APPENDIX A4. LINEAR STABILITY 844

(b) the set is closed with respect to scalar multiplication and vector addition

a(x + y) = ax + ay , a, b ∈ F , x, y ∈ V (a + b)x = ax + bx a(bx) = (ab)x Appendix A4 1 x = x , 0 x = 0 . (A4.1)

Here the field F is either R, the field of reals numbers, or C, the field of complex numbers. Given a subset V0 ⊂ V, the set of all linear combinations of elements of Linear stability V0, or the span of V0, is also a vector space.

A basis. {e(1), ··· , e(d)} is any linearly independent subset of V whose span is V. The number of basis elements d is the dimension of the vector space V. Mopping up operations are the activities that engage most scientists throughout their careers. — Thomas Kuhn, The Structure of Scientific Revolu- Dual space, dual basis. Under a general linear transformation g ∈ GL(n, F), the ( j) −1 j (k) tions row of basis vectors transforms by right multiplication as e = k(g ) k e , ′ and the column of xa’s transforms by left multiplication as x =P gx. Under left multiplication the column (row transposed) of basis vectors e(k) transforms † k † −1 ⊤ he subject of linear algebra generates innumerable tomes of its own, and is as e( j) = (g ) j e(k), where the dual rep g = (g ) is the transpose of the inverse way beyond what we can exhaustively cover. Here we recapitulate a few of g. This observation motivates introduction of a dual representation space V¯ , T essential concepts that ChaosBook relies on. The punch line is Eq. (A4.25): the space on which GL(n, F) acts via the dual rep g†.

Hamilton-Cayley equation (M − λi1) = 0 associates with each distinct root Definition. If V is a vector representation space, then the dual space V¯ is the set λi of a matrix M a projection ontoQ ith vector subspace of all linear forms on V over the field F.

M − λ j1 (1) (d) ¯ P = . If {e , ··· , e } is a basis of V, then V is spanned by the dual basis {e(1), ··· , e(d)}, i λ − λ Yj,i i j the set of d linear forms e(k) such that

(k) k e( j) · e = δ j ,

A4.1 Linear algebra k k where δ j is the Kronecker symbol, δ j = 1 if j = k, and zero otherwise. The components of dual representation space vectorsy ¯ ∈ V¯ will here be distinguished In this section we collect a few basic definitions. The reader might prefer going by upper indices straight to sect. A4.2. (y1, y2,..., yn) . (A4.2)

They transform under GL(n, F) as Vector space. A set V of elements x, y, z,... is called a vector (or linear) space F ′a † a b over a field if y = (g ) by . (A4.3)

For GL(n, F) no complex conjugation is implied by the † notation; that interpre- (a) vector addition “+” is defined in V such that V is an abelian group under tation applies only to unitary subgroups U(n) ⊂ GL(n, C). In the index notation, addition, with identity element 0; g can be distinguished from g† by keeping track of the relative ordering of the indices,

b b † b b (g)a → ga , (g )a → g a . (A4.4)

843 appendStability - 23jan2012 ChaosBook.org version15.9, Jun 24 2017 = i M P 846 ) , and ) i M ( ( (A4.15) (A4.13) (A4.11) (A4.14) (A4.17) (A4.16) (A4.12) (A4.18) e f d as R 1. Then it is ∈ M = x irreducible reps: c we delete one term subspace, i ations with numbers. P gonal, is the eigenvalue of ChaosBook.org version15.9, Jun 24 2017 onto the corresponding i single matrix as follows. λ x . ) on the 1 projection operators i λ = ( spectral decomposition i f P -dimensional vector 1 r = d i cients. We shall set X linearly independent vectors which can . ffi d ) . i ( ) e d . i ( . . In general, neither set is orthogonal, but by , are ) d x e is proportional to a right eigenvector ) i ) , : ) ) i ( j j P i d j ) e ( j d x P , the product of all such factors annihilates any λ ( ) factors yields the ) λ i e j + ( − ) λ + e i i λ λ − ··· ( ··· i i − complete . + λ satisfies its characteristic equation (A4.9), , j ) takes the scalar value i j , + is convention dependent and not worth fixing, unless you . Y λ P (no sum on 2 (2) c M ( (no sum on 1 ) j 0 M and j ( i e = P . λ f λ λ 2 = 2 = j ( i x x , λ ) ) − f ) δ j − j 1 ( i i 1 c j + + i ) annihilates and completeness relation (A4.14) we can rewrite e P λ P i λ λ i 1 1 M ) i j i = X (1) λ 1 P δ i λ i ) − − e 1 j

A4. LINEAR STABILITY , = λ j ( M 1 = 1 λ = Y − e ) x i M M orthogonal j − · = ( ( = and is thus easily evaluated through its = i M ) P 1 = M , i i i ( , d ( = M i j f i M MP M x P ( Y P Y e subspace: P i ) By (A4.10) every column of i P λ ( f If all eigenvalues are distinct e on Using be used as a (non-orthogonal) basis for any which are From (A4.10) it follows that It follows from the characteristic equation (A4.12) that This, of course, is the reason why anyone but a fool works with they reduce matrix (AKA “operator”) evaluations to manipul Any matrix function vector, and the matrix matrix ( the idempotence condition (A4.14), they are mutually ortho This humble fact has a name:from the Hamilton-Cayley this theorem. product, If we find that the remainder projects Dividing through by the ( eigenspace: its every row to a left eigenvector feel nostalgic about Clebsch-Gordan coe The non-zero constant convenient to collect all left and right eigenvectors into a APPENDIX appendStability - 23jan2012 845 (A4.8) (A4.7) (A4.9) (A4.6) distinct (A4.10) d is associative plication, one acts by scalar uted by some : , so that for any T matrix product M -dimensional vectors . They form a matrix d ChaosBook.org version15.9, Jun 24 2017 forms an algebra if, in . T ¯ 1 on which V T·T →T + i ⊗ λ M V also belongs to of Re : β ∈ ) ; (A4.5) t i 0, acting on ≥ ( · α . t C e i , α , γ γ λ t ∈ t ¯ t V i · · γ λ structure constants ⊗ β α t t αβ , V . a b distributive + + of the algebra itself. are the roots of its characteristic polynomial ) 0 of a vector space ∈ β γ β r t t t ( α α = M · · t t b c ) ) , , τ α α eigenvectors λ , with all α t t i ) t γ , the product ( t γ λ − M t γ . i · − , = = λ αβ ∈T are called the β . Typical examples of products are the c b c b ( are linearly independent. There are at most ] matrix τ t ) ) β ( γ β β d ) ) 0 γ 1 elements t j t t with respect to multiplication · t γ = ( ( , ( − Y × αβ t r · γ r b a b a X e α α τ d ) ) . t t ) , = + ) α α β = i t t γ ) t ( β = ( ( t β 1 e and i t ( αβ Lie algebra γ + closed λ = = t of a [ · · τ λ ) associative i α · ( c a c a − α t α ≡ ) ) ) =

A4. LINEAR STABILITY e t ( t A set of β β β ) erent types of algebras. For example, if the multiplication γ , t t t i M j ( β · · · ff ) λ α Lie product α α α t t t t ( det ( M e ( ( ( two elements , i Depending on what further assumptions one makes on the multi λ (a) the set is (b) the multiplication operation is , we would like to determine the algebra is and the Given a nonsingular matrix which defines a A4.2 Eigenvalues and eigenvectors Eigenvalues x multiplication by eigenvalue appendStability - 23jan2012 eigenvalues and eigenspaces, whichmethod, and we ordered assume by their have real been parts, Re comp If rep of the algebra, whose dimension is the dimension obtains di Algebra. addition to the vector addition and scalar multiplication, APPENDIX The set of numbers . } ) k ( e form exercise A4.1 A Im (A4.22) (A4.23) and the , )

based on ⊂ k T ( ′ e 848 A Re (A4.24) (A4.25) } → { ? Hamilton-

1) ). 1 + T k

sub-block − ( , so what is the meaning hood of the2 equilibrium α

e is a sum of the rescaling- −

eigenvalues. That is not d , 2] + either real or complex. All ) i N k 10 ( × λ e { [2 = , or esence of symmetries, one say about situation ChaosBook.org version15.9, Jun 24 2017 eigen-direction per a turn of the T ) ··· j . , the corresponding complex eigen- ( } e # =

has only real entries, it will in general ω , the block 1000 i 1 1 N A + − − i

, where λ →

As ′ 1 0 0 ω, µ (0) = i , A x i " t + # λ . J ω µ t { d , rotations t = The characteristic equation (A4.24) can = + ω ω = ) #

is a fundamental matrix of the system. (J. Halcrow) = α . t , see figure 4.3, with the rotation period (0) } ( (2) λ µ 1 x α )) sin x 0) t + T d cos ( At , k e − SO d e x (0 1 0 0 1 α , λ = (and not, let us say, k = X t t " λ = T ω { ω µ ) (0) ,..., ≈ radial y x ) = , t While for a matrix with generic real elements all eigen- 1 , given by sin cos Λ ( # x , − 2 we can write the solution for an arbitrary initial condition ( " x 0 , Nx ω , µ (0)

eigenvalues that lie within a diagonal t ) 0 = − t F e (0) = ( 1

d Complex eigenvalues. ) α 1 = t + x ˙ = x d k ( x ) ) ( eigenvector? µ is of order ω µ F N 1 t 1 π/ω, , λ e = k " α α 2 ,... 0) = A4.2 λ ) , λ λ ) = t = = t ( (0)

If t ( (0 − − 2 F

A4. LINEAR STABILITY x J N T x = , M M complex ) eigenvalues are degenerate, ( ( y 1 1 (0) , α identity and the generator of r r = = 1 x d x α α × ( Y Y its projection on to this set, where Example have either real eigenvalues,surprising, or but complex also conjugate the pairscoordinates corresponding used of eigenvectors in can defining a be dynamicalof flow a are real numbers a complex conjugate pair, vectors can be replaced by their realIn and this imaginary 2-dimensional parts, real representation, Trajectories of spiral in/out around radial expansion /contraction multiplierspiral: along the We learn that the typical turnover time scale in the neighbor We distinguish two cases: M can be brought tobe diagonal replaced by form. the minimal polynomial, Cayley (A4.12) now takes form appendStability - 23jan2012 Degenerate eigenvalues. APPENDIX values are distinct with probabilityor spacial 1, parameter that values is (bifurcation points). notwhere What true can in pr exercise A4.1 to of 0 847 t t J for this (A4.21) (A4.19) } ) t ( for a system independent ) d 0 ( d x e sis at time ) )) (A4.20) shearing caused t 0 a function is the ( x ) ) can be written as ( t d -dimensional space t ( ( ,..., d J independent initial conditions, ) e x l condition projected t rminant we refer to as fundamental matrix d ( ) of order 1 and dimen- ChaosBook.org version15.9, Jun 24 2017 =

t . As the system is linear, the sum ) ( ) (1) x t x e ,..., ( ) { ciently short times. ) t x t ( can thus be fashioned out of ( ffi exp( A

1 is constant in time, the system (4.2) is (1) − , i.e., for zero time the Jacobian e = A 1 ( ) ) is the linear approximation to a (0) If t 0 = = ( is a map from Φ x x ) is called the ) 1 ( T t t f t . Every solution ) − ( ( t ) J t ( Φ ( F F = Φ ) )]. t . t t (0), assemble them as columns of a matrix ( ( ) J ) d F d ( ( , x As the system is a linear, a superposition of any The set of solutions x , )) t . ( ) ..., 1 whose columns are the right eigenvectors of d ··· ( − ) , ), describing the orientation of a tangent plane to the ) e j (0) is also a solution. One can take any ) erential equations ˙ 0 0 ( i t x t , x (0) ff ( (0), e t ( . ( x t J fundamental system F ) 1 (2) (2) = f (0) t ) ,..., x x − ( x

1 Fundamental matrix. = 1 ) . Numerically this works for su ) t The Jacobian matrix , i

t is defined by the Taylor series for } − i j ( ( ) A t ) ( ) ) t (0) t e t e , F F ( fundamental matrix ( ( i 1 (1) (0), -dimensional vector space. A basis ) Φ x t A t j c = ) e = (1) d A4.1 ( t ( ) (1) x 1 ( x ) t d [ x = { ( 0 i = X exp( A4. LINEAR STABILITY x x = ( 1 = = Φ 0 − ) t ] matrix ) t ) ) − t d ( t t 1 ( t ( ( forms a × Φ J x x Example F autonomous, and the solution is where of any two solutions is also a solution. Therefore, given d d The Jacobian matrix can be written as transformation from ba erentiable function trajectories ff (0), and formally write the solution for an arbitrary initia is the inverse of a matrix is the identity. appendStability - 23jan2012 the basis at time The [ Jacobian matrix. function at a given point and the amount of local rotation and by the transformation.Jacobian The matrix inverse of of the inverse the function. Jacobian If matrix of to itself, the Jacobian matrixthe is ‘Jacobian.’ a square matrix, whose dete di Fundamental matrix (take 1). two solutions to Then the matrix form of (A4.18) is vector space is called a Φ initial states, d onto this basis, the system, and the Jacobian matrix sion Fundamental matrix (take 2). where APPENDIX of homogeneous linear di matrix ) can be (A4.30) (A4.28) (A4.31) (A4.29) M 2] t A Let’s illus-

× if we bring

850 (J. Halcrow) , with zeros t [2 exp( ln = t an be verified by J

ion in any dimension

Jordan form ove it; for a matrix

ithmic term 2] ight eigenvectors are (up to

, but is not an eigenvector, as × 0

ion operators . (A4.28) [2 ChaosBook.org version15.9, Jun 24 2017 . # = . 0

2 block there is only one eigenvector (2)

λ , Jacobian matrix ] = v 2 α 2 ) . , R ) d 0 0 0 0 λ # 1 λ . × " 0 1 − α # " , we distinguish two cases: (a) − d = λ [ A = can be brought to 1 )(1 # ( − λ = , λ (2) A # − 2 v 1 1 λ 3 3 1 λ is discussed in example A4.2. 1 , (5 1 0 0 1 λ − projection operator = − # = " − " 1 0 2 3 1 3 1 1 " 5 5 λ A 4 1

λ are the roots of (A4.27): " =

has already been described in sect. 4.8, and in the full + + } If = is the 4 1 = # 1 , λ i , 2 (1) ) # λ 6 5 e P = 1 { 1 · − ! ) − = 2 1 5 4 1 3 2 tv m m } λ . λ λ 2 v , − − " , + # = m # 6 , λ u 1 2

. For every such Jordan ) M M 1

2 λ ( ( − 1 λ 1 λ λ λ t

(1) Projection operator decomposition in 2 dimensions: { 1 4 4 1 m λ 2 − e 0 e # λ − 4 1 3 2 1 − 0 λ = + λ " A " = = ! " M v (2) u = A4.4 = =

v

= satisfies its secular equation (Hamilton-Cayley theorem) c 4 1 3 2 1 m 1 2 λ tA P A A P A4. LINEAR STABILITY P M det( " e M =

helps span the 2-dimensional space, (2) v (2) A Distinct eigenvalues case Degenerate eigenvalues. Example v generality for arbitrary dimensionoverall multiplicate in factors) sect. the rows/columns 5.1. of project The left/r Complex eigenvalues pair case brought to diagonal form.we continue This is in theeverywhere appendix except easy A7.2.1. for case the whose diagonal,the (b) discuss and Jordan some form is 1’s directly ab per block. Noting that we see that instead of acting multiplicatively on picks up a power-lowthe correction. extra term That into spells the trouble exponent). (logar trate how the distinct eigenvalues case works with the Its eigenvalues That explicit calculation: Associated with each root

APPENDIX appendStability - 23jan2012

has , B # (A4.27)

transfor- bt bt 849 acts sin is called the cos − d

M Enumeration of

such that R (A4.26) A t of basis bt bt sin cos

g solutions for the sim- . " he third is a combination forms to . What we have done is # at

This is the messy ary finite-dimensional ma- tor combination is beyond ir of eigenvalues. It follows 1 e − ce. Matrix ω having (1) distinct real eigen- = U 9): − ue to a finite or com- Bt A Bt ChaosBook.org version15.9, Jun 24 2017 e µ ω µ Ue " = , = . At # B e 0 t = 1 0 1 A , ) # " A λ t det λ 1 λ − e + 2 = λ λ 4det 0 λ . This is the easy case whose discussion we )( } A Bt − ) " λ e α tr 2 matrices can always be brought to such Hermi- d ) = − non-singular matrix − α, A , there is a similarity transformation 1 ( B 2 M 2 λ e 2] consist of the generalized eigenvectors of . ( λ R , (tr , × # # p U [2 t 1 2 = = , ··· µ 12 22 0 , e # A A , , ± 2) ) ) matrices as belonging to one of three classes of geometrical 0 µ A k 1 α, Decomposition of 2-dimensional vector spaces: t ( λ α, 0 AU λ tr 2] 11 21 e ( e 1 , 2 1 e A A 0 × λ − − i " 1) λ U " [2 " A = α, = A4.3 ( 2 = = = = , e 1 Bt { ) B B

A4. LINEAR STABILITY A λ det( e k α, ( -dimensional subspace spanned by a linearly independent se α Example M e every possible kind ofwhat linear we algebra can eigenvalue reasonably / undertakeplest here. eigenvec case, a However, general enumeratin takes us a longtrices. way toward The developing eigenvalues intuition about arbitr are the roots of the characteristic (secular) equation (A4. For any linear system in where the columns of one of the following forms: These three cases, called normalvalues, forms, (2) correspond degenerate to real eigenvalues, orthat (3) a complex pa where the corresponding Jacobianclassify all matrix is mations. The first case isof scaling, rotation the second and is scaling. aJordan shear, normal The and form. t generalization of these normal d appendStability - 23jan2012 where the product includes each distinct eigenvalue only on tian, diagonalizable form. M can only be broughtcase, to so upper-triangular, we Jordan only form. illustrate the key idea in example A4.3. eigenvectors continue in appendix A7.2.1. Luckily, if the degeneracy is d pact symmetry group, relevant multiplicatively APPENDIX on a . 852 (0)) cients x ( for plane ) (A4.33) (A4.34) ffi j 5 ( trix enjoy e EQ = as ‘covariant ) with respect ) t he two zero expo- ) j ( j per each period ). The continu- ( ( x j x (0), the object of e e ( δ x , Λ T ) 3 below in (5.9). f j s form, ( int 2 e s ues do not, so we may ) ChaosBook.org version15.9, Jun 24 2017 1 t s of equilibrium and ( . j refer to a full traversal of (the ‘starting’ point of the x ω ) x Thus each solution of the j , i ) x ( j δ ) . ( e t ± ( λ ) j µ P t ( (0) u from the two translational symmetries. j j = = x x 5 ) factor is not an eigenvalue of the = λ t δ δ ) ) j ) ) t ) is multiplied by j j EQ ( ( t ( T ( x ω = Λ j λ δ + x contraction by rewriting the coe j t δ / ( X (0) j : exponents j ) depends upon u u = x ) describes the variation of rT ( t ) ( j p , j ( , J ) e u . The exp( t , and we refer to eigenvectors t ( ) , T j x , ) 5 T j u ) ) ( j x ) EQ ( + e t ( ) µ ) ) t j j t ( ( ( j 0.072121610.062095260.06162059 0.040749890.020730750.009925378 SSS 0.009654012 0.073551430.009600794 0.045512741.460798e-06 SSS 0.2302166-0.0001343539 1.542103e-06 AAS-0.006178861 0.231129 SAA --0.007785718 SAA - ASA A -0.01064716 0.1372092 AAS-0.01220116 SAA -0.01539667 AAS-0.03451081 0.2774336-0.03719139 0.2775381 0.08674062 SSS ASA 0.215319 SAA ASA SAA SAA ( j e λ x j u δ t ) j ( j exp( j λ 3 4 7 =Λ 16 19 , it is only an interpolation between 5,6 8,9 1,2 X e t ) 10,11 12,13 14,15 17,18 20,21 22,23 24,25 26,27 = J x = 400. The exponents are ordered by the decreasing real part. T j in the (A4.33) eigenbasis, ( ) ) ) = X j t x ( ( A4. LINEAR STABILITY T j δ e = Re The first 27 least stable Floquet exponents x + ) are independent of in (A4.34) does not imply that eigenvalues of the Jacobian ma ) δ x t t j t ) periodic with period ( ( ( t p x x ( Λ j J δ δ A4.1: u ) as t Expand ( j x Even though the Jacobian matrix where write the eigenvalue equation as periodic orbit), we shall show in sect. 5.3 that its eigenval nents, to the numerical precisionFor of details, see our ref. computation, [13.43]. arise vectors’, or, for periodic orbits, as ‘Floquet vectors’. Taking into account (A4.32), we get that We can absorb this exponential growth δ Table Couette flow, equation of variations (4.2) may be expressed in the Floquet the periodic orbit. Indeed, while dynamical significance is the co-moving eigen-frame defined appendStability - 23jan2012 Jacobian matrix with to the stationary eigen-frame fixed by eigenvectors at the po any multiplicative property for APPENDIX ous time x = e

Two n

arac- A . i P

i by spec- ) t λ ( M x 851 =

imple: d on δ t eigenvector, i (A4.32) MP

th subspace is given . i

do not form an orthog-                 . } t ) d j # get the eigenvectors: ny function of λ ( , however, are orthogonal .3.

) is diagonal (the system is e e k { ( ready introduced in A e

o · If ChaosBook.org version15.9, Jun 24 2017 ) erential equations with . j . ( ff . e t 2 λ e 58591 19531 58593 19529 is the diagonal matrix of the eigenvalues , linearly independent eigenvectors, forcing p " t D 1 d

λ . Inserting this into the Taylor series for = 1 e 2 − . ∈ M                 P F p n ) 3) = t , (                , where ) −

satisfies the eigenvalue equation x FD 1 T t , i ∈ M − d (1 } } P = # # + x λ + 1 3 t 1 1 ) ( 1 − − may not have 1 ) satisfies the same equation: moreover the two FDF " " A P − , , T A . = 3) # # = . , , at point + 3 1 1 1 . ) ) − A " " T t t t FDF { { 7 ( ( is given by ( (

t . But x x A (5 1 2 δ tA δ − ... λ = = e ) = ) F Computing matrix exponentials. x 1 ( 1 t − } Dt in case at hand both subspaces are 1-dimensional. From the ch satisfies the eigenvalue equation, its every column is a righ } p · cients. Consider the equation of variations (4.2) evaluate 1 i , ; is the matrix of corresponding eigenvectors, the result is s J )). The periodicity of the stability matrix implies that if (2) (2) λ 3 of period i P Fe are orthonormal and complete, The dimension of the ffi t x e e ( i P , ,                p − = δ F A4.5 x FDF = P ) 7 ( ) tr (1) (1) t )( e e ( A At T A4. LINEAR STABILITY 1 { M { exp = − e A and + i

is diagonalizable, = t = d A ( ) A t x x FDF ˙ ( δ δ Example Matrices by teristic equation it followsconsequences that are immediate. First, wetral can decomposition, easily for evaluate example a Second, as and every row a left eigenvector. Picking first row/column we with overall scale arbitrary.The matrixonal is not basis. hermitian The , left-right s as eigenvector in (A4.18), dot by products inspection. (Continued in example 15.2.) uncoupled), then If of ( gives us to take a different, Jordan route, explained in example A4 A solutions are related by (4.5) is a solution, then also appendStability - 23jan2012 with chapter 4 inherit names from the Floquet theory of di A4.2.1 Floquet theory When dealing with periodic orbits, some of the quantities al a periodic orbit APPENDIX time-periodic coe ] ) ′ p q , 854 , q ′ Pois- ( P59.77 p } [ Q = q (A4.36) (A4.35) (A4.37) ctors. The , p = = p { ′ Q x ) are given by p . = , j i ′ ) q ch more than the t q p ) ( j ( on for 8.9) in a form con- ∂ p ( ∂ rves the 1 x Q G − D alluded to in sect. 8.1. = ChaosBook.org version15.9, Jun 24 2017 = of periodic orbit Q j i ′ ω p q i ∂ ∂ ± , µ . Writing zero eigenvalue, to the numerical pre-        , ) j tional SO(2) symmetry of this periodic = M p ( j i ′ ′ ′ p q p p λ p q ω ∂ ∂ ∂ ∂ ∂ ∂ (M.J. Feigenbaum and P. Cvitanovi´c) − ′ ′ p q q q = ∂ ∂ ∂ ∂ j i ′        q p ∂ dependent quantity ∂ = q . 1 ) j , − p ( i j , p µ 0 δ 0 j i ′ M 0.07212161 0.04074989 0.02073075 0.07355143 0.0096540120.009600794 0.04551274 0.2302166 q q ∂ ∂ = = = ) j = 1 0.06209526 ? p ( -1 0.06162059 -1 0.009925378 , j j j j i ′ k ′ ′ k ′ k σ 400, together with the symmetries of corresponding eigenve        p p p q q p q q ]. = ? ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ′ ′ p p i i p ′ k q i ′ k ′ k ∂ ∂ ∂ ∂ Re q p p q p p j 7 3 4 ∂ , ∂ ∂ ∂ ∂ ∂ 1,2 5,6 8,9 j i 10,11 ′ ′ ′ − q q p − − p p q ∂ ∂ ∂ ∂ j ∂ j j ∂ ′ k ′ k ′ k A4. LINEAR STABILITY p q p p p p        The first 13 least stable Floquet exponents = ∂ ∂ ∂ ∂ ∂ ∂ j i i = i i ′ k ′ k ′ ′ k p q q q q q q q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ M A4.2: In terms of the Poisson brackets, the time-evolution equati appendStability - 23jan2012 From (8.26) we obtain A4.4 Stability of Hamiltonian flows we can spell out the symplectic invariance condition (8.9): (19.28). Taken together, (A4.37) and (A4.36) imply that the flow conse is given by (19.30). We now recast the symplectic condition ( venient for using theand symplectic the constraints Jacobian on matrix and its inverse Table The symplectic structure ofincompressibility, or Hamilton’s the equations phase buys spaceThe us volume evolution conservation mu equations for any eigenvalues are ordered by thecision decreasing real of part. our computation, The one arisesorbit. from For the details, spanwise see ref. transla [ APPENDIX for plane Couette flow, 0.1 p s in 853 0.05 r Kura- 8 , and the EQ e bulate the , th Floquet j p . Λ 0 EQ (1) Q od choice is not it or relative pe- nts is aided by a ponents is always π/ω = contributes twice). uilibrium or relati- 2 p mbers which, due to = Λ ChaosBook.org version15.9, Jun 24 2017 0 , it is omitted from the , } 0.4 0.2 -0.05 p x, it is helpful to state the EQ e -0.2 -0.4 { T T , the product of expanding , the sign of the p ) plot. However, intelligent , , = } ) j 2 Λ 1 s ( , ) −− − j 1 . EQ ( ,µ s , ). ) j j + 1 p N ( Highway 61 Revisited G is the geometric weight of cycle + of the corresponding eigenfunction φ ◮ ω | p p ) j T ··· ). An example are the eigenvalues of EQ ( along the group orbit. In addition, if the Λ , |∈{ | O(2) wave-numbers, as well as take care j 2 G c / , p × µ,ω , φ Λ 1 | φ / j ( of the solution itself needs to be noted, and for symbols stand for sym- ], is an example of how to tabulate the leading , p ? — Bob Dylan, ± = EQ φ G = Λ Well Mack the Finger said toI Louie got the forty King red whiteAnd and a blue thousand shoe telephones strings thatDo don’t you ring know where I can get rid of these things? -invariant subspace, ) j S , where p ( + . σ , plotted according to their isotropy , the −− 8 is complex, its phase Eigenvalues of the plane Couette flow j N of the orbit; for a relative periodic orbit (12.23) one state , EQ p from ref. [13.42], plotted in figure A4.1. To decide how many +++ p Λ 8 G • A4.1: respectively, defined in ref. [13.42]. For ta- , and -axis units can be tricky for high-dimensional problems. Fo antisymmetric under symmetry operation j / − 3

A4. LINEAR STABILITY s EQ + − Figure and bles of numerical valuesChannelflow.org of stability eigenvalues see groups: equilibrium metric ◭ Table A4.2, taken from ref. [ Surveying this multitude of equilibrium and Floquet expone Table A4.1, taken from ref. [13.43], is an example of how to ta Floquet multipliers (5.6) is useful, as 1 addition the shift parameters in a cycle expansion (rememberWe that often each do complex eigenvalue care about of the wall-normal node counting. appendStability - 23jan2012 multiplier, or, if isotropy group table. The isotropy subgroup choice of the least stable (i.e., the most unstable)period eigenvalue of is the comple spiral-out motion (or spiral-in, if stable), moto-Sivashinsky system the correct choicethe are O(2) the symmetry, wave-nu come inknown pairs. as For yet; plane one needs Couette to flow group the O(2) go plot of the complex exponent plane ( Floquet exponents of theriodic monodromy orbit. matrix of For an a periodic periodic orb orbit one states the period should be indicated. If the isotropy is trivial, relative equilibrium (12.19) the velocity of the these are “physical” in the PDE case (where number of ex infinite, in principle), it is useful to look at the ( equilibrium leading Floquet eigenvalues ofve the equilibrium. stability The matrix isotropy subgroup of an eq APPENDIX A4.3 Eigenspectra: what to make out of them? es en 856 0. The ) (A4.45) (A4.44) (A4.42) (A4.43) = yu h x + ) )] yu uv xv h h h ) can be written t )( + − ( v is non singular and h U xv xy u h h ) . h U ChaosBook.org version15.9, Jun 24 2017 )( )( λ vv ) v u − ±               h h h e yy y y u 2 2 h + − h 2 2 h h q q x q q + / / xx h / / + − ˙ ˙ )] v y h ˙ ˙ x u with respect to the phase space v y xx 2( − − vv Λ= )( h h h h , 2 u x x + H λ )( h 1[. real. This is the most general case, h h ) 2 2 − ± 2 v , 2 2 q q e h + yv 2( 2( q ω q / . The matrix / uu ]0 | h / / 2 y ˙ ˙ v u ˙ + h x + − ˙ ∈ ˙ x y h | − − − ) 2 u ) ( 2] matrix, the eigenvalues are determined ν − Λ= and h , except the equilibrium points ˙ yv yv × ( = + xu x µ h h yy ˙ ˙ y u h ) ˙ ˙ | x v − h , if and − − − + )( uu ) H 2 v + h vv h πν xu xu is a [2 , i |∇ with ˙ ˙ ˙ ˙ x y v u h xx + h h ± 4 + ω h e )( )( M i               + = yy 2 u v v − ± )( h h h h v µ = 2 uu q y y ± ) h )( must span the space perpendicular to the flow on the ) h − . h h y Λ= e 2 v 2 2 1. | )( 2 y h 2 h M x − + 2 y ± h − , H + d h , if + u u E tr ( 2 − 2 h h are given (A4.45). One distinguishes 4 classes of eigenvalu u Λ= ) and x 2 x |∇ x + x x p v x l 2 h h , , , x x Λ= h 2 h h 1 = inverse hyperbolic [( u h h , if x ± 11 [( , l 2( ( 2( ( ˜ 2 ; q , ) 1 elliptic t y . q q 1 , if ,..., − + − − − is now the monodromy matrix and the equation of motion are giv x , 1 M ( 2 separates stable and unstable behavior. x x A4. LINEAR STABILITY or ( lM M is the derivative of the Hamiltonian ======tr ( matrix elements for the local transformation (A4.43) are ) ) = = i j t l = ( h M ˙ , 22 21 11 12 ⊤ i l l l l U M stable hyperbolic marginal possible only in systems with 3 or more degree of freedoms. loxodromic λ ˜ ˜ ˜ ˜ x h . The For a system with two degrees of freedom, the matrix • • • • M appendStability - 23jan2012 coordinates and with i.e., tr ( The matrix down explicitly, i.e., by The vectors energy manifold. symplectic at every phase space point with matrix elements for of For 2 degrees of freedom, i.e., APPENDIX by , 2 M D 855 time t δ (A4.38) (A4.41) . Setting t (G. Tanner) , preserving ) in form of with t ( t x δ ) t , ) is constant, i.e., itself is symplec- ( 1) elements of x t                     pendence of a dis- e symmetric under ( ndeed, we discover x U + orbit he symplectic group ∗ 0 t any time H if canonical = D ∇ monodromy matrix M ChaosBook.org version15.9, Jun 24 2017 ˜ L ) (2 l t ( d ...... on x x δ and = δ ˜ J 2 D . . . ∗ ∗ ∗ ∗ − . . . 2 0 0 00 0 0 / as local coordinates uncovers the two                     . An initial displacement in the direc- 1) t 0 J x = ) (A4.40) , − = )): ˙ ˜ t U L i j i ( ′ k D δ ( x (0)) (A4.39) q and q 2 constraints each; the last relation yields − ( x / ∂ ∂ = D ( 2 | U j 2 ′ k } ) 1) j U t p q ; LU of the flow (4.5), but the ( ( − q − ∂                     ∂ ) gives )) 1 , ˜ J t H 2 J i ⊤ − ( ) transfers according to ) D ∗ 0 − p 2 x x ˜ ( |∇ L U { − ( D ( j ′ k / d D J ) q q 2 H t x ∂ ∂ ( )) M = = ...... ∇ t i ′ k H ( ω q p , x ˙ ˜ ∇ . We get the equations of motion for the monodromy matrix ∂ J ( ˜ ∂ 0 L = 1 ,..., E = − ⊤ δ x . . . ∗ ∗ ∗ ∗ 1 U = = E x = x , = . . . ) 1 0 10 0 0

A4. LINEAR STABILITY ⊤ } } t j j ( E ))                     t p q x x ( , , , δ i x i = ( ⊤ q p )) ). ˜ ˜ with t { t J J { x ( D ( x Choosing ( interchange and yield (2 = j H , i.e., the transformations induced by a Hamiltonian flow are constraints. Hence only (2 are linearly independent, as itS p behooves group elements of t A4.5 Monodromy matrix for Hamiltonian flows the form of the equationsi of motion. The first two relations ar U appendStability - 23jan2012 son brackets It is not the Jacobian matrix which enters the trace formula.placement perpendicular This to matrix the flow givessome on the the trivial time energy parts de manifold. in I the Jacobian matrix trivial eigenvalues 1 of the transformed matrix in (A4.39) a independent. The projection of any displacement on These lead to tion of the flow ∇ the (non singular) transformation directly choosing a suitable local coordinate system on the Note that the properties a) – c) are only fulfilled for APPENDIX tic. . i ··· R Q h + # i i i − R Q complex eigenvalue pair 1 1 h # " ··· 2 1 ω 2] matrix + − × = 1 + i k 1 µ + ω µ P k ∗ k " P λ + + k k P P ··· h 857 k λ + or whatever you find the clearestreal way representation. to write this in the spectral decompositionplaced (A4.16) by is a real now [2 re- ··· (c) (d) (P. Cvitanovi´c) . ChaosBook.org version15.9, Jun 24 2017 . } ω i (Ver- . − # ω ω, µ i − + µ µ ω µ { " eigenvalues form a , = 1 1 = are matrices with real + + } # k k 1 for notational brevity. + Q P i k , λ i P k − = , λ λ , ∗ k and ) by λ P 1 1 1 { Q k + i k P # " P + + ∗ , R 0 ( k λ k 1 2 P P = 0 λ = = P P R # " can be written as where we denote conjugates of each other, P i i where elements. 1 1 − " (a) corresponding projection operators are complex (b) 2 1 complex conjugate pair, Show that ification of example A4.2.) Real representation of complex eigenvalues. soluAppStab - 1feb2008 Exercises Chapter A4 solutions: Linear stability Solution A4.1 - Real representation of complex eigenvalues

EXERCISES A4.1.