Planar

Abstract: This book deals with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced. From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems.

Abstract: The integrated planar technique has been considered as a reliable candidate for low-cost mass production of millimeter-wave circuits and systems. This paper presents new concepts that allow for a complete integration of planar circuits and waveguide filters synthesized on a single substrate by means of metallized post (or via-hole) arrays. Analysis of the synthesized integrated waveguide and design criteria are presented for the post pitch and diameter. A filter design method derived from a synthesis technique using inductive post is presented. An experimental three-pole Chebyshev filter having 1-dB insertion loss and return loss better than 17 dB is demonstrated. Integrating such planar and nonplanar circuits on a substrate can significantly reduce size, weight, and cost, and greatly enhance manufacturing repeatability and reliability.

Abstract: Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set - a two-dimensional zero dynamics submanifold of the full hybrid model $whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

Abstract: We report experimental results at 1.057 GHz that demonstrate the ability of a planar left-handed lens, with a relative refractive index of $\ensuremath{-}1$, to form images that overcome the diffraction limit. The left-handed lens is a planar slab consisting of a grid of printed metallic strips over a ground plane, loaded with series capacitors ($C$) and shunt inductors ($L$). The measured half-power beamwidth of the point-source image formed by the left-handed lens is 0.21 effective wavelengths, which is significantly narrower than that of the diffraction-limited image corresponding to 0.36 wavelengths.