Appendix C Finding cycles

chaosbook.dk

Appendix C Finding cycles

Appendix CFinding cycles(C. Chandre)C.1 Newton-Raphson methodC.1.1Contraction rateCONSIDER A d-DIMENSIONAL MAP x ′ = f (x) with an unstable fixed point x ∗ . TheNewton-Raphson algorithm is obtained by iterating the following mapx ′ = g(x) = x − (J(x) − 1) −1 ( f (x) − x) .The linearization of g near x ∗ leads tox ∗ + ɛ ′ = x ∗ + ɛ − (J(x ∗ ) − 1) −1 ( f (x ∗ ) + J(x ∗ )ɛ − x ∗ − ɛ) + O ( ‖ɛ‖ 2) ,where ɛ = x − x ∗ . Therefore,x ′ − x ∗ = O ( (x − x ∗ ) 2) .After n steps and if the initial guess x 0 is close to x ∗ , the error decreasessuper-exponentiallyg n (x 0 ) − x ∗ = O ( (x 0 − x ∗ ) 2n ).767


C.1.2Computation of the inverseThe Newton-Raphson method for finding n-cycles of d-dimensional mappingsusing the multi-shooting method reduces to the following equation⎛⎜⎝1 −D f (x n )−D f (x 1 ) 1· · · 1−D f (x n−1 ) 1⎞ ⎛⎟⎠ ⎜⎝δ 1δ 2· · ·δ n⎞⎟⎠ = − ⎛⎜⎝F 1F 2· · ·F n⎞⎟⎠ ,(C.1)where D f (x) is the [d × d] Jacobian matrix of the map evaluated at the point x,and δ m = x ′ m − x m and F m = x m − f (x m−1 ) are d-dimensional vectors. By somestarightforward algebra, the vectors δ m are expressed as functions of the vectorsF m :m∑( )n∑δ m = − β k,m−1 F k − β 1,m−1 1 − −1 β1,n⎛⎜⎝ β k,n F k⎞⎟⎠ ,k=1k=1(C.2)for m = 1, . . . , n, where β k,m = D f (x m )D f (x m−1 ) · · · D f (x k ) for k < m and β k,m = 1for k ≥ m. Therefore, finding n-cycles by a Newton-Raphson method with multipleshooting requires the inversing of a [d×d] matrix 1−D f (x n )D f (x n−1 ) · · · D f (x 1 ).C.2 Hybrid Newton-Raphson / relaxation methodConsider a d-dimensional map x ′ = f (x) with an unstable fixed point x ∗ .The transformed map is the following one:x ′ = g(x) = x + γC( f (x) − x),where γ > 0 and C is a d × d invertible constant matrix. We note that x ∗ is also afixed point of g. Consider the stability matrix at the fixed point x ∗A g = dgdx∣ = 1 + γC(A f − 1).x=x∗The matrix C is constructed such that the eigenvalues of A g are of modulus lessthan one. Assume that A f is diagonalizable: In the basis of diagonalization, thematrix writes:A˜g = 1 + γ ˜C( A˜f − 1),appendCycle - 3jun2008 ChaosBook.org version13, Dec 31 2009


2.521.510.5Figure C.1: Illustration of the optimal Poincaré surface.The original surface y = 0 yields a large distancex− f (x) for the Newton iteration. A much better choiceis y = 0.7.0xf(x)-0.5-1-1.50 0.2 0.4 0.6 0.8 1 1.2where A˜f is diagonal with elements µ i . We restrict the set of matrices ˜C to diagonalmatrices with ˜C ii = ɛ i where ɛ i = ±1. Thus A˜g is diagonal with eigenvaluesγ i = 1 + γɛ i (µ i − 1). The choice of γ and ɛ i is such that |γ i | < 1. It is easy to seethat if Re(µ i ) < 1 one has to choose ɛ i = 1, and if Re(µ i ) > 1, ɛ i = −1. If λ ischosen such that2|Re(µ i ) − 1|0 < γ < mini=1,...,d |µ i − 1| 2 ,all the eigenvalues of A g have modulus less that one. The contraction rate at thefixed point for the map g is then max i |1 + γɛ i (µ i − 1)|. If Re(µ i ) = 1, it is notpossible to stabilize x ∗ by the set of matrices γC.From the construction of C, we see that 2 d choices of matrices are possible. Forexample, for 2-dimensional systems, these matrices are{( 1C ∈0)0,1( −1001) ( 1,00−1) ( −1,00−1)}.For 2-dimensional dissipative maps, the eigenvalues satisfy Re(µ 1 )Re(µ 2 ) ≤ det D f 1, Re(µ 2 ) > 1) which is stabilized by ( −1 00 −1)has to be discarded.The minimal set is reduced to three matrices.C.2.1Newton method with optimal surface of section(F. Christiansen)In some systems it might be hard to find a good starting guess for a fixed point,something that could happen if the topology and/or the symbolic dynamics of theflow is not well understood. By changing the Poincaré section one might get abetter initial guess in the sense that x and f (x) are closer together. In figure C.1there is an illustration of this. The figure shows a Poincaré section, y = 0, aninitial guess x, the corresponding f (x) and pieces of the trajectory near these twopoints.If the Newton iteration does not converge for the initial guess x we might haveto work very hard to find a better guess, particularly if this is in a high-dimensionalsystem (high-dimensional might in this context mean a Hamiltonian system withappendCycle - 3jun2008 ChaosBook.org version13, Dec 31 2009

More magazines by this user
Similar magazines