The Bohl spectrum for nonautonomous differential equations

We develop the Bohl spectrum for nonautonomous linear differential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in general. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this not evident from the Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum

Quantum interference within the complex quantum Hamilton-Jacobi formalism

Quantum interference is investigated within the complex quantum Hamilton-Jacobi formalism. As shown in a previous work [Phys. Rev. Lett. 102, 250401 (2009)], complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection near vortical tubes, these structures being prominent features of quantum caves in space-time Argand plots. Here, we further analyze the divergence and vorticity of the quantum momentum function along streamlines near poles, showing the intricacy of the complex dynamics. Nevertheless, despite this behavior, we show that the appearance of the well-known interference features (on the real axis) can be easily understood in terms of the rotation of the nodal line in the complex plane. This offers a unified description of interference as well as an elegant and practical method to compute the lifetime for interference features, defined in terms of the average wrapping time, i.e., considering such features as a resonant process.Comment: revised version, 13 pages, 11 figures, 1 tabl

Dynamics of Numerics of Nonautonomous Equations with Periodic Solutions: Introducing the Numerical Floquet Theory

Nonautonomous systems with periodic solutions are encountered frequently in applications. In this paper, we will consider simple systems whose solutions are periodic with a known period. Their transformation under linearized collocation methods is investigated, using a technique called stroboscopic sampling, a discrete version of the well-known Poincaré map. It is shown that there is an inextricable relationship between AN stability (or BN stability) of the numerical methods and the correct qualitative behaviour of solutions

Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations

We consider 3×3 partially hyperbolic linear differential systems over an ergodic flow X^t and derived from the linear homogeneous differential equation x''(t)+β(X^t(t))x'(t)+ γ(t)x(t) = 0. Assuming that the partial hyperbolic decomposition E^s ⊕ E^c ⊕ E^u is proper and displays a zero Lyapunov exponent along the central direction E^c we prove that some C^0 perturbation of the parameters β(t) and γ(t) can be done in order to obtain non-zero Lyapunov exponents and so a chaotic behaviour of the solution.info:eu-repo/semantics/publishedVersio

Aspects of Invariant Manifold Theory and Applications

Recent years have seen a surge of interest in "data-driven" approaches to determine the equations governing complex systems. Yet in spite of modern computing advances, the high dimensionality of many systems --- such as those occurring in biology and robotics --- renders direct machine learning approaches infeasible. This dissertation develops tools for the experimental study of complex systems, based on mathematical concepts from dynamical systems theory. Our approach uses the fact that parsimonious assumptions often lead to strong insights from dynamical systems theory; such insights can be leveraged in learning algorithms to mitigate the “curse of dimensionality” and make these algorithms practical. Our first contribution concerns nonlinear oscillators. Oscillators are ubiquitous in nature, and usually associated with the existence of an "asymptotic phase" which governs the long-term dynamics of the oscillator. We show that asymptotic phase can be expressed as a line integral with respect to a uniquely defined closed differential 1-form, and provide an algorithm for estimating this "ToF" from observational data. Unlike all previously available data-driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within the entire region for which data convergent to the limit cycle is available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series. Our next contribution concerns locomotion systems which are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that in the ``Stokesian'' (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection'' model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime'' where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer connection-like. We derive this model using results from noncompact NHIM theory in a singular perturbation framework. Using a normal form derived from theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. Our algorithm has applications to the study of optimality of animal gaits, and to hardware-in-the-loop optimization to produce gaits for robots. Finally, we make fundamental contributions to NHIM theory: we prove that the global stable foliation of a NHIM is a $C^0$ disk bundle, and we prove that the dynamics restricted to the stable manifold of a compact inflowing NHIM are globally topologically conjugate to the linearized transverse dynamics at the NHIM restricted to the stable vector bundle. We also give conditions ensuring $C^k$ versions of our results, and we illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.PHDElectrical and Computer EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147642/1/kvalheim_1.pd