User:James Howard/Proposed/Stability of fixed points for symplectic maps

• 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.

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, \[\begin{array}{lcl} p^{\prime} &=& p-K\sin q \\ q^{\prime} &=& q + p^{\prime} \end{array}\] shown in Fig. 1. for \( K = 1\), has a Lyapunov stable fixed point at the origin for \( 0<K<4\).

Figure 1: Standard map for \( K = 1 \)

Introduction

Symplectic maps occur as descriptions of many physical problems, including the dynamics of a billiard (Birkhoff, 1927), orbits in particle accelerators (Courant and Snyder, 1959), magnetic field line configurations in plasmas (Rechester, 1976), the dynamics of asteroids (Wisdom,1983), mixing of passive tracer in incompressible fluids (Aref, 1984), and more generally as integrators for Hamiltonian flows (Marsden, 1996).

Any periodically time-dependent Hamiltonian system with \(n\) degrees of freedom generates a 2n-dimensional symplectic map by following the flow for one period. Similarly, an autonomous Hamiltonian system with \(n + 1\) 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 Poincaré 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 multipliers that define linear stability of the fixed points (and periodic orbits) of the mappings.

A mapping \( S \) of a 2n-dimensional manifold is symplectic (Arnold, 1990) if its Jacobian matrix \(L = DS\) preserves the two-form (skew-scalar product), \[\tag{1} [ L\xi,L\eta] = [\xi,\eta] \]

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 \[\tag{2} [\xi,\eta] = \xi^T J\eta, \quad J = \left ( {\begin{array}{*{20}c} 0 & I_n \\ -I_n & 0\\ \end{array}} \right ) \] with \(I_n\) the \(n\times n\) identity. Equivalently a mapping \(S\) is symplectic if \(L\) satisfies \[\tag{3} L^TJL = J. \] It can be shown that \(\det L = + 1\), so that symplectic maps are volume and orientation preserving. For two-dimensions, this is also sufficient: every 2D mapping that preserves oriented area is symplectic; however, in higher dimensions, symplectic maps also preserve a hierarchy of other invariants (the Poincare invariants).

Hamiltonian systems possess a natural symplectic structure. Indeed, Hamilton's equations in \(n\) degrees of freedom can be written \[\tag{4} \frac{dz}{dt} = J\,\nabla H(z,t) \] where \(z = (q,p)\) is the 2n-dimensional phase space point and \( H(z,t) \) is the Hamiltonian. 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 the invariance 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, in principle, be reduced to that of symplectic maps.

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, \[\tag{5} (\delta z)^n = DS^n\cdot \delta z_0 \]

where \( \delta z = z-z_0\) and \( DS \) is the Jacobian of \( S \ .\) For distinct \( \lambda_i \ ,\) eq.(4) has the fundamental solution \[\tag{6} \delta z = \lambda_i^n \xi_i \]

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 \[\tag{7} \det (L - \lambda I) = 0. \]

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.

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..

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, \[\tag{8} P(\lambda) = \lambda^{2n} - A_1 \lambda^{2n-1} + A_2 \lambda ^{2n-2} - \cdots + A_2 \lambda^2 - A_1 \lambda + 1. \]

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) \[\tag{x:label exists!} \rho = \lambda + 1/\lambda \]

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 \[\tag{9} \lambda^2 - \rho\lambda + 1 = 0 \]

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...

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 \[\tag{:label exists!} L = \left ( {\begin{array}{*{20}c} 1-K\cos q_0 & 1 \\ -K\cos q_0 & 1\\ \end{array}} \right ) \]

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

Stability in Arbitrary Dimension

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 \[\tag{:label exists!} P(\lambda) =\lambda^4 - A\lambda^3 + B \lambda^2 - A\lambda + 1 \]

with \( A = tr L \) and \( 2B = (tr L)^2 - tr (L^2) \ .\) Dividing by \( \lambda^2 \) gives the RCP, \[\tag{:label exists!} Q(\rho) = \rho^2 - A\rho + B - 2 \]

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.

Six-Dimensional Maps

The characteristic equation is \[\tag{:label exists!} P(\lambda) = \lambda^6 - A\lambda^5 + B\lambda^4 - C\lambda^3 + B\lambda^2 - A\lambda + 1 \]

with coefficients given by \[\tag{:label exists!} A = tr A,\qquad 2B = (tr ~L)^2 - tr (L^2) \]

\[ 3C = tr(L^3) - A \;tr (L^2) + B tr L. \] Dividing by \( \lambda^3 \) yields the RCP \[ Q(\rho) = \rho^3 - A\rho^2 + D\rho - E \] where \[\tag{:label exists!} D = B-3,\qquad E = C - 2A. \]

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).

References

Aref, H. (1984). “Stirring by Chaotic Advection.” J. Fluid Mech. 143: 1-21.

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, J. M., I. I. Satija and B. Hu (1986). “Period doubling in four-dimensional symplectic maps.” Phys. Rev. A 34: 4325.

Marsden, J. E., G. W. Patrick and W. F. Shadwick, Eds. (1996). Integration Algorithms and Classical Mechanics. Providence, American Mathematical Society.

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.

Rechester, A. B. and S. T.H. (1976). “Magnetic Braiding Due to Weak Asymmetry.” Physical Review Letters 36: 587-591.

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

Wisdom, J., S. J. Peale and F. Mignard (1983). “The Chaotic Rotation of Hyperion.” Icarus 58: 137-152.

Wisdom, J. (1982). “The Origin of the Kirkwood Gaps: A Mapping for Asteroidal Motion Near the 3/1 Commensurability.” Astron. J. 87: 577-593.

See Also

Stability, Stability of Hamiltonian Equilibria, Hamiltonian Systems, Symplectic Maps,Stability,Bifurcations,Periodic Orbits Symplectic Maps