A note on linear differential equations with periodic coefficients.

Size: px
Start display at page:

Download "A note on linear differential equations with periodic coefficients."

Transcription

1 A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, Lleida, Spain. E mail: mtgrau@matematica.udl.es (2) Departamento de Matemáticas. Universidad Carlos III Leganés (Madrid), Spain. E mail: dperalta@math.uc3m.es Abstract We consider a linear homogeneous differential equation of the form ẋ = A(t)x where A(t) is a square matrix of C 1, real and T -periodic functions, with T >. We give several criteria on the matrix A(t) to prove asymptotic stability of the trivial solution to equation ẋ = A(t)x. These criteria allow us to show that any finite configuration of cycles in R n can be realized as hyperbolic limit cycles of a polynomial vector field. 2 AMS Subject Classification: 34A3, 34D8, 37D5, 34C7. Key words and phrases: linear differential equations, characteristic multipliers, hyperbolicity, limit cycles. 1 Introduction In this note we are concerned with stability properties of the trivial solution to linear differential equations with periodic coefficients ẋ = A(t) x, (1) where x R n and A(t) is an n n square matrix of C 1 real functions, which is T -periodic in t, with T >. As usual the dot means derivation with respect to the real independent variable t. Let us briefly recall the definition of Lyapunov stability of the trivial solution of (1) as given in [1]. We denote by x the modulus or distance of the vector x to the origin. The solution x = is called Lyapunov stable if for any ε >, there exists a δ = δ(ε) > such that if x < δ, the solution x(t) of (1) with The first author is partially supported by a DGICYT grant number MTM C2-2. The second author acknowledges financial support from the Spanish MEC through the Juan de la Cierva program. 1

2 initial condition x() = x satisfies x(t) < ε for any t. The solution x = is called asymptotically stable if there exists a b > such that x < b implies x(t) as t +. The solution x = is unstable if it is not stable. The stability of the trivial solution to (1) can be studied by the modulus of the eigenvalues associated to its monodromy matrix. Let Φ(t) be the fundamental matrix solution of (1) such that Φ() is the identity matrix. Usually, Φ(t) is called the matriciant of (1). The monodromy matrix of (1) is given by Φ(T ). As described in [1], if all the eigenvalues of Φ(T ) have modulus lower or equal to 1, then the trivial solution of (1) is Lyapunov stable and if all the eigenvalues of Φ(T ) have modulus strictly lower than 1, then the trivial solution of (1) is asymptotically stable. If all the eigenvalues of Φ(T ) have modulus different from 1, then the trivial solution is structurally stable and it is said to be hyperbolic. We recall that a real square matrix A is negative (positive) definite if x T A x < (> ) for any nonzero vector x, where the superscript T denotes transposition. We observe that if A is negative definite, then the real part of the eigenvalues of A is strictly negative. The reciprocal is not true: there are matrices with the real part of all their eigenvalues negative but which do not satisfy x T A x < for any x, for instance, the matrix A = ( ), has its both eigenvalues 1 and 2 real and negative, but the quadratic form (x 1, x 2 ) A (x 1, x 2 ) T = x x 1 x 2 2x 2 2, is not negative for all nonzero (x 1, x 2 ) (e.g. take the vector (1, 1)). Of course a symmetric matrix whose eigenvalues are negative satisfies to be negative definite. The stability properties of the matrix A(t) and the monodromy matrix Φ(T ) are not generally related. In fact there are several examples which show that the sign of the real part of the eigenvalues of the matrix A(t) does not determine the stability of the solution x = of (1), see e.g. [4, 5, 6]. Therefore the only general procedure to study the stability of x = is solving Eq. (1), a task which is generally extremely difficult. This fact suggests the interest of providing sufficient conditions on the matrix A(t) ensuring that the trivial solution of Eq. (1) is hyperbolic and asymptotically stable. The idea of this work is to obtain some of these sufficient conditions over the matrix A(t). Apart from some theorems that we will review in sections 2 and 3 the literature on this problem is rather scarce. Furthermore no results in this direction appear in the classical textbooks on dynamical systems, as Chicone [4], Hale [1], Lefschetz [11] or Perko [14]. The goal of this paper is to fill this gap and to bring the reader s attention to this important problem. Equations of type (1) are very important in applications because they appear when studying the stability of limit cycles of vector fields in R n+1. Indeed let X be a C 1 vector field with a periodic orbit γ. Let us endow a neighborhood of γ with a local coordinates system defined by: s S 1 is a coordinate on the cycle and x R n are coordinates on a section Σ orthogonal to γ, in particular γ = {x = }. The normal variational equation expressed in these coordinates 2

3 has the form (1) and it is well known [1] that the stability of the trivial solution of Eq. (1) determines the stability and hyperbolicity of the limit cycle γ of X. Combining Theorem 2 with some ideas introduced in Reference [13] and a Theorem obtained in Reference [8] we can prove the following result, which solves the open problem stated in [13]: Can any finite configuration of cycles be realized as hyperbolic limit cycles of a polynomial vector field? Theorem 1. Any finite configuration of (smooth) cycles in R n can be realized (up to global diffeomorphism) as hyperbolic and asymptotically stable limit cycles of a polynomial vector field. This paper is organized as follows. Section 2 contains some preliminary material on the problem that we study. Several results giving sufficient conditions over the matrix A(t) in order to ensure the hyperbolicity and asymptotic stability of the corresponding Eq. (1) are proved in Section 3. Finally, in Section 4 we prove Theorem 1. 2 Preliminary results The following theorem appears in some books related to control theory and Liapunov stability and is contained in Persidskiĭ and Malkin s theorems, see [12]. Theorem 2. [12] satisfying: Let Q(t) and P (t) be two square, of class C 1, real matrices Q = P + P A + A T P, (2) and such that Q(t) Q, Q being a negative definite constant matrix, and P 1 P (t) P 2, P 1, P 2 being two positive definite constant matrices, for any t R. Then the trivial solution to Eq. (1) is hyperbolic and asymptotically stable. The proof of Theorem 2 goes through considering the function G = x T P (t)x which gives a strict Liapunov function for Eq. (1). That is, we have that Ġ = x T Q(t)x < for all nonzero x. The existence of a strict Liapunov function gives the hyperbolicity and asymptotic stability of Eq. (1), because all its nonzero solutions x(t) need to satisfy that x(t ) < x(). The assumptions on P (t) and Q(t) are necessary in order to apply Liapunov s stability theorem for non-autonomous systems, see e.g. [11]. These assumptions are automatically satisfied if P (t) and Q(t) are T -periodic matrices, positive definite and negative definite respectively for any t [, T ). In the particular case in which the matrix A is constant, Liapunov Theorem ensures that if the eigenvalues of A have negative real part then given any matrix Q constant and negative definite, there exists a matrix P positive definite such that Q = P A + A T P, see for instance [9, 11]. Moreover, this matrix P can be 3

4 computed through: P = e sat Q e sa ds. The following theorem is stated and proved in paper [7] and is a generalization of Liapunov theorem to matrices A(t) with periodic entries. Theorem 3. [7] Suppose A(t) is a continuous and T -periodic matrix. 1. If all the eigenvalues of the monodromy matrix of Eq. (1) lie in the unit disk { λ < 1} then, for every continuous matrix Q(t) on [, T ], there is a unique solution P (t) to the boundary value problem: P + P A + A P = Q, P () = P (T ). < t < T Moreover, if Q(t) = Q (t) is negative definite, then P (t) = P (t) is positive definite, for t [, T ]. 2. Assume Q(t) is a continuous and negative definite matrix on [, T ] and the solution P (t) to (3) is such that P (t) = P (t) and P () is positive definite, then all the eigenvalues of the monodromy matrix for Eq. (1) lie in the unit disk { λ < 1}. This theorem is stated for complex matrices and A is the conjugate transpose of A. Since we are only considering real matrices, we have that A = A T. We remark that in the proof of Theorem 3 the matrix P (t) is constructed in terms of the fundamental matrix Φ(t) solution to Eq. (1). It establishes the equivalence between asymptotic stability of the trivial solution to Eq. (1) and the existence of matrices Q(t) (negative definite), P (t) (positive definite) satisfying Eq. (3). However this theorem does not give a way to construct these matrices unless Eq. (1) is solved. Note that the second part of Theorem 3 complements Theorem 2, although the matrices Q(t) and P (t) do not need to be symmetric and periodic to apply Theorem 2. In light of these theorems an effective technique to study the stability of the trivial solution to Eq. (1) is to prove the existence of such matrices P (t) and Q(t). In the next section we will exploit this idea to obtain several stability criteria. 3 Criteria for hyperbolicity and asymptotic stability of Eq. (1) This section contains several criteria to ensure the existence of the matrices Q(t) and P (t) described in Theorem 2 and, thus, we can deduce that all the eigenvalues of the monodromy matrix associated to Eq. (1) have modulus strictly lower than 1. (3) 4

5 A simple observation, which appears in Hahn s stability book [9], although it is not well known, is the following Proposition. Proposition 1. If A(t) is negative definite for any t [, T ) then x = is hyperbolic and asymptotically stable. Proof. Just note that P (t) = 1 and Q(t) = A(t) + A(t) T verify the assumptions in Theorem 2. Most of the known sufficient conditions on the matrix A(t) to ensure the asymptotic stability of the trivial solution to (1) assume a decomposition of the matrix A(t) in the form A(t) = A +A 1 (t), where A is a constant matrix whose eigenvalues have all negative real part and A 1 (t) is generally a small matrix satisfying certain conditions, see [1, 3, 7, 11, 12]. Our criteria do not involve such a decomposition and concern the relation between A(t) and the solutions of (1). Proposition 2. If there exists a real and constant matrix Q, symmetric and negative definite such that Ȧ T Q + QA = and the eigenvalues of A(t) have all negative real part for t [, T ), then the solution to (2) is: P (t) = e sa(t)t Qe sa(t) ds, which is symmetric and positive definite. Accordingly x =, solution to (1), is hyperbolic and asymptotically stable. Proof. Let us define the function φ(s; t) = e sa(t)t Qe sa(t). We note that φ(; t) = Q and, since the eigenvalues of A(t) have all negative real part for t [, T ), lim s φ(s; t) =. Therefore, P (t) is well-defined for all t (just apply Liapunov theorem to each value of t). Let us consider P + P A + A T P. Note that, under the hypotheses: P (t) = e sa(t)t s[ȧ(t)t Q + QȦ(t)]esA(t) ds =. On the other hand, we have that s φ(s; t)e sa(t)t [A(t) T Q+QA(t)]e sa(t). Therefore, P A + A T P = = e sa(t)t [QA(t) + A(t) T Q]e sa(t) ds s φ(s; t) ds = φ(; t) lim φ(s; t) = Q. s We have that the matrix P satisfies Eq. (2). The fact that it is symmetric is proved by: P (t) T = (e sa(t)t Qe sa(t)) T ds = e sa(t)t Q T e sa(t) ds P (t), 5

6 because Q T = Q. In order to show that P (t) is positive definite, we consider x and the product x T P x = (esa(t) x) T ( Q)(e sa(t) x)ds. We take y(s; t) = e sa(t) x and we have x T P x = y(s; t) T ( Q)y(s; t)ds. Since the matrix Q is negative definite, we have that y(s; t) T ( Q)y(s; t) > for all s and t. When integrating, we get that x T P x = y(s; t) T ( Q)y(s; t)ds >. So, P (t) is positive definite. Proposition 3. Assume that x T A(t)x ɛ (t) + ɛ 1 (t)x 2 and x T C(t)x ɛ (t) + ɛ 1 (t)x 2 for x small enough, where C(t) := Ȧ(t) + A2 (t), ɛ (t), ɛ (t) and ɛ 1 (t) < ɛ 1 (t)/(2k) with K := δ + max t [,T ) ɛ 1 (t), for any t [, T ) and some δ >, then x = is hyperbolic and asymptotically stable as solution of Eq. (1). Proof. Let P (t) = k 1 I + k 2 A(t) with k 1 R + and k 2 R. We have that G = x t P (t)x = k 1 x 2 + k 2 x t A(t)x (k 1 k 2 ɛ 1 (t))x 2 k 2 ɛ (t) (k 1 k 2 ɛ 1 (t))x 2. If we take k 1 = k 2 K, we have that G >. Hence, G is a Lyapunov function (depending on t). We study now the matrix Q(t) = P (t)+p (t)a(t)+a T (t)p (t). Since x T A T Ax = (Ax) T Ax we deduce that x T ( A + A T A + A 2 )x x T (Ȧ + A 2 )x ɛ (t) + ɛ 1 (t)x 2. We get that: x T Q(t)x 2k 1 x T A(t)x + k 2 x T C(t)x (2k 1 ɛ (t) k 2 ɛ (t)) + (2k 1 ɛ 1 (t) k 2 ɛ 1 (t))x 2 (2k 1 ɛ 1 (t) k 2 ɛ 1 (t))x 2. Taking into account the definition of k 1 and the hypotheses, we conclude that x T Q(t)x k 2 (2Kɛ 1 (t) ɛ 1 (t)) <. Applying Theorem 2 the result follows. Corollary 1. If A is negative semidefinite, i.e. x t Ax, and Ȧ + A2 is positive definite, then x = is hyperbolic and asymptotically stable as solution to Eq. (1). Corollary 2. If the minimum of ɛ 1 (t) in t [, T ) is positive, then for ɛ 1 (t) small enough (in t [, T )), x = is hyperbolic and asymptotically stable as solution to Eq. (1). We note that ɛ 1 (t) small enough in Corollary 2 means that A(t) is not negative definite but it is nearly so. The following examples illustrate some of the criteria previously obtained and show that they are easily verifiable in particular cases. 6

7 Example. We consider the periodic matrix cos A(t) = 4 t sin 2 t 2 cos 4 t + 2 sin 2 t 1/2 2 cos 4 t + 2 sin 2 t 1/2 2 It is symmetric and its eigenvalues are both real and negative for any value of t, hence it is negative definite. Proposition 1 implies that {x = } is a hyperbolic and asymptotically stable equilibrium of Eq. (1). However, obtaining this result solving Eq. (1) cannot be done by any elementary method. A similar discussion applies to the periodic matrix A(t) = 1 cos 2 t 1+cos 4 t 2+3 cos 2 t cos 4 t sin t/2 1+cos 4 t e 2+3 cos 2 t cos 4 t 1+cos 4 t e sin t/2 3e sin t Example. Let us consider any T -periodic function a(t) of class C 1, and let σ, σ 1, σ 2 be three real numbers such that σ <, σ 1 <, σ 2 < and σ 1 σ 2 > 1. We define the matrices ( ) ( ) σ a(t) σ A(t) = 2 a(t) σ1 1, Q =, σ 1 a(t) σ + a(t) 1 σ 2 and we note that A(t) has eigenvalues σ ± a(t) 1 σ 1 σ 2, which have negative real part, and that Q is symmetric and negative definite. Moreover, A(t) satisfies that Ȧ T Q + QȦ =. Therefore, we deduce by Proposition 2, that the trivial solution to the system ẋ = A(t)x is hyperbolic and asymptotically stable. We note that this result cannot be obtained directly solving ẋ = A(t)x due to the generality in the form of the matrix A(t). A particular case of Eq. (1) which usually appears in applications is the second order differential equation ẍ + p(t)ẋ + q(t)x =, (4) where p(t) and q(t) are C 1 and T -periodic functions. Let us finish this section establishing a criterion for asymptotic stability of second order differential equations. An example is also provided to show that the assumptions of the criterion can be easily verified. Proposition 4. Let p(t) and q(t) be C 1 and T -periodic functions which verify 1. T p(t)dt > 2. T r(t)dt 3. T r(t) dt 4 T where r(t) q(t) 1 2ṗ(t) 1 4 p(t)2 is non-identically zero. Then the solution x = to Eq. (4) is hyperbolic and asymptotically stable... 7

8 Proof. First let us simplify Eq. (4) with the following change of variable x(t) = y(t)e 1 2 This transforms the equation into the form t p(t)dt. ÿ + r(t)y =, (5) where r(t) is defined in the statement of the proposition. Under hypotheses 2 and 3 Bellman proves ([1], pages ) that all solutions y(t) to Eq. (5) are bounded when t ±. Since p(t) satisfies hypothesis 1 we get that e 1 t 2 p(t)dt when t +. Combining both results we conclude that any solution x(t) to Eq. (4) verifies x(t) as t + and hence all the eigenvalues of the monodromy matrix are 1. Hyperbolicity follows because otherwise there would be some solutions for which x(t) is constant for any t R. Example. In order to illustrate Proposition 4, let us consider Eq. (4) with p(t) = cos 2 (t) and q(t) = q R. The first hypothesis of Proposition 4 is fulfilled since π p(t) dt π/2. The value of the corresponding function r(t) gives that π r(t)dt = π(q 3/32). Therefore, the second hypothesis is satisfied when q 3/32. Moreover, an easy numerical analysis shows that if q 343/75, the third hypothesis is also satisfied. Hence, we deduce that if 3/32 q 343/75, the trivial solution is hyperbolic and asymptotically stable, as a consequence of Proposition 4. 4 Proof of Theorem 1 Let C be any finite configuration of smooth cycles in R n+1, possibly linked and knotted when n = 2. Following [13] we apply a modification of Nash-Tognoli s theorem [2] to show that there is a global diffeomorphism H : R n+1 R n+1 such that H(C) is a non-singular algebraic set, so it is defined by f 1 (x 1,..., x n+1 ) =,. f n (x 1,..., x n+1 ) =, where f 1,..., f n are polynomials satisfying that rank(df 1,..., df n ) H(C) = n. (6) Define the following polynomial vector field in R n+1 : X = [ (df 1... df n )] i F, 8

9 where is the Hodge star operator and i denotes the index raising operator. The function F is defined as F = 1 2 (f f 2 n). This vector field is a slight modification of the one used in [13]. Furthermore, following the procedure of [13], it is not difficult to prove that the cycles in H(C) are the only periodic orbits, in fact asymptotically stable limit cycles, of X. Let us now show that these limit cycles are hyperbolic and hence structurally stable under small perturbations of X. First compute the following derivative with respect to the variable t: f i = X f i = n ( f i f j )f j, j=1 i = 1,..., n. This equation can be written in matrix notation as where and Df(x)X(x) = k(x)f(x), f(x) = f 1 (x).. f n (x), ( f 1 ) 2 f 1 f n k(x) =.., f 1 f n ( f n ) 2 which is called the cofactor matrix. Lemma 1. The matrix k(x) evaluated at any limit cycle γ H(C) is negative definite. Proof. Condition (6) implies that the functions A k := det( f i f j ) H(C), 1 i, j k have ( 1) k sign for k = 1,..., n. Since k(x) is symmetric it follows that the matrix k(γ(t)) is negative definite for any limit cycle γ H(C). In [8] it was proved that the stability properties of the trivial solution to ẋ = k(γ(t))x, (7) determine the stability properties of the limit cycle γ of X. Applying Lemma 1 and Theorem 2, with P = I/2 and Q(t) = k(γ(t)), we get that the trivial solution to (7) is hyperbolic, thus concluding that the limit cycles of X are hyperbolic. 9

10 References [1] R. Bellman, Stability Theory of Differential Equations. McGraw-Hill, New York, [2] J. Bochnak, M. Coste and M. Roy, Real Algebraic Geometry. Springer, Berlin, [3] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer, Berlin, [4] C. Chicone, Ordinary Differential Equations with Applications. Springer, New York, [5] W. A. Coppel, Dichotomies in stability theory. Lecture Notes in Mathematics, Vol. 629 Springer Verlag, Berlin New York, [6] H. A. De Kleine, A note on the asymptotic stability of periodic solutions of autonomous differential equations. SIAM Rev. 26 (1984), [7] G. V. Demidenko and I.I. Matveeva, On stability of solutions to linear systems with periodic coefficients. Siber. Math. J. 42 (21), [8] A. Gasull, H. Giacomini and M. Grau, On the stability of periodic orbits for differential systems in R n. CRM Preprint, 26. [9] W. Hahn, Stability of motion. Springer-Verlag, New York, [1] J. K. Hale, Ordinary differential equations. Wiley, New York, [11] S. Lefschetz, Differential Equations: Geometric Theory, Dover, New York, [12] I. G. Malkin, Theory of stability of motion, Nauka, Moscow, [13] D. Peralta-Salas, Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles. J. Differential Equations 217 (25), [14] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 21. 1

6 Linear Equation. 6.1 Equation with constant coefficients

6 Linear Equation. 6.1 Equation with constant coefficients 6 Linear Equation 6.1 Equation with constant coefficients Consider the equation ẋ = Ax, x R n. This equating has n independent solutions. If the eigenvalues are distinct then the solutions are c k e λ

More information

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct

More information

Math Ordinary Differential Equations

Math Ordinary Differential Equations Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x

More information

1. Find the solution of the following uncontrolled linear system. 2 α 1 1

1. Find the solution of the following uncontrolled linear system. 2 α 1 1 Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +

More information

Period function for Perturbed Isochronous Centres

Period function for Perturbed Isochronous Centres QUALITATIE THEORY OF DYNAMICAL SYSTEMS 3, 275?? (22) ARTICLE NO. 39 Period function for Perturbed Isochronous Centres Emilio Freire * E. S. Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos

More information

Lecture 9. Systems of Two First Order Linear ODEs

Lecture 9. Systems of Two First Order Linear ODEs Math 245 - Mathematics of Physics and Engineering I Lecture 9. Systems of Two First Order Linear ODEs January 30, 2012 Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 1 / 15 Agenda General Form

More information

ZERO-HOPF BIFURCATION FOR A CLASS OF LORENZ-TYPE SYSTEMS

ZERO-HOPF BIFURCATION FOR A CLASS OF LORENZ-TYPE SYSTEMS This is a preprint of: Zero-Hopf bifurcation for a class of Lorenz-type systems, Jaume Llibre, Ernesto Pérez-Chavela, Discrete Contin. Dyn. Syst. Ser. B, vol. 19(6), 1731 1736, 214. DOI: [doi:1.3934/dcdsb.214.19.1731]

More information

Problem List MATH 5173 Spring, 2014

Problem List MATH 5173 Spring, 2014 Problem List MATH 5173 Spring, 2014 The notation p/n means the problem with number n on page p of Perko. 1. 5/3 [Due Wednesday, January 15] 2. 6/5 and describe the relationship of the phase portraits [Due

More information

EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS

EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS

More information

Chapter III. Stability of Linear Systems

Chapter III. Stability of Linear Systems 1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,

More information

PERIODIC SOLUTIONS WITH NONCONSTANT SIGN IN ABEL EQUATIONS OF THE SECOND KIND

PERIODIC SOLUTIONS WITH NONCONSTANT SIGN IN ABEL EQUATIONS OF THE SECOND KIND PERIODIC SOLUTIONS WITH NONCONSTANT SIGN IN ABEL EQUATIONS OF THE SECOND KIND JOSEP M. OLM, XAVIER ROS-OTON, AND TERE M. SEARA Abstract. The study of periodic solutions with constant sign in the Abel equation

More information

There is a more global concept that is related to this circle of ideas that we discuss somewhat informally. Namely, a region R R n with a (smooth)

There is a more global concept that is related to this circle of ideas that we discuss somewhat informally. Namely, a region R R n with a (smooth) 82 Introduction Liapunov Functions Besides the Liapunov spectral theorem, there is another basic method of proving stability that is a generalization of the energy method we have seen in the introductory

More information

8 Periodic Linear Di erential Equations - Floquet Theory

8 Periodic Linear Di erential Equations - Floquet Theory 8 Periodic Linear Di erential Equations - Floquet Theory The general theory of time varying linear di erential equations _x(t) = A(t)x(t) is still amazingly incomplete. Only for certain classes of functions

More information

AVERAGING THEORY AT ANY ORDER FOR COMPUTING PERIODIC ORBITS

AVERAGING THEORY AT ANY ORDER FOR COMPUTING PERIODIC ORBITS This is a preprint of: Averaging theory at any order for computing periodic orbits, Jaume Giné, Maite Grau, Jaume Llibre, Phys. D, vol. 25, 58 65, 213. DOI: [1.116/j.physd.213.1.15] AVERAGING THEORY AT

More information

A NON-AUTONOMOUS KIND OF DUFFING EQUATION JAUME LLIBRE AND ANA RODRIGUES

A NON-AUTONOMOUS KIND OF DUFFING EQUATION JAUME LLIBRE AND ANA RODRIGUES This is a preprint of: A non-autonomous kind of Duffing equation, Jaume Llibre, Ana Rodrigues, Appl. Math. Comput., vol. 25, 669 674, 25. DOI: [.6/j.amc.24..7] A NON-AUTONOMOUS KIND OF DUFFING EQUATION

More information

Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.

Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4. Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the

More information

1 Lyapunov theory of stability

1 Lyapunov theory of stability M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

More information

7 Planar systems of linear ODE

7 Planar systems of linear ODE 7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution

More information

2.10 Saddles, Nodes, Foci and Centers

2.10 Saddles, Nodes, Foci and Centers 2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one

More information

Department of Mathematics IIT Guwahati

Department of Mathematics IIT Guwahati Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,

More information

Lecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.

Lecture 4. Chapter 4: Lyapunov Stability. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University. Lecture 4 Chapter 4: Lyapunov Stability Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 4 p. 1/86 Autonomous Systems Consider the autonomous system ẋ

More information

Nonlinear Control Lecture 5: Stability Analysis II

Nonlinear Control Lecture 5: Stability Analysis II Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41

More information

CDS Solutions to the Midterm Exam

CDS Solutions to the Midterm Exam CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2

More information

Z i Q ij Z j. J(x, φ; U) = X T φ(t ) 2 h + where h k k, H(t) k k and R(t) r r are nonnegative definite matrices (R(t) is uniformly in t nonsingular).

Z i Q ij Z j. J(x, φ; U) = X T φ(t ) 2 h + where h k k, H(t) k k and R(t) r r are nonnegative definite matrices (R(t) is uniformly in t nonsingular). 2. LINEAR QUADRATIC DETERMINISTIC PROBLEM Notations: For a vector Z, Z = Z, Z is the Euclidean norm here Z, Z = i Z2 i is the inner product; For a vector Z and nonnegative definite matrix Q, Z Q = Z, QZ

More information

REAL AND COMPLEX HOMOGENEOUS POLYNOMIAL ORDINARY DIFFERENTIAL EQUATIONS IN n-space AND m-ary REAL AND COMPLEX NON-ASSOCIATIVE ALGEBRAS IN n-space

REAL AND COMPLEX HOMOGENEOUS POLYNOMIAL ORDINARY DIFFERENTIAL EQUATIONS IN n-space AND m-ary REAL AND COMPLEX NON-ASSOCIATIVE ALGEBRAS IN n-space Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 8, 1996, 327 333 REAL AND COMPLEX HOMOGENEOUS POLYNOMIAL ORDINARY DIFFERENTIAL EQUATIONS IN n-space AND m-ary REAL

More information

Periodic solutions of the perturbed symmetric Euler top

Periodic solutions of the perturbed symmetric Euler top Periodic solutions of the perturbed symmetric Euler top Universitatea Babeş-Bolyai (Cluj-Napoca, Romania) abuica@math.ubbcluj.ro and Universitat de Lleida (Lleida, Spain) Plan of the talk 1 Our problem

More information

Lyapunov Stability Theory

Lyapunov Stability Theory Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous

More information

(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);

(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively); STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend

More information

An introduction to Birkhoff normal form

An introduction to Birkhoff normal form An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

More information

8.1 Bifurcations of Equilibria

8.1 Bifurcations of Equilibria 1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations

More information

Symmetric Spaces Toolkit

Symmetric Spaces Toolkit Symmetric Spaces Toolkit SFB/TR12 Langeoog, Nov. 1st 7th 2007 H. Sebert, S. Mandt Contents 1 Lie Groups and Lie Algebras 2 1.1 Matrix Lie Groups........................ 2 1.2 Lie Group Homomorphisms...................

More information

Course 216: Ordinary Differential Equations

Course 216: Ordinary Differential Equations Course 16: Ordinary Differential Equations Notes by Chris Blair These notes cover the ODEs course given in 7-8 by Dr. John Stalker. Contents I Solving Linear ODEs 1 Reduction of Order Computing Matrix

More information

ON THE DYNAMICS OF THE RIGID BODY WITH A FIXED POINT: PERIODIC ORBITS AND INTEGRABILITY

ON THE DYNAMICS OF THE RIGID BODY WITH A FIXED POINT: PERIODIC ORBITS AND INTEGRABILITY ON THE DYNAMICS OF THE RIID BODY WITH A FIXED POINT: PERIODIC ORBITS AND INTERABILITY JUAN L.. UIRAO 1, JAUME LLIBRE 2 AND JUAN A. VERA 3 Abstract. The aim of the present paper is to study the periodic

More information

THE NUMBER OF POLYNOMIAL SOLUTIONS OF POLYNOMIAL RICCATI EQUATIONS

THE NUMBER OF POLYNOMIAL SOLUTIONS OF POLYNOMIAL RICCATI EQUATIONS This is a preprint of: The number of polynomial solutions of polynomial Riccati equations, Armengol Gasull, Joan Torregrosa, Xiang Zhang, J. Differential Equations, vol. 261, 5071 5093, 2016. DOI: [10.1016/j.jde.2016.07.019]

More information

VERSAL DEFORMATIONS OF BILINEAR SYSTEMS UNDER OUTPUT-INJECTION EQUIVALENCE

VERSAL DEFORMATIONS OF BILINEAR SYSTEMS UNDER OUTPUT-INJECTION EQUIVALENCE PHYSCON 2013 San Luis Potosí México August 26 August 29 2013 VERSAL DEFORMATIONS OF BILINEAR SYSTEMS UNDER OUTPUT-INJECTION EQUIVALENCE M Isabel García-Planas Departamento de Matemàtica Aplicada I Universitat

More information

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems

Linear ODEs. Existence of solutions to linear IVPs. Resolvent matrix. Autonomous linear systems Linear ODEs p. 1 Linear ODEs Existence of solutions to linear IVPs Resolvent matrix Autonomous linear systems Linear ODEs Definition (Linear ODE) A linear ODE is a differential equation taking the form

More information

3 Stability and Lyapunov Functions

3 Stability and Lyapunov Functions CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

More information

An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system

An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 6, No. 4, 2018, pp. 438-447 An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian

More information

ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION

ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION NAEEM M.H. ALKOUMI

More information

Lecture 16. Theory of Second Order Linear Homogeneous ODEs

Lecture 16. Theory of Second Order Linear Homogeneous ODEs Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda

More information

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling

Examples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling 1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples

More information

BESSEL MATRIX DIFFERENTIAL EQUATIONS: EXPLICIT SOLUTIONS OF INITIAL AND TWO-POINT BOUNDARY VALUE PROBLEMS

BESSEL MATRIX DIFFERENTIAL EQUATIONS: EXPLICIT SOLUTIONS OF INITIAL AND TWO-POINT BOUNDARY VALUE PROBLEMS APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 11 23 E. NAVARRO, R. COMPANY and L. JÓDAR (Valencia) BESSEL MATRIX DIFFERENTIAL EQUATIONS: EXPLICIT SOLUTIONS OF INITIAL AND TWO-POINT BOUNDARY VALUE PROBLEMS

More information

Matrices A(t) depending on a Parameter t. Jerry L. Kazdan

Matrices A(t) depending on a Parameter t. Jerry L. Kazdan Matrices A(t depending on a Parameter t Jerry L. Kazdan If a square matrix A(t depends smoothly on a parameter t are its eigenvalues and eigenvectors also smooth functions of t? The answer is yes most

More information

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes

22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one

More information

Control, Stabilization and Numerics for Partial Differential Equations

Control, Stabilization and Numerics for Partial Differential Equations Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua

More information

MATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018

MATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018 MATH 57: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 18 1 Global and Local Optima Let a function f : S R be defined on a set S R n Definition 1 (minimizers and maximizers) (i) x S

More information

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,

More information

ENERGY DECAY ESTIMATES FOR LIENARD S EQUATION WITH QUADRATIC VISCOUS FEEDBACK

ENERGY DECAY ESTIMATES FOR LIENARD S EQUATION WITH QUADRATIC VISCOUS FEEDBACK Electronic Journal of Differential Equations, Vol. 00(00, No. 70, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp ENERGY DECAY ESTIMATES

More information

The complex periodic problem for a Riccati equation

The complex periodic problem for a Riccati equation The complex periodic problem for a Riccati equation Rafael Ortega Departamento de Matemática Aplicada Facultad de Ciencias Universidad de Granada, 1871 Granada, Spain rortega@ugr.es Dedicated to Jean Mawhin,

More information

Sliding Vector Fields via Slow Fast Systems

Sliding Vector Fields via Slow Fast Systems Sliding Vector Fields via Slow Fast Systems Jaume Llibre Paulo R. da Silva Marco A. Teixeira Dedicated to Freddy Dumortier for his 60 th birthday. Abstract This paper concerns differential equation systems

More information

Higher Order Averaging : periodic solutions, linear systems and an application

Higher Order Averaging : periodic solutions, linear systems and an application 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,

More information

On the dynamics of strongly tridiagonal competitive-cooperative system

On the dynamics of strongly tridiagonal competitive-cooperative system On the dynamics of strongly tridiagonal competitive-cooperative system Chun Fang University of Helsinki International Conference on Advances on Fractals and Related Topics, Hong Kong December 14, 2012

More information

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T. Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where

More information

Reduction to the associated homogeneous system via a particular solution

Reduction to the associated homogeneous system via a particular solution June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one

More information

A sensitivity result for quadratic semidefinite programs with an application to a sequential quadratic semidefinite programming algorithm

A sensitivity result for quadratic semidefinite programs with an application to a sequential quadratic semidefinite programming algorithm Volume 31, N. 1, pp. 205 218, 2012 Copyright 2012 SBMAC ISSN 0101-8205 / ISSN 1807-0302 (Online) www.scielo.br/cam A sensitivity result for quadratic semidefinite programs with an application to a sequential

More information

LOCALLY POSITIVE NONLINEAR SYSTEMS

LOCALLY POSITIVE NONLINEAR SYSTEMS Int. J. Appl. Math. Comput. Sci. 3 Vol. 3 No. 4 55 59 LOCALLY POSITIVE NONLINEAR SYSTEMS TADEUSZ KACZOREK Institute of Control Industrial Electronics Warsaw University of Technology ul. Koszykowa 75 66

More information

BASIC MATRIX PERTURBATION THEORY

BASIC MATRIX PERTURBATION THEORY BASIC MATRIX PERTURBATION THEORY BENJAMIN TEXIER Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity

More information

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N. ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that

More information

First and Second Order Differential Equations Lecture 4

First and Second Order Differential Equations Lecture 4 First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence

More information

L 1 criteria for stability of periodic solutions of a newtonian equation

L 1 criteria for stability of periodic solutions of a newtonian equation Math. Proc. Camb. Phil. Soc. (6), 14, 359 c 6 Cambridge Philosophical Society doi:1.117/s354158959 Printed in the United Kingdom 359 L 1 criteria for stability of periodic solutions of a newtonian equation

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Invariant Manifolds of Dynamical Systems and an application to Space Exploration

Invariant Manifolds of Dynamical Systems and an application to Space Exploration Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated

More information

The goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T

The goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T 1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.

More information

APPPHYS217 Tuesday 25 May 2010

APPPHYS217 Tuesday 25 May 2010 APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag

More information

LOCAL STABILITY IMPLIES GLOBAL STABILITY IN SOME ONE-DIMENSIONAL DISCRETE SINGLE-SPECIES MODELS. Eduardo Liz. (Communicated by Linda Allen)

LOCAL STABILITY IMPLIES GLOBAL STABILITY IN SOME ONE-DIMENSIONAL DISCRETE SINGLE-SPECIES MODELS. Eduardo Liz. (Communicated by Linda Allen) DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS SERIES B Volume 7, Number 1, January 2007 pp. 191 199 LOCAL STABILITY IMPLIES GLOBAL STABILITY IN SOME ONE-DIMENSIONAL DISCRETE

More information

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class

ODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is

More information

LIMIT CYCLES COMING FROM THE PERTURBATION OF 2-DIMENSIONAL CENTERS OF VECTOR FIELDS IN R 3

LIMIT CYCLES COMING FROM THE PERTURBATION OF 2-DIMENSIONAL CENTERS OF VECTOR FIELDS IN R 3 Dynamic Systems and Applications 17 (28 625-636 LIMIT CYCLES COMING FROM THE PERTURBATION OF 2-DIMENSIONAL CENTERS OF VECTOR FIELDS IN R 3 JAUME LLIBRE, JIANG YU, AND XIANG ZHANG Departament de Matemàtiques,

More information

1.4 The Jacobian of a map

1.4 The Jacobian of a map 1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p

More information

On the Representation of Orthogonally Additive Polynomials in l p

On the Representation of Orthogonally Additive Polynomials in l p Publ. RIMS, Kyoto Univ. 45 (2009), 519 524 On the Representation of Orthogonally Additive Polynomials in l p By Alberto Ibort,PabloLinares and José G.Llavona Abstract We present a new proof of a Sundaresan

More information

Converse Lyapunov theorem and Input-to-State Stability

Converse Lyapunov theorem and Input-to-State Stability Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts

More information

Solution of Linear State-space Systems

Solution of Linear State-space Systems Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state

More information

ECE504: Lecture 8. D. Richard Brown III. Worcester Polytechnic Institute. 28-Oct-2008

ECE504: Lecture 8. D. Richard Brown III. Worcester Polytechnic Institute. 28-Oct-2008 ECE504: Lecture 8 D. Richard Brown III Worcester Polytechnic Institute 28-Oct-2008 Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 1 / 30 Lecture 8 Major Topics ECE504: Lecture 8 We are

More information

Ordinary Differential Equations II

Ordinary Differential Equations II Ordinary Differential Equations II February 9 217 Linearization of an autonomous system We consider the system (1) x = f(x) near a fixed point x. As usual f C 1. Without loss of generality we assume x

More information

Mathematical Economics. Lecture Notes (in extracts)

Mathematical Economics. Lecture Notes (in extracts) Prof. Dr. Frank Werner Faculty of Mathematics Institute of Mathematical Optimization (IMO) http://math.uni-magdeburg.de/ werner/math-ec-new.html Mathematical Economics Lecture Notes (in extracts) Winter

More information

PH.D. PRELIMINARY EXAMINATION MATHEMATICS

PH.D. PRELIMINARY EXAMINATION MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 207 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem

More information

Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN

Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 198-8 Construction of generalized pendulum equations with prescribed maximum

More information

ON THE SOLUTION OF DIFFERENTIAL EQUATIONS WITH DELAYED AND ADVANCED ARGUMENTS

ON THE SOLUTION OF DIFFERENTIAL EQUATIONS WITH DELAYED AND ADVANCED ARGUMENTS 2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 57 63. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu

More information

LINEAR ALGEBRA: THEORY. Version: August 12,

LINEAR ALGEBRA: THEORY. Version: August 12, LINEAR ALGEBRA: THEORY. Version: August 12, 2000 13 2 Basic concepts We will assume that the following concepts are known: Vector, column vector, row vector, transpose. Recall that x is a column vector,

More information

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1 Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium

More information

Chini Equations and Isochronous Centers in Three-Dimensional Differential Systems

Chini Equations and Isochronous Centers in Three-Dimensional Differential Systems Qual. Th. Dyn. Syst. 99 (9999), 1 1 1575-546/99-, DOI 1.17/s12346-3- c 29 Birkhäuser Verlag Basel/Switzerland Qualitative Theory of Dynamical Systems Chini Equations and Isochronous Centers in Three-Dimensional

More information

Applied Differential Equation. November 30, 2012

Applied Differential Equation. November 30, 2012 Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Structural Consensus Controllability of Singular Multi-agent Linear Dynamic Systems

Structural Consensus Controllability of Singular Multi-agent Linear Dynamic Systems Structural Consensus Controllability of Singular Multi-agent Linear Dynamic Systems M. ISAL GARCÍA-PLANAS Universitat Politècnica de Catalunya Departament de Matèmatiques Minería 1, sc. C, 1-3, 08038 arcelona

More information

THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS

THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS GUILLAUME LAJOIE Contents 1. Introduction 2 2. The Hartman-Grobman Theorem 2 2.1. Preliminaries 2 2.2. The discrete-time Case 4 2.3. The

More information

Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.

Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M. 5 Vector fields Last updated: March 12, 2012. 5.1 Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that

More information

Lecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases.

Lecture 11. Andrei Antonenko. February 26, Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Lecture 11 Andrei Antonenko February 6, 003 1 Examples of bases Last time we studied bases of vector spaces. Today we re going to give some examples of bases. Example 1.1. Consider the vector space P the

More information

Optimization Theory. A Concise Introduction. Jiongmin Yong

Optimization Theory. A Concise Introduction. Jiongmin Yong October 11, 017 16:5 ws-book9x6 Book Title Optimization Theory 017-08-Lecture Notes page 1 1 Optimization Theory A Concise Introduction Jiongmin Yong Optimization Theory 017-08-Lecture Notes page Optimization

More information

Asymptotic behavior of Ginzburg-Landau equations of superfluidity

Asymptotic behavior of Ginzburg-Landau equations of superfluidity Communications to SIMAI Congress, ISSN 1827-9015, Vol. 3 (2009) 200 (12pp) DOI: 10.1685/CSC09200 Asymptotic behavior of Ginzburg-Landau equations of superfluidity Alessia Berti 1, Valeria Berti 2, Ivana

More information

Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then

Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then 1. x S is a global minimum point of f over S if f (x) f (x ) for any x S. 2. x S

More information

A REPRESENTATION THEOREM FOR ORTHOGONALLY ADDITIVE POLYNOMIALS ON RIESZ SPACES

A REPRESENTATION THEOREM FOR ORTHOGONALLY ADDITIVE POLYNOMIALS ON RIESZ SPACES A REPRESENTATION THEOREM FOR ORTHOGONALLY ADDITIVE POLYNOMIALS ON RIESZ SPACES A. IBORT, P. LINARES AND J.G. LLAVONA Abstract. The aim of this article is to prove a representation theorem for orthogonally

More information

Autonomous systems. Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous.

Autonomous systems. Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous. Autonomous equations Autonomous systems Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous. i f i(x 1, x 2,..., x n ) for i 1,..., n As you

More information

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations.

We shall finally briefly discuss the generalization of the solution methods to a system of n first order differential equations. George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 1 Ordinary Differential Equations In this mathematical annex, we define and analyze the solution of first and second order linear

More information

1 The Observability Canonical Form

1 The Observability Canonical Form NONLINEAR OBSERVERS AND SEPARATION PRINCIPLE 1 The Observability Canonical Form In this Chapter we discuss the design of observers for nonlinear systems modelled by equations of the form ẋ = f(x, u) (1)

More information

Math 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.

Math 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses

More information

Math 273, Final Exam Solutions

Math 273, Final Exam Solutions Math 273, Final Exam Solutions 1. Find the solution of the differential equation y = y +e x that satisfies the condition y(x) 0 as x +. SOLUTION: y = y H + y P where y H = ce x is a solution of the homogeneous

More information

Weighted balanced realization and model reduction for nonlinear systems

Weighted balanced realization and model reduction for nonlinear systems Weighted balanced realization and model reduction for nonlinear systems Daisuke Tsubakino and Kenji Fujimoto Abstract In this paper a weighted balanced realization and model reduction for nonlinear systems

More information

Convergence Rate of Nonlinear Switched Systems

Convergence Rate of Nonlinear Switched Systems Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the

More information

Generalized Riccati Equations Arising in Stochastic Games

Generalized Riccati Equations Arising in Stochastic Games Generalized Riccati Equations Arising in Stochastic Games Michael McAsey a a Department of Mathematics Bradley University Peoria IL 61625 USA mcasey@bradley.edu Libin Mou b b Department of Mathematics

More information

arxiv: v1 [math.ds] 27 Jul 2017

arxiv: v1 [math.ds] 27 Jul 2017 POLYNOMIAL VECTOR FIELDS ON THE CLIFFORD TORUS arxiv:1707.08859v1 [math.ds] 27 Jul 2017 JAUME LLIBRE AND ADRIAN C. MURZA Abstract. First we characterize all the polynomial vector fields in R 4 which have

More information

PERIODIC SOLUTIONS OF EL NIÑO MODEL THROUGH THE VALLIS DIFFERENTIAL SYSTEM

PERIODIC SOLUTIONS OF EL NIÑO MODEL THROUGH THE VALLIS DIFFERENTIAL SYSTEM This is a preprint of: Periodic solutios of El niño model thorugh the Vallis differential system, Rodrigo D. Euzébio, Jaume Llibre, Discrete Contin. Dyn. Syst., vol. 34(9), 3455 3469, 214. DOI: [1.3934/dcds.214.34.3455]

More information