Stability of fixed points for symplectic maps

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Author: Dr. James Howard, Center for Integrated Plasma Studies and Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder

Dr. James Howard accepted the invitation on 20 April 2007 (self-imposed deadline: 19 November 2007).

UNDER CONSTRUCTION!

A fixed point $z_0 = (q_0,p_0)$ of a symplectic map S is Lyapunov stable if all nearby orbits remain close to $z_0$ for all iterations of S, linearly stable if all orbits of the tangent map DS are bounded, and spectrally stable if all eigenvalues of DS lie on the unit circle. For example, the standard map,

$p^{\prime} = p-K\sin q$

$q^{\prime} = q + p^{\prime}$

shown in Fig. 1. for $K = 1$ , has a stable fixed point at the origin for $0<K<4$ .

Stability of fixed points for symplectic maps

Figure 1: Standard map for $K = 1$

1 Introduction
2 Types of Stability
3 Krein's Theorem
4 Stability Boundaries
- 4.1 Two-Dimensional Maps
- 4.2 Example: The Fermi Map
5 Stability in Arbitrary Dimension
6 Natural Maps
- 6.1 Example: A Froeschle-Type Map
7 References
8 See Also

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Introduction

Symplectic maps occur in many physical problems, including orbits in particle accelerators (Courant and Snyder, 1959), magnetic field line mapping (xx, 19xx), guiding center motion (ref, 19xx), and plasma wave heating (ref, 19xx) Any periodic Hamiltonian system of $n$ degrees of freedom generates a 2n-dimensional symplectic map by following the flow for one period. Similarly, an autonomous Hamiltonian system of $n + 1$ degrees of freedom induces a family of 2n-dimensional symplectic maps parametrized by the value of the Hamiltonian, by considering the first return to a surface of section. Return maps provide much useful information on the behavior of continuous time systems, including the existence or nonexistence of invariant tori, and the location and stability of resonances and periodic orbits. In all these investigations it is important to have analytic formulas for the linear stability limits of the fixed points (and periodic orbits) of the mappings.

A mapping $S$ of a 2n-dimensional manifold is symplectic (Arnold, 1990) if its tangent map $L = DS$ preserves the two-form (skew-scalar product),

(1)

$[L\xi,L\eta] = [\xi,\eta]$

for all $\xi, \eta \in R^{2n}$ . By Darboux's theorem (Arnold, 1980) local coordinates can always be found such that the skew-scalar product takes the standard form

(2)

$[\xi,\eta] = \xi^T J\eta$

where

$J = \left ( {\begin{array}{*{20}c} 0 & I_n \\ -I_n & 0\\ \end{array}} \right )$

with $I_n$ the $n\times n$ identity. Equivalently a mapping $S$ is symplectic if its Jacobian matrix $L$ satisfies

(3)

$L^TJL = J.$

It can be shown that $\det L = + 1$ , so that symplectic maps are volume preserving. In dimension $2n = 2$ a mapping is symplectic iff it preserves oriented area; in higher dimension an hierarchy of other invariants (the Poincare invariants) are preserved.

Hamiltonian systems possess a natural symplectic structure. Indeed, Hamilton's equations in $n$ degrees of freedom can be written

(5)

$\frac{dz}{dt} = J\cdot DH(z,t)$

where $z = (q,p)$ is the 2n-dimensional phase space point, $H(z,t)$ is the Hamiltonian, and DH is its derivative with respect to z. Canonical transformations, those which preserve the form of Hamilton's equations and the value of the Hamiltonian, may then be recognized as symplectic maps in the standard basis. Invariance of the skew-scalar product (2) corresponds to conservation of the Lagrange bracket (Goldstein, 1980). Moreover, the time evolution of a Hamiltonian system may be viewed as a symplectic map from arbitrary initial to final states. Hence the study of Hamiltonian systems can often be reduced to that of symplectic maps.

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Types of Stability

A mapping $S:M\mapsto M$ on a manifold $M$ has a fixed point at $z_0$ if $S:z_0 \mapsto z_0$ and a periodic orbit of length N if...

DEFINITION: A fixed point $z_0$ is Lyapunov stable if for every neighborhood $U$ of $z_0$ there exists a subneighborhood $V \in U$ such that $z_0 \in U \implies z \in V$ .

The motion near a fixed point is given by the variational equations,

(5)

$(\delta z)^n = DS^n\cdot \delta z_0$

where $\delta z = z-z_0$ and $DS$ is the Jacobian of $S$ . For distinct $\lambda_i$ , eq.(4) has the fundamental solution

(6)

$\delta z = \lambda_i^n \xi_i$

where the $\lambda_i$ are the eigenvalues (multipliers) of $DS$ and $\xi_i$ the associated eigenvector.

DEFINITION: A fixed point $z_0$ is linearly stable if the tangent map $DS:\mathbb{R}^m \mapsto \mathbb {R}^m$ is bounded, i.e. $\vert DS^N (\delta { z_0}) \vert < \infty$ for all integers $N>0$ , where $\vert \cdot \vert$ is some suitable norm.

The eigenvalues $\lambda_i$ are given by the characteristic equation

(7)

$\det (L - \lambda I) = 0.$

It is not difficult to show that the eigenvalues of a symplectic matrix come in reciprocal pairs $(\lambda,1/\lambda)$ (Howard and MacKay, 1987).

DEFINITION: A symplectic map is spectrally stable if all its eigenvalues $\lambda$ lie on the unit circle ( $S^1$ ), i.e. $\vert\lambda\vert=1$

A symplectic map is linearly stable iff it is spectrally stable and all its Jordan blocks are simple. Thus,

Stability $\implies$ Linear stability $\implies$ Spectral stability,

but not vice versa. A periodic orbit is linearly stable iff it is spectrally stable and all Jordan blocks corresponding to eigenvalues on the unit circle are one dimensional. Since the boundaries of linear and spectral stability are identical for symplectic maps, the concept of spectral stability allows one to describe stability limits without continually excluding the case of multiple eigenvalues.

Since $L$ is real its eigenvalues also come in complex conjugate pairs. Hence, eigenvalues occur in the following configurations:

complex conjugate pairs $\lambda,\lambda^*,~ \vert\lambda \vert=1$
real pairs $(\lambda, 1/\lambda)$
complex quadruplets $\lambda, 1/\lambda, \lambda^*, 1/\lambda^*,\vert\lambda\ne 1$
$\lambda = \pm 1$

Moreover, $\lambda, 1/\lambda,~ \lambda^*, 1/\lambda^*$ all have the same multiplicity and Jordan block structure, while eigenvalues $\pm 1$ have even multiplicity.

Now consider a symplectic map which depends smoothly on parameters $\mu$ , so that its eigenvalues also vary continuously with $\mu$ . It follows that a periodic orbit can lose spectral (and therefore linear) stability in only three ways:

Saddle-node bifurcation: a pair of eigenvalues collide at $\lambda = +1$ and move off along the real axis.
Period-doubling bifurcation: a pair of eigenvalues collide at $\lambda = -1$ and move off along the real axis
Krein bifurcation: two eigenvalue pairs collide on $S^1$ and move off into the complex plane, forming

a complex quadruplet.

[edit]

Krein's Theorem

When two pairs of eigenvalues $\lambda$ merge on $S^1$ (Krein collision) they may either move out into the complex plane (Krein bifurcation) or simply pass through each other, remaining on the unit circle. The outcome depends on a special invariant peculiar to symplectic matrics:

define signatures..

[edit]

Stability Boundaries

Since the multipliers of $L$ occur in reciprocal pairs, the characteristic polynomial is reflexive, $P(1/\lambda) = \lambda^{-2n} P(\lambda)$ , so that the coefficients forma a palindrome,

(9)

$P(\lambda) = \lambda^{2n} - A_1 \lambda^{2n-1} + A_2 \lambda ^{2n-2} - \cdots + A_2 \lambda^2 - A_1 \lambda + 1.$

The coefficients of $P$ may be expressed in terms of the elements of the matrix $L$ (Gantmacher, 1959). Now define the stability index (Broucke, 1969)

(9)

$\rho = \lambda + 1/\lambda$

and divide $P$ by $\lambda^{2n}$ to get the reduced characteristic polynomial (RCP) of degree n,

$Q(\rho) = \rho^n - A_1^{\prime}\rho^{n-1} +\cdots + (-)^n A_n^{\prime}$

where the $A_i^{\prime}$ are affine combinations of the $A_i$ . For given $\rho$ there are two multipliers, given by

$\lambda^2 - \rho\lambda + 1 = 0$

from which we have the

Lemma: A fixed point of a real symplectic matrix is spectrally stable iff all its stability indices $\rho_i$ are real, with $\vert\rho\vert <2$ .

Thus, the calculation of the multipliers of symplectic matrix has been reduced from solving a polynomial of degree 2n to solving a polynomial of degree n plus the quadratic (x). It follows that

Theorem: A fixed point of a symplectic matrix is spectrally stable iff all the zeroes of its reduced characteristic polynomial are real and lie in the interval $[-2,2]$ .

Discuss stability bounds here...

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Two-Dimensional Maps

Two-dimensional maps are common in physical problems (Lichtenberg and Lieberman, 1980). The characteristic polynomial is

$P(\lambda) = \lambda^2 - A\lambda + 1$

with $A = tr L$ . Dividing by $\lambda$ gives the RFP.

$Q(\rho) = \rho - A$

so that L is spectrally stable iff $\vert A\vert < 2$ . The stability boundary consists of the two points $A = 2$ (saddle-node bifurcation) and $A = -2$ (period-doubling bifurcation). Since Q is of degree one Krein collisions cannot occur.

For the standard map (1) the Jacobian is

$L = \left ( {\begin{array}{*{20}c} 1-K\cos q_0 & 1 \\ -K\cos q_0 & 1\\ \end{array}} \right )$

so that, at $q_0 = 0,~ A = tr L = 2-K$ and the stability condition is $0<K<4 ,$ .

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Example: The Fermi Map

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Stability in Arbitrary Dimension

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Sturm's Theorem

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Four-Dimensional Maps

Four-dimensional maps occur in the three-body problem (Broucke, 1969), orbits in particle accelerators (Dragt and Finn, 1999), and plasma wave heating (xx,xx). They are of particular interest in dynamics, as they are the lowest dimensional system for which Arnold diffusion (Arnold, 1980) can occur. The characteristic equation reads

$P(\lambda) =\lambda^4 - A\lambda^3 + B \lambda^2 - A\lambda + 1$

with $A = tr L$ and $2B = (tr L)^2 - tr (L^2)$ . Dividing by $\lambda^2$ gives the RCP,

$Q(\rho) = \rho^2 - A\rho + B - 2$

so that

$\rho = \frac{1}{2} (A\pm\sqrt{A^2 - 4B + 8})$

More here... Figure 3 shows the stable region for 4D maps in the space of polynomial coefficients.

Figure 2: Stability diagram for 4D map.

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Six-Dimensional Maps

The characteristic equation is

$P(\lambda) = \lambda^6 - A\lambda^5 + B\lambda^4 - C\lambda^3 + B\lambda^2 - A\lambda + 1$

with coefficients given by

$A = tr A,\qquad 2B = (tr ~L)^2 - tr (L^2)$

$3C = tr(L^3) - A \;tr (L^2) + B tr L.$

Dividing by $\lambda^3$ yields the RCP

$Q(\rho) = \rho^3 - A\rho^2 + D\rho - E$

where

$D = B-3,\qquad E = C - 2A.$

Figure 3 is a perspective view of the stable region for 6D maps in the space of polynomial coefficients A, B, C.

Figure 3: Stability diagram for 6D map.

The 8-dimensional case is worked out in detail in (Howard and Mackay, 1987).

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Natural Maps

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Example: A Froeschle-Type Map

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References

Arnold, V. I. and Avez, A. (1968). Ergodic Problems of Classical Mechanics, New York, Benjamin.

Arnold, V. I. (1990). Mathematical Methods of Classical Mechanics, 2nd Ed., New York, Springer.

Courant, E. D. and Snyder, H. S. (1958). "Theory of the alternating-gradient synchrotron" Ann. Phys. (N.Y.) 3, 1 (NY)

Dickson, L. D. (1939). New First Course in The Theory of Equations, New York, Wiley.

Dragt, A. (1982). Lectures on Nonlinear Orbit Dynamics, AIP Conference Proceedings, Vol 87, New York, AIP.

Goldstein, H. (1980) Classical Mechanics, 2nd Ed., New York, Addison, Wesley.

Howard, J. E., Lichtenberg, A. J., Lieberman, M. A., and Cohen, R. H. (1986). "Four-dimensional mapping model for two-frequency ECRH," Physica D. 20, 259.

Howard, J. E. and MacKay, R. S. (1987). "Linear stability of symplectic maps," J. Math. Phys. 28, 1038-1051.

Howard, J. E. and Dullin, H. R. (1998). "Stability of Natural Maps," Phys. Lett. A.

Lichtenberg, A. J. and Lieberman, M. L. (1980). Regular and Chaotic Dynamics, 2nd Ed., New York, Springer.

MacKay, R. S. (1992). Renormalization in Area Preserving Maps, London, World Scientific.

Mao, ......

Meiss, J. D., (2004). "Symplectic Maps," in Encyclopedia of Nonlinear Science, Ed. A. Scott, New York, Rutledge.

Meyer, K. R. and Hall, G. R. (1992). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, New York, Springer.

Roberts, J. G. R. and Quispel, R. 1992. Chaos and time-reversal symmetry in dynamical systems, Phys. Rep. 216, 63.

Wisdom, J.

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Stability of fixed points for symplectic maps

From Scholarpedia

Contents

Introduction

Types of Stability

Krein's Theorem

Stability Boundaries

Two-Dimensional Maps

Example: The Fermi Map

Stability in Arbitrary Dimension

Sturm's Theorem

Four-Dimensional Maps

Six-Dimensional Maps

Natural Maps

Example: A Froeschle-Type Map

References

See Also

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