Unbounded Orbits Research Articles

On the Nonlinear Ddynamics of Fast Filtering Algorithms

The main purpose of this paper is to address a fundamental open problem in linear filtering and estimation, namely, what is the steady-state or asymptotic behavior of the Kalman filter, or the Kalman gain, when the observed stationary stochastic process is not generated by a finite-dimensional stochastic system, or when it is generated by a stochastic system having higher-dimensional unmodeled dynamics. For example, some time ago Kalman pointed out that the usual positivity conditions assumed in the classical situation are not in fact necessary for the Kalman filter to converge. Using a “fast filtering” algorithm, which incorporates the statistics of the observation process as initial conditions for a dynamical system, this question is analyzed in terms of the phase portrait of a “universal” n;onlinear dynamical system. This point of view has additional advantages as well, since it enables one to use the theory of dynamical systems to study the sensitivity of the Kalman filter to (small) changes in initial conditions; e.g., to changes in the statistics of the underlying process. This is especially important since these statistics are often either approximated or estimated. In this paper, for a scalar observation process, necessary and sufficient conditions for the Kalman filter to converge are derived using methods from stochastic systems and from nonlinear dynamics-especially the use of stable, unstable, and center manifolds. It is also shown that, in nonconvergent cases, there exist periodic points of every period p, $p \geq 3$ that are arbitrarily close to initial conditions having unbounded orbits, rigorously demonstrating that the Kalman filter can also be “sensitive to initial conditions.”

Capture orbits and melnikov integrals in the planar three-body problem

The separatrix between bounded and unbounded orbits in the three-body problem is formed by the manifolds of forward and backward parabolic orbits. In an ideal problem these manifolds coincide and form the boundary between the sets of bounded and unbounded orbits. As a mass parameter increases the movement of the parabolic manifolds is approximated numerically and by Melnikov's method. The evidence indicates that for positive values of the mass parameter these manifolds no longer coincide, and that capture and oscillatory orbits exist.

Infinity manifold of a swinging Atwood's machine

The unbounded motions of a swinging Atwood's machine are analysed by blowing up the singularity at infinity. The asymptotic motion is reduced to a gradient flow on an ellipsoid. By studying the flow on this ellipsoid it is shown that the unbounded orbits oscillate either an infinite number of times or not at all, depending only on a critical value of the mass ratio.

Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

In the previous papers of this series we discussed the representation in quantum mechanics of canonical transformations leading to action and angle variables, for Hamiltonians with bounded or unbounded orbits, i.e., whose spectra is either discrete, continuous, or mixed. In the present paper the results are extended to Hamiltonians with periodic potentials which have a band spectra. Again the canonical transformations are non-linear and non-bijective and the classical analysis shows that the angle variable φ (always in the interval 0≤φ≤2π) and action J can be defined for energies both below and above the maximum height of the potential. In all of the original phase space the variables ( q, p) are then periodic functions of φ. Inversely, because of the invariance of the Hamiltonian under translations q → q + a, the (φ, J) are also periodic functions of q. thus to recover bijectiveness we require an infinite sheet structure in both the ( q, p) and (φ, J) phase spaces. In turn the sheet structure can be replaced by appropriate ambiguity groups and spins, with the help of which we propose an explicit expression for the representation in quantum mechanics of the canonical transformation, and recover the latter when we pass to the classical limit with the help of the WKB approximation. The present analysis corroborates the previous surmise that the nature of the spectra of a quantum mechanical Hamiltonian, i.e., continuous, discrete, mixed, or of bands, is related to the ambiguity group and spin of the problem. As the latter originates in classical mechanics when we discuss the canonical transformations from ( q, p) to (φ, J), we conclude that some quantum features, such as the nature of spectra of operators, are already implicit in the classical picture.

Chaos and nonlinear dynamics of single‐particle orbits in a magnetotaillike magnetic field

The properties of charged‐particle motion in Hamiltonian dynamics are studied in a magnetotaillike magnetic field configuration. It is shown by numerical integration of the equation of motion that the system is generally nonintegrable and that the particle motion can be classified into three distinct types of orbits: bounded integrable orbits, unbounded stochastic orbits, and unbounded transient orbits. It is also shown that different regions of the phase space exhibit qualitatively different responses to external influences. The concept of “differential memory” in single‐particle distributions is proposed. Physical implications for the dynamical properties of the magnetotail plasmas and the possible generation of non‐Maxwellian features in the distribution function are discussed.

On orbits of unipotent flows on homogeneous spaces

AbstractLet G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of (ut) on G/Γ is finite. For ‘arithmetic lattices’ this was proved in [2]. The present generalization is obtained by studying the ‘frequency of visiting compact subsets’ for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of ℝ-rank 1 and Γ is any (not necessarily arithmetic) lattice in G.

Canonical transformations to action and angle variables and their representations in quantum mechanics: III. The general problem

The representation in quantum mechanics of canonical transformations to action and angle variables is discussed for a general class of Hamiltonians including some that have both bounded and unbounded orbits where, in the latter case, the definition of the transformation is suitably extended. Again, as in the particular examples discussed in the previous papers of this series, the canonical transformations are nonlinear and nonbijective. We can recover though the bijectiveness, i.e., the one-to-one onto mapping, either by introducing a sheet structure in the original phase space or using the concepts of ambiguity group and ambiguity spin. With the help of these concepts we obtain an expression for the representation in quantum mechanics of the canonical transformation and recover the latter when we pass to the classical limit with the help of the WKB approximation. Furthermore, we establish in this paper a one-to-one correspondance between the arbitrariness in the phase of the representation and in the choice of the variable conjugate to the action.

Alignment errors in the strong-focusing synchrotron

A method is presented for the treatment of errors in alignment between the sectors of a strong focusing accelerator. Motion in one lateral coordinate only is treated, and the individual sectors are assumed to be ideal. It is found that if the ideal orbits (without misalignments) are periodic in one revolution the effect of the misalignments is to produce unbounded orbits; but that away from, such resonances stable orbits exist. Estimates are made of the alignment tolerances required for reasonable amplitudes of the perturbed motion. The essential features of the results are independent of the detailed nature of the guide field so lorg as the ideal orbits are stable