A New Invariance Property of Lyapunov Characteristic Directions S. Bharadwaj and K.D. Mease Mechanical and Aerospace Engineering University of California, Irvine, California, 92697-3975 Email: sanjay@eng.uci.edu, kmease@uci.edu Abstract Lyapunov exponents and direction æelds are used to characterize the time-scales and geometry of general linear time-varying LTV systems of diæerential equations. Lyapunov exponents are already known to correctly characterize the time-scales present in a general LTV system; they reduce to real parts of eigenvalues when computed for linear time-invariantlti systems and real parts of Floquet exponents when computed for periodic LTV systems. Here, we bring to light new invariance properties of Lyapunov direction æelds to show that they are analogous to the Schur vectors of an LTI system and reduce to the Schur vectors when computed for LTI systems. We also show that the Lyapunov direction æeld corresponding to the smallest Lyapunov exponent when computed for an LTI system with real distinct eigenvalues reduces to the eigenvector corresponding to the smallest eigenvalue and when computed for a periodic LTV system with real distinct Floquet exponents, reduces to the Floquet direction æeld corresponding 1 Introduction In the eæort towards understanding the geometric structure and qualitative behavior of the state-space æow of smooth ænite-dimensional nonlinear dynamical systems of the form _x = fx ; x 2 R n 1 where fæ : R n! TR n is a smooth vector æeld, a commonly adopted ærst step is to study the linearized dynamics about a reference orbit ææ of Eq.1, i.e., _æ = f æt. Linear time-varying LTV systems of interest to us typically arise from such a linearization about a reference orbit which we assume to be smooth and bounded. The resulting LTV system is given by @fæt æ _x = @x where æx 2 T x R n and where At 2 R næn to have bounded entries. = Atæx 2 is assumed Recall that the transition matrix æt; associated with the LTV system is given by @æ @t = Atæ ; æ;=i n 3 where I n is the identity matrix of order n. Unlike LTI systems, no closed form expression exists in general relating the transition matrix æ of an LTV system to At. Also note that æxt =æt; æx for all t; 2 R. The LTV system determines how tangent vectors are mapped between the diæerent tangent spaces T æt R n along the reference trajectory. Knowledge of the directions of tangent vectors that growdecay at extremal rates when mapped by the æow determined by the LTV system and their corresponding rates of expansioncontraction yields information about convergencedivergence of state trajectories neighboring ææ and their corresponding rates. A complete spectral characterization of the LTV dynamics aims at uncovering the diæerent asymptotic rates of growthdecay of solutions of the LTV system along with their corresponding directions. Geometrically speaking, we seek linearly independent time-varying direction æelds along ææ that identify the diæerent characteristic directions on each tangent space T æt R n along the reference trajectory. When the reference trajectory is an isolated equilibrium point of Eq.1, i.e., æt = æ 0 for all t 2 R, the resulting LTV system becomes linear time-invariant LTI and is determined by At = A = constant for all t 2 R. The spectral structure of an LTI system, æ _x = Aæx, is completely characterized by the eigenvalues and eigenvectors of the matrix A. When the reference trajectory is an isolated periodic orbit of Eq.1, i.e, æt + kt =æt for all t 2 R;k 2Zwhere T 2 R + is the minimal period, the spectral structure of the resulting periodic LTV system is determined by Floquet theory ë3ë. Avast body of literature ë1, 8, 10, 11, 12ë exists on eæorts to characterize the spectral structure of general LTV systems which arise when the reference trajectory is not necessarily either an equilibrium point or a periodic orbit, beginning with the pioneering work
of Lyapunov ë7ë in his 1892 thesis. Lyapunov introduced the notion characteristic exponents to characterize solutions of LTV dynamics with extremal evolution rates which have since received a lot of attention from theoretical and computational perspectives ë1, 2ë. The same cannot be said of the associated Lyapunov direction æelds. By investigating the invariance properties of Lyapunov direction æelds we show that in addition to identifying directions of extremal growthdecay of solutions, they span invariant distributions see ë6ë for a description of distributions analogous to the invariant subspaces spanned by Schur vectors of LTI dynamics. 1.1 Lyapunov Exponents The theory of Lyapunov exponents is described in wellknown texts ë5, 9ë. For regular LTV systems ë1, 2, 9ë, the Lyapunov exponents are well-deæned limits on tangent subbundles of ææ on which the evolution rates are extremal. They are given by ë5ë 1 i ëæx i ë = lim ln kæx i tk t!1 t 1 i ëæx i ë = lim ln kæt; æx i k t!1 t, where æx i t ; 1 i n constitute a set of normal basis solutions ë5ë of Eq.2. In the sequel, we assume that LTV systems of interest possess the property of regularity. It is important to note that Lyapunov exponents are constants over the entire trajectory ææ and are independent of the starting point æ. Also, the exponents are known to be invariant under Lyapunov transformations ë5ë. Lyapunov exponents computed for an LTI system yield the real parts of the eigenvalues and when computed for a periodic LTV system yield the real parts of the Floquet exponents see ë5ë,thm. 63.4. 1.2 Lyapunov Directions The Lyapunov directions at any given time along the reference trajectory ææ point in the direction of extremal average rates of growthdecay. So, they can be determined at any initial time by vectors æx i which extremize kæt; æx i k as t!1. The solutions æx i to this extremization problem is given by the eigenvectors of lim t!1 æ T t; æt; 1=2t ë2ë. Greene and Kim ë4ë have shown that when the timescales given by the Lyapunov exponents are distinct, the corresponding Lyapunov directions depend only on the position in state space given by æ. The Lyapunov direction æelds are everywhere orthogonal and determine directions at each point along the reference trajectory along which the average growthdecay rate is extremal. However, Lyapunov direction æelds in general are not solutions of the LTV system. For example, when the Lyapunov directions are propagated forward from some time they do not in general remain orthogonal. In other words, not all of the Lyapunov direction æelds are invariant under the linear æow of the LTV dynamics. In the following section we investigate the invariance properties of Lyapunov direction æelds for regular LTV systems with distinct Lyapunov exponents. 2 Invariance Properties of Lyapunov Direction æelds The following lemma shows that when a particular Lyapunov direction is propagated forward by the linear æow, the time-evolved direction can develop components only along Lyapunov direction æelds associated with smaller Lyapunov exponents. Lemma 1 Consider a regular LTV system of the form 2 which has the distinct ordered Lyapunov exponents i ;i=1;::: ;n where 1 é:::é n and corresponding time-varying directions l i t;i=1;::: ;n for all t 2 R. Propagate these directions l i forward from an initial time to t; t é using the transition matrix so that m i t =æt; l i. Then 0 if iéj; hm i t;l j ti= 4 nonzero if i j where hæ; æi denotes the inner product and 1 i; j n. Proof: Recall that the Lyapunov exponents i are computed by 1 i ël i ë= lim ln kæt; l i k 5 t!1 t, where the Lyapunov directions l i t are mutually orthogonal at each t 2 R because they arise from the computation of eigenvectors of a symmetric matrix. Therefore 0 if i 6= j; hl i ;l j i =æ ij = 1 if i = j for all 1 i; j n. However, when propagated forward from time to t 1 é using æt 1 ;, the propagated vectors m i t 1 =æt 1 ;l i are in general no longer mutually orthogonal. Let us assume that hm i t 1 ;l j t 1 i6= 0 for all iéjand show that we arrive at a contradiction. Since vectors l j t 1 are linearly independent in fact orthogonal, we can express the vectors m i t 1 as linear combinations of l j t 1. For each i, let m i t 1 = nx j=1 c ij l j t 1 6
The above assumption then corresponds to c ij 6= 0 for all iéj. Eq. 6 implies that for each i, æt; t 1 m i t 1 = nx j=1 c ij æt; t 1 l j t 1 As t becomes large, the right hand side is dominated by the term corresponding to the vector l j ;j = j æ for which æt; t 1 l j æt 1 is the largest and c ij æ 6= 0. Since we have assumed the Lyapunov exponents which determine the rates of change in size of vectors when propagated by the linear æow to be arranged in descending order, the dominant term corresponds to when j æ is the smallest positive integer for which c ij æ 6= 0. Recall that the Lyapunov exponents i are constants along the reference trajectory ææ and independent of the starting point æ on the reference trajectory. Compute the Lyapunov exponents from the initial directions m i t 1 using 1 i = lim ln kæt; t 1 m i t 1 k t!1 t, t 1 7 In the light of the earlier argument, we can see that kæt; t 1 m i t 1 k!c ij ækæt; t 1 l j æk as t!1and that i = j æ. This contradicts the fact that Lyapunov exponents are constants independent of the starting point on the reference trajectory unless j æ = i. The condition j æ = i holds if and only if c ij = 0 for all i é j. Since this discussion is valid for any time t 1 é, we drop the subscript and conclude that hm i t;l j ti= for all times té. 0 if iéj; nonzero if i j A direct consequence of the above lemma is the following theorem. Theorem 1 For any 1 k n, the Lyapunov direction æelds corresponding to the k smallest Lyapunov exponents of the LTV system 2 deæne an invariant distribution ææt = spanfl n,k+1 t;::: ;l n tg. Proof: Consider the Lyapunov directions l n,k+1 ;::: ;l n at some initial time which correspond to the k smallest Lyapunov exponents. When propagated forward to time t, let m i t =æt; l i for all n, k + 1 i n. Lemma 1 implies that at time t, spanfm n,k+1 t;::: ;m n tg = spanfl n,k+1 t;::: ;l n tg. This is because, the directions m n,k+1 t;::: ;m n t cannot have components along any of the directions l 1 t;::: ;l n,k t. Consequently, ifv2ææ then æt; v 2 ææt for all té. Therefore æ is an invariant distribution. The above theorem and and lemma show that for regular LTV systems with distinct Lyapunov exponents the Lyapunov direction æelds deæne a collection of invariant distributions æ k æ = spanfl k ;::: ;l n g such that dimæ k = n, k +1 and æ 1 ::: æ n. When æxt 2 æ i then kæxtk Ke it where K is a positive constant. When these results are applied to an LTI system typically resulting from linearizing nonlinear dynamics about an equilibrium point æ 0, the distributions æ i become subspaces W i of T æ0 R n. So the Lyapunov directions reduce to Schur vectors that span the respective nested invariant subspaces W 1 ::: W n. This establishes that the Lyapunov direction æelds for an LTV system are time-varying analogs of Schur vectors for an LTI system. Since a special case of the above theorem where k =1 has many important consequences, we present it as a separate result. Theorem 2 The direction æeld l n æ corresponding to the smallest Lyapunov exponent n is invariant when propagated using the transition matrix, i.e., æt; l n = n t; l n t. Proof: For the LTV system 2 with Lyapunov exponents 1 é ::: é n, l n t is the direction æeld corresponding to the smallest Lyapunov exponent n. Applying Lemma 1 to the direction l n results in hæt; l n ;l j ti=0; 8j6=n 8 This implies that at time t, the direction l n t points in the same direction as m n t =æt; l n, i.e, æt; l n = n t; l n t 9 where n t; is the scaling factor that determines that rate of growth of the size of l n. Hence wehave shown that the Lyapunov direction æeld l n æ corresponding to the smallest Lyapunov exponent is invariant when propagated by the transition matrix. The following important corollaries and examples further justify the validity of the direction information provided by this Lyapunov direction æeld as an accurate generalization by showing that it is indeed consistent with the known results in the special cases of LTI systems and periodic LTV systems. Corollary 1 The Lyapunov direction corresponding to the smallest Lyapunov exponent, computed for an LTI
system æ _x = Aæx which is assumed to have real and distinct eigenvalues ordered from largest to smallest, points in the same direction as the eigenvector direction corresponding to the smallest eigenvalue. Proof: As stated earlier, it is already known that the Lyapunov exponents of an LTI system are the same as real parts of the eigenvalues of A. Theorem 2 states that the Lyapunov direction l n corresponding to the smallest Lyapunov exponent satisæes the condition of invariance when propagated using æt; = expa æ t,. Therefore æt; l n = n t, l n where n = exp n æ t,, i.e., l n is an eigenvector of æ with eigenvalue n which is the same as the eigenvector of A corresponding to the smallest eigenvalue. The following example shows that the Lyapunov directions computed for an LTI system reduce to the Schur vectors and in addition illustrates the above corollary. Example 1 Consider the LTI system æ _x = Aæx where,6 5 A = 10 4,5 The eigenvalues of A are 1 =,1 ; 2 =,10 and the corresponding eigenvectors are e 1 = ë1; 1ë and e 2 = ë5;,4ë respectively. The Schur vectors obtained by orthogonalizing the eigenvectors are s 2 = e 2 and s 1 = ë4; 5ë. The transition matrix of this LTI system is given by æt; = expa æ t, = 1=9ëæ ij ë where æ 11 = 4e,t, +5e,10t, æ 12 =,5e,10t, +5e,t, æ 21 =,4e,10t, +4e,t, æ 22 = 5e,t, +4e,10t, The eigenvalues of æ T t; æt; are q 1;2 t; =e,2t, w æ w 2, 6561e,18t, where w =41e,18t,, e,9t, +41. The Lyapunov exponents are given by 1 1;2 = lim t!1 2t, ln 1;2 11 As t!1, 1 t;! a exp,2t, and 2 t;! b exp,20t, where a and b are some constants whose true value is unimportant for our purpose. Therefore we ænd the Lyapunov exponents 1 =,1 and 2 =,10 which are the same as the eigenvalues 1 and 2 of A. The corresponding Lyapunov directions are given by the eigenvectors of æ T t; æt; as t!1. They are l 1;2 = lim t!1,9e,9t, +9æ p w 2,6561e,18t, 41e,18t,, e,9t,, 40 Therefore, l 2 = ë50;,40ë and l 1 = ë,32;,40ë which point in the same directions as the Schur vectors s 2 and s 1 respectively. Also note that l 2 points in the same direction as e 2. We shall now show through the following corollary that the result of Theorem 2 is also consistent with the results of Floquet theory when applied to periodic LTV systems. Corollary 2 The Lyapunov direction æeld corresponding to the smallest Lyapunov exponent, computed for a periodic LTV system æ _x = Atæx with At+T =At which is assumed to have real and distinct Floquet exponents ordered from largest to smallest points in the same direction as the Floquet direction æeld corresponding Proof: As stated earlier, it is already known that the Lyapunov exponents of a periodic LTV system are the same as real parts of the Floquet exponents. Theorem 2 states that the Lyapunov direction æeld l n æ corresponding to the smallest Lyapunov exponent satisæes the invariance condition. Therefore æt; l n = n t; l n t 12 where n = exp n æ t,. We know that the Lyapunov direction æelds are dependent only on the position in state space along the reference trajectory ææ. In this case, ææ is periodic, i.e., æ + T = æ. Consequently, l n + T =l n, i.e., l n æ is periodic. Together with the invariance condition, we get æ + T;l n = exp n T l n 13 This implies that l n is an eigenvector of æ + T; corresponding to Floquet exponent n = n. This shows that l n æ is a periodic direction æeld pointing in the same direction as the Floquet direction æeld corresponding We shall illustrate this corollary with the following example. Example 2 Consider the periodic LTV system æ _x =
Atæx where At =ëa ij të so that A 11 t =, 1 2 cos 2t + 9 2 sin 2t, 11 2 A 12 t = 1 2 sin 2t + 9 2 cos 2t + 3 2 A 21 t = 1 2 sin 2t + 9 2 cos 2t, 3 2 A 22 t =, 9 2 sin 2t + 1 2 cos 2t, 11 2 with period T =. This system was constructed from the LTI system in example 1represented here asæ_x= Bæx with a periodic transformation æx = Stæx where,6 5 cos t, sin t B = ; St = 14 4,5 sin t cos t so that At =S,1 tbst, S,1 t _St. The Floquet exponents turn out to be 1 =,1 and 2 =,10. The corresponding periodic Floquet direction æelds are given by e 1 t = cos t + sin t, sin t + cos t ; e 2 t = 5 cos t, 4 sin t,5 sin t, 4 cos t The transition matrix for this periodic LTV system is given by æt; =S T tæt; S 15 where æt; is the transition matrix in example 1. The Lyapunov exponents and associated directions are computed from the eigenvalues and eigenvectors of æ T t; æt; as t!1. Note that æ T t; æt; =S T æ T t; æt; S 16 because StS T t =I n. Consequently, the eigenvalues of æ T æ are the same as the eigenvalues of æ T æ and its eigenvectors are related to that of æ T æ by the transformation S T. Therefore, the Lyapunov exponents of the periodic LTV system turn out to be 1 =,1 and 2 =,10 which are the same as the Floquet exponents. The direction æeld corresponding to the smallest Lyapunov exponent 2 computed from the eigenvector of æ T t; æt; as t!1is given by 5 l 2 =S T =,4 5 cos, 4 sin,5 sin, 4 cos 17 which is the same as the Floquet direction æeld corresponding 3 Conclusions Lyapunov direction æelds span invariant distributions on which the average asymptotic evolution rates of solutions of the LTV dynamics are extremal. They reduce to Schur vectors when computed for an LTI system thereby establishing that they are appropriate timevarying analogs of Schur vectors. In particular, the Lyapunov direction æeld corresponding to the smallest Lyapunov exponent is invariant under the linear æow and this direction æeld is consistent with the corresponding eigenvector when computed for LTI dynamics and with the corresponding Floquet direction æeld when computed for periodic LTV dynamics. Illuminating examples were presented to illustrate these results. References ë1ë F. Colonius and W. Kliemann, ëthe Lyapunov Spectrum of Families of Time-Varying Matrices," Transactions of The American Mathematical Society, Vol. 348, No. 11, pp. 4389í4408, 1996. ë2ë L. Dieci, R. D. Russell and E. S. Van Vleck, ëon the Computation of Lyapunov Exponents for Continuous Dynamical Systems," SIAM Journal of Numerical Analysis, Vol. 34, No. 1, pp. 402í423, 1997. ë3ë M. Farkas, Periodic Motions, Springer-Verlag, New York, 1994. ë4ë J. M. Greene, and J.-S. Kim, ëthe Calculation of Lyapunov Spectra," Physica D, Vol. 24, pp. 213í225, 1987. ë5ë W. Hahn, Stability of Motion, Springer-Verlag, New York, 1967. ë6ë A. Isidori, ënonlinear Control Systems", Second edition, Springer, 1994. ë7ë A. M. Lyapunov, ëthe General Problem of Stability of Motion," International Journal of Control, Vol. 55, No. 3, pp. 531-773, 1992. ë8ë R. J. Sacker and G. R. Sell, ëa Spectral Theory for Linear Diæerential Systems," Journal of Diæerential Equations, Vol. 27, pp. 320-358, 1978. ë9ë G. Sansone and R. Conti, Nonlinear Diæerential Equations. Pergamon Press, New York, 1964. ë10ë W. E. Wiesel, ëoptimal Pole Placement in Time- Dependent Linear Systems," Journal of Guidance, Control and Dynamics, Vol. 18, No. 5, pp. 995í999, 1995. ë11ë M.-Y. Wu, ëa New Concept of Eigenvalues and Eigenvectors and its Applications," IEEE Transactions on Automatic Control, Vol. 25, No. 4, pp. 824í826, 1980. ë12ë J. J. Zhu, ëa Uniæed Spectral Theory for Linear Time-Varying Systems - Progress and Challenges," Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, IEEE, New York, Vol. 3, pp. 2540í2546, 1995.