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This book gives a modern differential geometric treatment of linearly nonholonomically constrained systems. It discusses in detail what is meant by symmetry of such a system and gives a general theory of how to reduce such a symmetry using the concept of a differential space and the almost Poisson bracket structure of its algebra of smooth functions. The above theory is applied to the concrete example of Carathéodory's sleigh and the convex rolling rigid body. The qualitative behavior of the motion of the rolling disk is treated exhaustively and in detail. In particular, it classifies all motions of the disk, including those where the disk falls flat and those where it nearly falls flat.

We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system. We suppose that the origin is a parabolic fixed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connection associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincare-Melnikov function.

We introduce a geometric mechanism for diffusion in a priori unstable nearly integrable dynamical systems. It is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. We argue that these objects created by resonances can be incorporated in transition chains taking the place of the destroyed primary KAM tori. We establish rigorously the existence of this mechanism in a simple model that has been studied before.The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. This toolkit is based on extending and unifying standard techniques. A new tool used here is the scattering map of normally hyperbolic invariant manifolds. The model considered is a one-parameter family, which for $\\varepsilon = 0$ is an integrable system. We give a small number of explicit conditions the jet of order $3$ of the family that, if verified imply diffusion. The conditions are just that some explicitely constructed functionals do not vanish identically or have non-degenerate critical points, etc. An attractive feature of the mechanism is that the transition chains are shorter in the places where the heuristic intuition and numerical experimentation suggests that the diffusion is strongest.

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

This paper presents a study of analysis of minimum-time trajectories for a differential drive robot equipped with a fixed and limited field-of-view camera, which must keep a given landmark in view during maneuvers. Previous works have considered the same physical problem and provided a complete analysis/synthesis for the problem of determining the shortest paths. The main difference in the two cost functions (length vs. time) lays on the rotation on the spot. Indeed, this maneuver has zero cost in terms of length and hence leads to a 2D shortest path synthesis. On the other hand, in case of minimum time, the synthesis depends also on the orientations of the vehicle. In other words, the not zero cost of the rotation on the spot maneuvers leads to a 3D minimum-time synthesis. Moreover, the shortest paths have been obtained by exploiting the geometric properties of the extremal arcs, i.e., straight lines, rotations on the spot, logarithmic spirals and involute of circles. Conversely, in terms of time, even if the extremal arcs of the minimum-time control problem are exactly the same, the geometric properties of these arcs change, leading to a completely different analysis and characterization of optimal paths. In this paper, after proving the existence of optimal trajectories and showing the extremal arcs of the problem at hand, we provide the control laws that steer the vehicle along these arcs and the time-cost along each of them. Moreover, this being a crucial step toward numerical implementation, optimal trajectories are proved to be characterized by a finite number of switching points between different extremal arcs, i.e., the concatenations of extremal arcs with infinitely many junction times are shown to violate the optimality conditions.

Perturbation to Noether symmetries and adiabatic invariants of discrete nonholonomic nonconservative mechanical systems on an uniform lattice are investigated. Firstly, we review Noether symmetry and conservation laws of a nonholonomic nonconservative system. Secondly, we study continuous Noether symmetry of a discrete nonholonomic system, give the Noether symmetry criterion and theorem of discrete corresponding holonomic system and nonholonomic system. Thirdly, we study perturbation to Noether symmetry of the discrete nonholonomic nonconservative system, give the criterion of perturbation to Noether symmetry for this system, and based on the definition of adiabatic invariants, we construct the theorem under which can lead to Noether adiabatic invariants for this system, and the forms of discrete Noether adiabatic invariants are given. Finally, we give an example to illustrate our results.

This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

This paper is devoted to modeling and theoretical analysis of dynamic control systems subject to a class of rheonomous affine constraints, which are called A -rheonomous affine constraints. We first define A -rheonomous affine constraints and explain their geometric representation. Next, a necessary and sufficient condition for complete nonholonomicity of A -rheonomous affine constraints is shown. Then, we derive nonholonomic dynamic systems with A -rheonomous affine constraints (NDSARAC), which are included in the class of nonlinear control systems. We also analyze linear approximated systems and accessibility for the NDSARAC. Finally, the results are applied to some physical examples in order to check the application potentiality.

This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov.It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov's work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov's original work.Also included is an article by A. Katok that presents Anosov's scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform.