Stability of Hamiltonian equilibria
| James Howard (2013), Scholarpedia, 8(10):3627. | doi:10.4249/scholarpedia.3627 | revision #143451 [link to/cite this article] |
An equilibrium \(z_0=(q_0,p_0) \in \mathbb{R}^{2n} \) of an autonomous Hamiltonian flow is Lyapunov stable if all nearby orbits remain close to \( z_0\) for all forward time, linearly stable if all orbits of the tangent flow are bounded, and spectrally stable if all eigenvalues of the tangent flow are pure imaginary. For example, consider the nonlinear pendulum of length \( l\) and mass \( m\), as shown in Fig. 1 and described by the Hamiltonian
- <math 1>
H(\theta,p_{\theta}) = \frac{1}{2m l^2} p_{\theta}^2 + mgl(1-\cos \theta), </math>
where \( p_{\theta}= m l^2\dot \theta\) is the momentum canonically conjugate to the angle \( \theta \). Setting \( \partial H /\partial \theta = \partial H/\partial p_{\theta} = 0 \) gives the two equilibria, \( z_1 = (0,0)\) and \( z_2 = (\pi,0) \). Figure 2 depicts the phase space \( (\theta,p_{\theta}) \) for this simple system, showing the equilibria \( z_1\) and \( z_2\). As we shall see, the central equilibrium \( z_1 \) is not only linearly but also Lyapunov stable, while \( z_2 \) is always unstable.
Contents |
Introduction
Stability of motion is one of the oldest problems in mathematical physics, with important contributions dating back to the eighteenth century. Indeed, an historical account of progress in stability analysis reads like a Who's Who in Mathematics and Physics. Important early contributions on the zeros of polynomials were made by Cauchy, Lagrange, Fourier, and Hermite, culminating in the great theorems of Sturm, which provided necessary and sufficient conditions that a given real polynomial have all its zeros in a specified real interval. The relevance of these seminal studies in the theory of equations to orbital stability was recognized and developed further by Airy and Clifford, and subsequently carried to completion by Maxwell and Routh. Maxwell, in his celebrated Adams Prize Essay of 1857 on the stability of the rings of Saturn obtained linear stability criteria in two and three degrees of freedom. Routh (1877) devised two different methods, the first based on Clifford's idea of forming the polynomial whose zeros are the twofold sums of the zeros of the characteristic polynomial, the second based on the Cauchy Index Theorem, coupled with Sturm's Theorem. Routh's investigations on the stability of governors evolved into the field now known as control theory. His second method is still widely used for dissipative systems, together with the equivalent method of Hurwitz.
In contrast, the powerful techniques of Lyapunov are nonlinear and involve the use of so-called comparison functions, for which there are no general constructive methods. While there is considerable activity in the russian literature (Rumyantsev and Sosnitskii, 1992), Lyapunov techniques will not be covered in this article. The Routh-Hurwitz scheme explicity assumes that no eigenvalues lie on the imaginary axis. Lyapunov's techniques yield only sufficient conditions for stability and have limited appicability to Hamiltonian systems. The familiar Lagrangian method (Goldstein, 1980) is applicable only when the kinetic energy is positive definite, excluding such important examples as the restricted three-body problem (Broucke, 1969).
Recently a new method was devised which overcomes all the above problems, yielding explicit linear stability bounds for Hamiltonian equilibria in arbitrary dimension. The method is based on two innovations: (i) introduction of a "reduced characteristic equation" of degree half that of the usual characteristic equation, (ii) use of Sturm's theorem to relate the eigenvalues of the reduced characteristic equation to orbital stability. Nonlinear stability can be proven for natural flows, for which \( H = T + U \), where the kinetic energy \( T(q,p) \) is positive definite and the potential energy \( U(q) \) possesses a quadratic local minimum. More generally, nonlinear stability can be proven in the more general case where \( H(q,p) \) is not separable, but definite in \( q \) and \( p \). Significant progress has also been made on converse stability, where conditions are obtained for instability of non-natural systems. See (Rumyantsev and Sosnitskii) for a recent review.
Equilibria of Hamiltonian flows differ markedly from those encountered in dissipative systems (Stability), where the existence of attractors and repellors results in asymptotic stability. In contrast, stability of Hamiltonian systems is neutral rather than hyperbolic. A vector field of an autonomous Hamiltonian flow can be written as \( \dot z = J\cdot DH \), where \( DH \) is the Jacobian of \( H \) and
- <math 2>
J = \left ( {\begin{array}{*{20}c} 0 & I_n \\ -I_n & 0\\ \end{array}} \right ) </math> is the fundamental symplectic two-form, with \( I_n \) the \( n\times n \) unit matrix.
Types of Stability
There are many notions of stability of motion, but only three of these are of primary importance for Hamiltonian systems.
DEFINITION: An equilibrium \( z_0 \in \mathbb{R}^{2n} \) is Lyapunov stable (nonlinearly stable) if for every neighborhood \( V\) of \( z_0 \),there exists a subneighborhood \( U\in V\) such that\( z(0) \in U \implies z(t) \in V\) for all forward time.
The motion near an equilbrium is given by the variational equations,
- <math 3 >
\delta \dot z = L \delta z </math> where \( \delta z = z-z_0\) and \( L = J\cdot D^2 H\) is the Linearization, with \( D^2 H\) the Hessian matrix of second derivatives. Since \( D^2 H \) is symmetric it follows that \( L \) is Hamiltonian, i.e. \( L^T J + JL = 0 \). The solution of (3) is called the tangent flow, and for distinct eigenvalues takes the form \[ \delta z(t) = \sum_i a_i e^{\sigma_i t} \] where \( \sigma_i \) are the eigenvalues of \( L \).
DEFINITION: An equilibrium is linearly stable if all orbits of the tangent flow are bounded for all forward time.
Thus, nonlinear stability is a much stronger property than linear stability, as the sets \( U\) and \( V \) do not have to be infinitesimally small.
The eigenvalues \( \sigma \) are given by the characteristic equation \[ P(\sigma) = \det(L -\sigma I) = 0. \] It is not difficult to show that the eigenvalues of a Hamiltonian matrix come in pairs \( \pm \sigma\) (Howard and MacKay, 1987), which leads to exponential growth unless all its eigenvalues lie on the imaginary axis.
DEFINITION: An equilibrium is spectrally stable if all eigenvalues of its linearization are pure imaginary.
Note however, that spectral stability does not imply linear stability. Nevertheless, it can be shown that the boundaries of linear and spectral stability coincide. The precise relation is that an equilibrium is linearly stable if and only if it is spectrally stable and all its Jordan blocks are one-dimensional (MacKay, 1986). Thus,
Nonlinear stability \( \implies \) Linear stability \( \implies \) Spectral stability,
but not vice versa. A famous counterexample is the Cherry Hamiltonian (Cherry, 1926)
- <math 4>
H = -\frac{\omega_1}{2}(p_1^2+q_1^2) +\frac{\omega_2}{2} (p_2^2+q_2^2)+\frac{\alpha}{2}[2 q_1 p_1 p_2-q_2 (q_1^2-p_1^2)] </math> where \( \omega_1 \) and \( \omega_2 \) are adjustable frequencies and \( \alpha \) is a nonlinearity parameter. In spite of the linear stability of the origin (\( \sigma = i\omega \)), for \( \omega_2 = 2\omega_1 \) an explicit solution shows that the nonlinear terms lead to explosive growth. We shall return to this example below.
Since \( L \) is a real matrix, its eigenvalues also come in complex conjugate pairs. Hence, eigenvalues occur in the following configurations:
- nonzero pairs \( \pm i\omega,~\omega\in \mathbb{R} \)
- nonzero pairs \( \pm \sigma \)
- nonzero complex quadruplets \( \pm a\pm ib,~a,b\in \mathbb{R} \)
- \( \sigma=0 \).
Moreover, \( \pm\sigma, \pm\sigma^* \) all have the same multiplicity and Jordan block structure, while a zero eigenvalue has even multiplicity (MacKay, 1986)
Now consider a Hamiltonian which depends smoothly on parameters \( \mu \) so that its eigenvalues also vary continuously with \( \mu \). It follows that an equilibrium can lose spectral (and therefore linear) stability in only two ways:
- a pair of imaginary eigenvalues merge at 0 and split onto the real axis (saddle-node bifurcation)
- a pair of imaginary eigenvalues collide at a nonzero point and split off into the complex plane, forming a complex quadruplet (Krein bifurcation)
Of course, combinations of these configurations or eigenvalues of multiplicity greater than one are possible. The essential fact is that every equilibrium on the boundary of spectral stability must have a multiple eigenvalue which is either zero or nonzero imaginary. However, not every equilibrium satisfying one of these conditions is actually on the boundary of spectral stability! This is a consequence of the existence of additional invariants for Hamiltonian flows which may prevent an eigenvalue pair from leaving the imaginary axis. These are the Krein Signatures, which we now describe.
Krein's Theorem
When two pairs of eigenvalues \( \sigma=\pm i\omega \) meet on the imaginary axis (Krein collision) they may either move out into the complex plane (Krein bifurcation) or simply pass through each other, remaining on the imaginary axis. Which eventually actually occurs depends on a special invariant peculiar to Hamiltonian flows:
Let \( \xi_k \) be an eigenvector corresponding to the eigenvalue \( \sigma_k \). Then the Krein signature (Arnold and Avez, 1968, Meiss, 2007) is
- <math 5>
s_k = sgn (\xi_k^T JL \xi_k). </math> If \( s_1=s_2 \) the eigenvalues remain on the imaginary axis; if \( s_1 \ne s_2 \) the signature is mixed and it is possible for the eigenvalues to leave the imaginary axis, forming a complex quadruplet. For more information on the mixed signature case see (Mackay, 1986). In general a pair of periodic orbits are formed and one speaks of a Hamiltonian-Hopf bifurcation. For example, suppose \( \alpha = 0 \) in the Cherry Hamiltonian (4), with both \( \omega_{1,2} \) positive. It is then easily seen that the Krein signature is mixed, yet the two independent counter-rotating oscillators are completely unperturbed when \( \omega_1 \mapsto \omega_2 \). In the important special case of two degree of freedom natural flows, where the kinetic energy is positive definite, it can be shown that Krein bifurcations are impossible. Whether this extends to arbitrary dimension is an open question.
Other types of stability encountered in Hamiltonian flows are orbital stability, which describes the divergence of two neighboring orbits, regarded as point sets, and structural stability, which describes the sensitivity (or insensitivity) of the qualitative features of a flow to changes in parameters. See Stability for details.
Stability Boundaries
Since its eigenvalues occur in \( \pm \) pairs the characteristic polynomial is even,
- <math 6>
P_{2n}(\sigma) = \det (L - \sigma I) = \sigma^{2n} + A_1 \sigma^{2n-2} +\cdots + A_n </math> where the \( A_k \) may be expressed in terms of the elements of \( L \) (Gantmacher, 1960). Introducing the new variable \( \tau = -\sigma^2 \) then gives the reduced characteristic polynomial,
- <math 7>
Q_n(\tau) = (-1)^n P_{2n} = \tau^n - A_1 \tau^{n-1} + \cdots + (-1)^n A_n. </math> Hence, a Hamiltonian equilibrium is spectrally stable iff all zeros of its reduced characteristic polynomial are positive. An equilibrium is on the boundary of spectral stability iff there is a zero at \( \tau=0 \) or a multiple zero at \( \tau>0 \) with mixed Krein signature. For example, for the pendulum (1),
- <math 8>
L = \left( {\begin{array}{*{20}c} 0 & 1/ml^2 \\ -mgl\cos{\theta_0} & 0 \\ \end{array}} \right )\implies Q_1(\tau) = \tau - \det L=\tau - \omega_0^2 \cos{\theta_0}
</math> where \( \omega_0 = \sqrt{g/l} \) is the frequency of small oscillations. For \( \theta_0 = 0 \) setting <math > Q_1 = 0 </math> gives \( \tau = \omega_0^2 \), indicating spectral stability, while for \( \theta_0 = \pi \) the result is \( \tau = -\omega_0^2 \), indicating an unstable equilibrium.
Two Degrees of Freedom
The characteristic equation is quartic, but the reduced characteristic equation is an easily-solved quadratic,
- <math 9>
Q_2(\tau) = \tau^2 - A \tau + B = 0 </math> where (Gantmacher, 1959)
- <math 10>
A = -\frac{1}{2} Tr (L^2) ,\qquad B = \det L. </math> The zeroes of \( Q_2 \),
- <math 11>
\tau = \frac{1}{2} A\pm \sqrt{\frac{1}{4}A^2 - B} </math> are both non-negative iff \( A \ge 0\) and \( 0\le B\le A^2/4 \). The stability boundaries are thus
- Zero Root\[ B = 0, A\ge 0\] (saddle-node)
- Multiple Root\[ B = \frac{1}{4} A^2 \] (Krein collision)
Figure 3 shows the stable region in the space of polynomial coefficients\( (A,B) \). In general a saddle-node bifurcation occurs upon crossing the horizontal boundary \( B = 0 \), although a pitchfork bifurcation (MacKay, 1986) is possible when \( H \) possesses certain spatial symmetries. If a locus of equilibria crosses the parabolic upper boundary a Krein bifurcation occurs. This is possible only if the Krein signatures of the merging eigenvalues are mixed. If \( H \) is natural (positive definite kinetic energy) a Krein bifurcation cannot occur.
Axisymmetric Systems
Consider a natural Hamiltonian \( H = T + U \), with \( U(\mathbf{r}) \) independent of the azimuth \( \phi \). Then \( p_{\phi} \) is conserved, so that part of the kinetic energy may be incorporated into an effective potential
- <math 12>
U^e(\rho,z) = \frac{p_{\phi}^2}{2m\rho^2} + U(\rho,z) </math> Formally, this process reduces the dimension of the system from three to two (Abraham and Marsden, 1978). The relative equilibria are then circular orbits in 3D and points in 2D and are given by \( \nabla U^e = 0 \). One then speaks of relative stability.
Example: The Stark Problem
An example of a relative equilibrium is the classical Stark problem (Howard, 1995a), in which a hydrogen atom is perturbed by a uniform electric field \({\mathbf F}\). The Hamiltonian is, in cylindrical coordinates \((\rho, \phi, z)\) and scaled atomic units,
- <math 13>
H = \frac{1}{2} (p_{\rho}^2 + p_z^2) + U^e(\rho,z) </math> where \( p_{\rho} = \dot\rho,~p_z = \dot z \), and effective potential
- <math 14 >
U^e(\rho,z) = \frac{\mu} {2\rho^2} - \frac{1}{r} - z
</math> with \( r = \sqrt{\rho^2 + z^2} \), and \( \mu \) is the scaled conserved \( \phi\)-component of angular momentum. The electric field has been taken in the z-direction, which is a symmetry axis for the problem. This integrable system is an example of a relative equilibrium, in which the critical points of the effective potential are circular orbits, whose stability is determined by the type of each critical point. The equilibria are then just the critical points of \( U^e(\rho,z) \), whose type is readily determined from the Hessian determinant, \(\Delta = \det D^2 U^e \). It is not necessary to calculate the eigenvalues although this is implicit in the analysis of \( D^2 U^e \). Figure 4 shows level sets of \( U^e \) in scaled coordinates for three values of the control parameter \( \mu \). The elliptic fixed point corresponds to a stable circular orbit and the saddle point to an unstable circular orbit. Since \( H \) is a natural 2 dof system, Krein bifurcations are impossible and stability can only be lost via a saddle-node or pitchfork bifurcation.
Setting \( \nabla U^e = 0 \) shows that the equilibria are given by \[ \rho_0^4 = \mu r_0^3,~~z_0 = r_0^3 \implies r_0(1-r_0^4)^2 = \mu \]. If \(\Delta_0 < 0 \) there is a saddle; if \( \Delta_0 >0\) there is a local minimum (stable circular orbit) if \( U_{\rho\rho}^e >0 \) and a local maximum if \( U_{\rho\rho}^e <0 \). A critical point changes type when \( \Delta_0 \) passes through zero. It follows that \( \Delta_0 \) changes sign when \( \mu = \mu^* = 64\sqrt{3}/243 \), at which point a stable-unstable pair of equilibria merge and disappear for \( \mu > \mu^* \), as seen in Fig. 4. Since \( H \) is natural one obtains as an added bonus the Lyapunov stability of the elliptic points.
Other axisymmetric systems possessing relative equilibria include ion motion in Paul traps (Bluemel, 1995), planetary dust dynamics (Howard et al., 1999), the problem of two fixed centers, and galactic dynamics.
Example: Microwave Ionization
As a example of a nonaxisymmetric system, consider a hydrogen atom perturbed by a circularly polarized microwave field with the electron orbit lying in the plane of polarization, which is described in a co-rotating frame by the autonomous Hamiltonian (Howard, 1992)
- <math 15>
H = \frac {1}{2} p_{\rho}^2 + \frac {p_{\phi}^2}{2 \rho^2} -\omega p_{\phi}-\frac{1}{\rho} + F\rho \cos\phi </math>, where \( p_{\phi} = \rho^2 (\dot\phi + \omega) \), and \( \omega \) and \( F \) are the frequency and amplitude of the microwave field. Again we have a relative equilibrium, but with the additional complication of nonconstant \( p_{\phi} \) and consequent indefinite kinetic energy. In this case stability does not correspond to the type of the critical point and a more sophisticated approach is needed. The equilibria are obtained from \( \nabla H = 0 \implies \phi_0 = (0,\pi),~p_{\phi}=\omega \rho_0^2 \), where
- <math 16>
\omega^2 \rho_0^3 - \sigma \rho_0^2 - 1 = 0 </math>
with \( \sigma = \cos\phi_0 =\pm 1 \). Working out the linearization then shows that the coefficient \( B \) cannot change sign, while the discriminant \( \Delta \) changes sign when \( \mu = \mu^* = 648 \), signalling a Krein bifurcation. In fact it is possible to work out the equilibrium locus \( B = B(A) \) in closed form for this system, showing that \( \lim_{\mu\to 0} \Delta = 0\) and \( \lim_{\mu\to \infty} B = 0\). Thus, the equilibrium is linearly stable for \( \mu > \mu^* \). It is not necessary to calculate the Krein signature or solve the cubic (16). Since this system in not natural the question of nonlinear stability remains unanswered.
Although an effective potential does not exist for this problem, it is useful to employ the zero-velocity function as the locus of points where the kinetic energy vanishes, with scaled radius \( \rho \to \omega^2 \rho/F \),
- <math 15a>
Z(\rho,\phi) = -\frac{1}{2} \rho^2 -\frac{\mu}{\rho} + \rho\cos\phi </math> with dimensionless parameter \( \mu = \omega^4/F^3 \). Setting \( Z = E \) then yields the zero-velocity curves, which form boundaries for trapped and untrapped motion. It is also easy to see that the critical points of \( Z \) are identical to those of \( H \), with the caveat that a stable orbits can occur at a maximum of ZVC! Figure 5 shows level sets of \( Z \) for \( \mu = 800 \). An orbit with energy \( E \) cannot cross a ZVC with the same energy.
Stability in Arbitrary Dimension
When \( n = 3 \) it is possible to use Descarte's rule of signs to obtain stability bounds. When \( n > 3 \) however, it is preferable to employ Sturm's method (Dickson, 1939), which yields a full set of necessary and sufficient conditions for spectral stability.
Sturm's Theorem
This method (magic bullet), widely used in the 19th century, consists in defining the Sturm sequence \( \{F_k(\tau)\} \) by \( F_0(\tau) = Q(\tau),~F_1(\tau) = Q^{\prime}(\tau) \), and for \( k\ge 2 \) by division:
- <math 17>
F_{k-2}(\tau) = G_{k-1}(\tau) F_{k-1}(\tau) - F_k(\tau),\qquad deg F_k < deg F_{k-1} </math> until a constant \( F_t \) is obtained. There is a multiple root iff \( F_t = 0 \). That is, at each stage one divides \( F_{k-1} \) by \( F_{k-2} \) to get the quotient \( G_{k-1} \) and remainder \( -F_k \). Let \( V(\tau) \) be the total number of variations in sign in proceeding through the Sturm sequence at \( \tau \) (ignoring zeroes). Then the number of distinct real roots in the interval \( (a,b] \) is exactly \( V(a) - V(b) \). In the present case all the zeroes of \( Q(\tau) \) are required to be non-negative real. By Sturm's theorem, this is true iff \( V(0) - V(\infty) = n \), which can be achieved iff \( t = n \). Note that the same conditions guarantee nonlinear stability for natural systems.
Three Degrees of Freedom
The reduced characteristic equation reads
- <math 18>
\tau^3 - A\tau^2 + B\tau - C = 0 </math> where
- <math 19>
A = -\frac{1}{2} Tr (L^2) ,\qquad 8B = [Tr(L^2)]^2 - 2 Tr(L^4) ,\qquad C = \det L. </math> Working out the Sturm sequence for the cubic (19) shows that the motion is spectrally stable iff
- <math 21>
A>0,\qquad B>0,\qquad C> 0,\qquad \Delta > 0. </math> where
- <math 20>
\Delta = 4(A^2-3B)(B^2-3AC) - (AB-9C)^3 </math> is the discriminant (Dickson, 1939).
The same conditions result upon applying Descarte's rule of signs to (19) to conclude that there are either 1 or 3 positive zeros and no negative zeros, provided that \( A, B, C \) are all positive. Further requiring that the discriminant be non-negative excludes the possibility of complex roots. The 3D stable region is depicted in Fig. 6. The stability boundaries are given by the plane \( C = 0 \) (saddle-node) and the quartic surface \( \Delta = 0 \) (Krein collision), with \( A,B > 0 \). Whether a Krein collision actually occurs when \( \Delta = 0 \) of course depends on the Krein signatures. If H is natural Krein collisions are impossible. For \( n \ge 4 \) Descarte's rule does not suffice but Sturm's theorem still applies. The quartic case is worked out in detail in (Howard and MacKay, 1987).
Natural Flows
While the above methods yield explicit linear stability bounds for arbitrary Hamiltonian equilibria in arbitrary dimension, they provide absolutely no information about nonlinear stability. Indeed, as the Cherry problem demonstrates, linear stability is no guarantee of nonlinear stability. However, there is an important subclass of Hamiltonians for which a sufficient condition for nonlinear stability may be obtained.
- Dirichlet's Theorem. Let \( z_0 \) be a locally quadratic equilibrium of the natural Hamiltonian, \( H = T(q,p) + U(q)\), where \( T \) is positive definite. Then \( z_0 \) is Lyapunov stable.
In such cases linear stability is tantamount to nonlinear stability. For a proof see (MacKay, 1986). Examples of natural flows include the nonlinear pendulum and the Stark problem, both described above. A very important example of a non-natural flow is the restricted three-body problem (Broucke, 1969).
The polynomial coefficients are especially simple for 2D natural flows: \[ A = Tr D^2 U , \qquad B = \det D^2U \] from which it can be shown that \( \Delta>0 \), so that Krein bifurcations cannot occur. Whether this extends to higher dimension is not known. A stronger result applies to non-separable Hamiltonians (Krechetnikov and Marsden, 2007):
Lagrange-Dirichlet Theorem: Let the second variation of the Hamiltonian \( \delta^2 H \) be definite at an equilibrium \( z_0 \). Then \( z_0 \) is stable.
For more information see (Krechetnikov and Marsden, 2007).
The Elliptic Paul Trap
As an example of a 3D natural flow consider two-ion motion in an elliptic Paul trap, whose averaged motion is described by the pseudopotential (Howard and Farrelly, 2007)
- <math 22>
U(x,y,z) = \frac{1}{2} (\lambda_1^2 x^2 + y^2 + \lambda_2^2 z^2) + \frac{1}{r} </math> where \( r = \sqrt{x^2 + y^2 + z^2}\) is the inter-ion distance and \( \lambda_{1,2} \) are dimensionless parameters depending on the trap geometry. This highly symmetric potential possesses three pairs of critical points, which form a set of Morse saddles (Poston and Stewart, 1980). It turns out that in general only one of the three pairs is stable. Figure 7 is a 3D contour plot of \( U(x,y,z) \) for \( \lambda_1 = 0.4, \lambda_2 = 0.6 \), for which the pair of equilibria on the x-axis are stable.
References
Abraham, R. and Marsden, J. E. (1978). Foundations of Mechanics, 2nd Ed., New York, Benjamin.
Arnold, V. I. and Avez, A. (1968). Ergodic Problems of Classical Mechanics, New York, Benjamin.
Arnold, V. I. (1980). Mathematical Methods of Classical Mechanics, 2nd Ed., New York, Springer.
Bluemel, R. (1995). "Nonlinear Dynamics of Trapped Ions," Phys. Rev. A 51, 620.
Broucke, R. (1969). "Stability of periodic orbits in the elliptic, restricted three-body problem," AIAA J. 7, 1003-9.
Cherry, T. M. (1926). "On periodic solutions of Hamiltonian systems of differential equations," Phil. Trans. R. Soc. A227, 137-221.
Dickson, L. E. (1939). New First Course in The Theory of Equations, New York, Wiley.
Gantmacher, F. R. (1959) The Theory of Matrices, New York, Chelsea.
Goldstein, H. (1980). Classical Mechanics, 2nd Ed., Reading, Addison-Wesley.
Howard, J. E. and MacKay, R. S. (1987). "Calculation of linear stability boundaries for equilibria of Hamiltonian systems," Phys. Lett. A122, 331-334.
Howard, J. E. (1992). "Stochastic ionization of hydrogen atoms in a circularly polarized microwave field," Phys. Rev. A46, 364-372.
Howard, J. E. (1995). "Saddle-point ionization and the Runge-Lenz invariant," Phys. Rev. A51, 3934-46.
Howard, J. E., Horanyi, M., and Stewart, G. R. (1999). "Global Dynamics of Charged Dust Particles in Planetary Magnetospheres," Phys. Rev. Lett. 83, 3993.
Howard, J. E. and Farrelly, D. (2007). "Two-ion motion in an ellipsoidal Paul trap," submitted to Phys. Rev. A.
Krechnetnikov, R. and Marsden, J. E. (2007). Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, 519.
MacKay, R. S. (1986). "Stability of equilibria of Hamiltonian systems," in Nonlinear Phenomena and Chaos, Ed. S. Sarkar, Bristol, Hilger.
Meyer, K. R. and Hall, G. R. (1992). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, New York, Springer.
Meiss, J. D. (2007). Differential Dynamical Systems, Philadelphia, SIAM.
Poston, T. and Stewart, I. (1996). Catastrophe Theory and its Applications, New York, Dover.
Routh, E. J. (1877). A Treatise on the Stability of A Given State of Motion, London, MacMillan.
Rumyantsev, V. V. and Sosnitskii, S. P. (1992). J. Appl. Maths. Mechs. 57, 1102.
See Also
Hamiltonian Systems, Stability, Stability of Symplectic Maps,Bifurcation,Periodic Orbit, Three Body Problem
