We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman’s controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems ’ standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of planar curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from nonlinear physics. A high frequency control strategy is proposed such that the averaged systems become flat. ∗This work was partially supported by the G.R. “Automatique ” of the CNRS and by the D.R.E.D. of the “Ministère de l’Éducation Nationale”. 1 1

. We give a proof of the persistence of invariant tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions. With respect to the case of anisochronous systems, there is the additional problem to find the set of allowed rotation vectors for the invariant tori, which can not given a priori simply by looking at the unperturbed system, and which leads to a sort of singular implicit function problem. Albeit the solutions are not analytic in the size of the perturbation, an analytic expansion for the solution can be envisaged and successfully used in order to explicitly construct the solution as an absolutely convergent power series. 1. Introduction sec.1 p.1.1 1.1. Lindstedt series and KAM theorem. The KAM theorem assures the persistence of a large number of invariant tori under perturbations of integrable systems. For analytic Hamiltonians a posteriori one can consider the equations of motion and look directly for analytic quasiperiodi...

It has long been known that a simple rigid pendulum can be stabilized against small disturbances in its upside-down state by rapid vertical oscillations of its pivot. Over part of the parameter range for which this happens we find that there is a second, quite different way in which the pendulum can avoid falling over, namely by getting trapped in a limit cycle oscillation about the upward vertical. Each upside-down oscillation of this kind has a distinctive 'multiplenodding' character. 1.

The constrained Lagrangian and constrained Hamiltonian equations of motion for a general nonrelativistic classical mechanical system subject to rheonomous holonomic constraints are derived in an easy and straightforward manner. The numerical integration of the constrained equations of motion are discussed. It is shown how constraint errors introduced by the numerical integration can be avoided by introducing simple constraint correction schemes. As an example, the developed constrained methods are applied to the periodically driven inverted n-linked pendulum. It is demonstrated how the constrained methods leads to very efficient numerical algorithms. In the case of the n-linked pendulum the computational complexity using the constrained methods is O(n) compared to O(n³) using the conventional unconstrained approach.

The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering and science, including the modeling of chemical reactions, heat transfer, and diffusion, and the solution of optimal control problems. A TPBVP may have no solution, a single solution, or multiple solutions. A new strategy is presented for reliably locating all solutions of a TPBVP. The method determines narrow enclosures of all solutions that occur within a specified search interval. Key features of the method are the use of a new solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with intervalvalued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method. Keywords: Two-point boundary value problem; Ordinary differential equations; Initial value Systems of ordinary differential equations (ODEs) arise in mathematical models throughout science and engineering. When an explicit condition (or conditions) that a solution must satisfy is specified at one value of the independent variable, usually its lower bound, this is referred to as

Dynamical systems may experience undesirable behavior or instability, which can be eliminated using feedback control means. However, in the absence of feedback control, the stability of some systems may be increased by imposing parametric excitation. In other cases, the exit time of the system response from the stable to the unstable domain may be prolonged by imposing external noise, a phenomenon termed noise-enhanced stability (NES). This article presents an assessment of the mechanisms of stabilization via multiplicative noise and noise-enhanced stability. The first part deals with stabilization via deterministic parametric excitation of gravity-defying systems such as the inverted simple and spherical pendulums, aeroelastic structures, human walking, and quantum nonlinear couplers. The second part introduces the concept of noise-induced transition (NIT) in one-dimensional nonlinear systems and ship roll motion. Stabilization of originally unstable systems via multiplicative noise is treated in the third part. The fourth part addresses the influence of additive noise in delaying the exit time of system response to an unstable domain. This topic is

By 1807 the scale of the task the Huddersfield Canal Company had taken on must have been painfully evident. The canal involved the construction of the longest man-made tunnel of the time, but this was only one of the problems faced by the canal company as costs spiralled and a mixture of poor workmanship and the harsh conditions of the Pennine moorlands created additional delays. Construction had started in 1794 following and Act of Parliament passed in the same year. The engineer in charge was initially Benjamin Outram, a protegé of William Jessop (one of the most influential engineers of the age of the canal and secretary of the Smeatonian Society of Civil Engineers for twenty years), and co-founder with Jessop of the Butterley ironworks, but Outram was over committed and left the project in 1802; his career ended early and in tragedy. A third Act to raise finance was passed in 1806, and Thomas Telford was brought in to report on the condition and prospects for the canal. His report of January 1807 was generally up-beat and characteristically detailed. Despite further disasters such as the Black Flood, where a reservoir dam embankment burst with the loss of five lives, the canal was completed in 1811. It never made the profits promised in the early days of the canal boom, but as a monument to the stubborn pigheadedness of early industrial Britain it has few equals. The Standedge Tunnel itself was completed in 1809, having cost over fifty lives and £124,000, about a quarter of the entire costs of construction. It was 5,477 yards long, though alterations mean that it is now a little longer. By 1846 further tunnels had been created alongside the canal tunnel to carry the railroads which rapidly out-competed the riven canal network. The tunnel re-opened in 2001 as a tourist attraction rather than as part of the industrial transport network it had originally been intended to serve. 1807 was, of course, the centenary of Euler’s birth. This year is the tercentenary. The world moves quickly, but I am sure that the engineers of the Huddersfield Narrow Canal would have benefitted from the clear thinking which is so evident in Euler’s work. They may even have

Two algorithms for deriving the constrained Lagrangian and Hamiltonian equations of motion are presented. They are used to derive the constrained equations of motion for various pendulum systems in an easy and straightforward manner. The advantages of the algorithms are discussed. Computer experiments are performed by solving numerically the constrained equations of motion.

This year we have had both frogspawn and toadspawn in the pond. The tadpoles emerged first and very quickly started to eat the toadspawn. Since the toad tadpoles are poisonous to many creatures it is not clear whether the tadpoles were eating the developing toad tadpoles or just the jelly, but whichever combination they were eating, the carefully roped toadspawn was rapidly destroyed. It is possible that the frog tadpoles were getting their retaliation in early, and that if they fail to destroy the toad tadpoles before they leave the spawn they might become food for the toads-to-be in their turn. To preserve a few toad tadpoles we removed them from the pond and kept them in a jar. One characteristic I had not appreciated is that whilst frog tadpoles rest horizontally on the bottom of the pond, young toad tadpoles rest vertically with their heads up; a position which would be very unstable in the absence of the supporting relatively high density water. Indeed, I have not discovered the source of the stability of this pose, which should be closer to an inverted pendulum than anything else I can think of, but I suppose that a light inverted pendulum could be stable in treacle, though had I been thinking about it at the time I would have looked to see whether there was any periodic oscillation in the tadpoles ’ tails. My thoughts turn to small oscillations because of recent work on stabilizing an inverted pendulum, and even highly articulated multiple pendula such as the bicycle chain, using periodic or random perturbations [1, 2, 6]. Experimentalists and applied mathematicians produced a series of wonderful experiments and applications in the mid 1990s which reproduce the effect of the classic Indian rope trick – a flexible wire can be stabilized in the upwards direction by the right sort of oscillation at the base [5, 7]. This provides a great example of the power of applied mathematics to explain surprises (it took no new physics to understand what was going on, just the time and expertise required to analyze the equations).

On the dynamics of a vertically driven