaries We consider system (1) under the assumptions (A 1 ) and (A 2 ). hypothesis (A 3 ) we suppose Instead of det g y (x, h(x, t), t, 0) 0 (x, t) G R, (78) that is, y = h(x, t) is not a simple root of the degenerate equation g(x, y, t, 0) = 0. (79) Under this assumption we cannot apply Proposition 1.1 to system (1) in order to establish the existence of a slow integral manifold near M 0 for small ε. Our goal is to derive conditions which imply that for sufficiently small ε system (1) has at least one integral manifold M ε with the representation y = ψ i (x, t, ε) = h(x, t) + ε q i h 1,i (x, t) + ε 2q i h 2,i (x, t) where q i, 0 < q i < 1, is a rational number. The key idea to solve this problem consists in looking for scalings and transformations of the type ε = µ r, y = ỹ(µ, z, x, t), t = t(µ, τ) such that system (1) can be reduced to a system 30

31 dx dτ µ dz dτ = f(x, z, τ, µ), = g(x, z, τ, µ) (80) to which Proposition 1.1 can be applied. In this process the method of gauge function plays an important role. We illustrate our approach by considering a simple example, at the same time we recall the method of undetermined gauges. Example 4.1. Let us consider the system dx ε dy = y, = y 2 y 3 + εφ 2 (x, t), (81) where φ is a smooth positive function. The degenerate equation to (81) reads 0 = y 2 y 3 (82) and has the isolated but multiple root y = 0. To find a transformation reducing system (81) to a system to which Proposition 1.1 can be applied we look for an approximation of the roots of the equation 0 = y 2 y 3 + εφ 2 (x, t) (83) by means of the method of undetermined gauges (see e.g. [14]). To this purpose we represent a solution of (83) in the form y = δ 1 (ε)y 1 (x, t) + δ 2 (ε)y 2 (x, t) (84) The functions δ i (ε), called gauges, must be determined along with the functions y i (x, t). Concerning the gauge functions δ i (ε) we suppose that they are monotone in the interval I 0 and satisfy δ i (ε) 0 and δ i+1 (ε)/δ i (ε) 0 as ε 0 for all i. Substituting y = δ 1 (ε)y 1 into (83) leads to the equation 0 = y 2 1δ 2 1(ε) + y 3 1δ 3 1(ε) + εφ 2 (x, t). (85) 31

32 As δ 3 1(ε) δ 2 1(ε) for sufficiently small ε we simplify (85) to 0 = y 2 1δ εφ 2 (x, t). (86) Now we have to compare the order functions δ 1 (ε) and ε. Supposing that δ 2 1(ε) is the leading term in (86) we get y 1 = 0, if we assume that εφ 2 (x, t) is leading we are not able to determine y 1. If we suppose that δ 2 1(ε) and εφ 2 (x, t) are of equal significance for small ε, then we can set δ 1 (ε) := ε. (87) We note that this is not the only possible choice for δ 1 (ε) (see [14]). Putting (87) into (86) we obtain y 1 (x, t) = ±φ(x, t). (88) Similarly we can determine higher order gauges and coefficients. Now we use the relations (84) and (87) to scale the parameter ε and the variable y by ε = µ 2, y = µz. Substituting these relations into (81) we get dx µ dz = µz = z 2 + φ 2 (x, t) µz 3. (89) Taking into account that the degenerate equation to (89) has the two isolated simple solutions z = ±φ(x, t) we may apply Proposition 1.1 to system (89) with respect to these roots and get that system (81) has two integral manifolds with the representation y = ±φ(x, t) ε + O(ε). In the following sections we study the existence and approximation of slow integral manifolds of system (1) in some degenerate cases. 4.2 Quasi-homogeneous degenerate equations We study system (1) under the assumption (A 1 ). We replace the assumptions (A 2 ) and (A 3 ) by the the following hypotheses. (H 1 ). The function g(x, y, t, 0) can be represented in the form g(x, y, t, 0) g 1 (x, y, t) + g 2 (x, y, t) (90) where the functions g 1 and g 2 have the following properties 32