Piecewise Polynomial Systems Research Articles

Limit Cycles of a Class of Piecewise Smooth Liénard Systems

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Liénard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

Detection of critical points of multivariate piecewise polynomial systems

We propose a general scheme for detecting critical locations (of dimension zero) of piecewise polynomial multivariate equation systems. Our approach generalizes previously known methods for locating tangency events or self-intersections, in contexts such as surface–surface intersection (SSI) problems and the problem of tracing implicit plane curves. Given the algebraic constraints of the original problem, we formulate additional constraints, seeking locations where the differential matrix of the original problem has a non-maximal rank. This makes the method independent of a specific geometric application, as well as of dimensionality. Within the framework of subdivision based solvers, test results are demonstrated for non-linear systems with three and four unknowns.

Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems

The multivariate spline is a piecewise polynomial with certain smoothness, and the piecewise algebraic hypersurfaces with certain smoothness (i.e. the set of all common zeros of multivariate splines) have become useful tools for representing or approximating surfaces. Studying an effective method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology is one of the important ways to solve the problem of how to represent or approximate geometric objects with certain topological structure (especially the complex topology structure), and also a new and important topic of computational geometry and algebraic geometry. Parametric piecewise polynomial systems are not only closely related to a series of reasearch, such as intersection of surfaces; blending curves and surfaces; generation of transition surfaces, but also essential generalization of the parametric semi-algebraic systems. The purpose of this paper is to introduce some recent research progress on the construction of real piecewise algebraic hypersurfaces, and parametric piecewise polynomial systems.

Topologically guaranteed bivariate solutions of under-constrained multivariate piecewise polynomial systems

We present a subdivision based algorithm to compute the solution of an under-constrained piecewise polynomial system of n−2 equations with n unknowns, exploiting properties of B-spline basis functions. The solution of such systems is, typically, a two-manifold in Rn. To guarantee the topology of the approximated solution in each sub-domain, we provide subdivision termination criteria, based on the (known) topology of the univariate solution on the domain’s boundary, and the existence of a one-to-one projection of the unknown solution on a two dimensional plane, in Rn. We assume the equation solving problem is regular, while sub-domains containing points that violate the regularity assumption are detected, bounded, and returned as singular locations of small (subdivision tolerance) size. This work extends (and makes extensive use of) topological guarantee results for systems with zero and one dimensional solution sets. Test results in R3 and R4 are also demonstrated, using error-bounded piecewise linear approximations of the two-manifolds.

Limit cycle bifurcations in a class of perturbed piecewise smooth systems

In this paper, we concern with the number of limit cycles in a piecewise polynomial system. First, we give 42 different phase portraits of the unperturbed system with at least one closed orbit. Then, we perturb one phase portrait of them by piecewise polynomials, and consider lower bounds for the maximal numbers of limit cycles emerging from the origin and generalized homoclinic loop, respectively.

Convex relaxations for L2-gain analysis of piecewise affine/polynomial systems

This paper proposes some sufficient conditions based on the computation of polynomial and piecewise polynomial storage functions for -gain analysis of discrete-time piecewise affine or piecewise polynomial systems. The computation of such storage functions is performed by means of convex optimisation techniques via the sum-of-squares decomposition of multivariate polynomials.

A high-order harmonic balance method for systems with distinct states

A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct states, i.e. where the ODEs involve piecewise defined functions. An event-driven scheme is used which is based on the direct calculation of the state transition time instants between these states. An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems. Moreover, it is shown that derivatives of the solution of up to second order can be calculated analytically, making the method especially attractive for design studies.The methodology is applied to several structural dynamical systems with conservative and dissipative nonlinearities in externally excited and autonomous configurations. Great performance and robustness of the proposed procedure was ascertained.

A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

SUMMARY This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non-convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.

A class of orthonormal complete piecewise polynomial systems and applications thereof

In the application of geometric graphs and image shape analysis, the Gibbs phenomenon appears if we approximate discontinuous geometric graphs using trigonometric functions, while the approximation effect of Walsh functions is not very good because of its slow convergence. This paper constructs a class of piecewise polynomials systems (referred to as quaternary U-Systems), whose breakpoints only appear at quaternary ratio- nal numbers. Such quaternary U-Systems are a class of complete orthonormal systems in L2 0,1 . In addition, we also investigate their properties, formulae for basis values and Fourier-QU coefficients, and present a set of explicit expressions for a quaternary U-system of degree r (r=2,3,4). Next, we apply a finite Fourier-QU series to represent image edges, and propose using the finite Fourier-QU coefficients to depict geometric graphs and image shapes. As a result, we obtain a new class of polynomial descriptors, called QU descriptors, and prove that unified QU descriptors are invariant under translation, scale, and rotation. Finally, we verify experimentally that the convergence rate of Fourier-QU series is faster than that of Fourier series, Walsh series, and Fourier-BU series in terms of the approximation of the function of a single variable. Furthermore, the experimental results prove that the QU descriptors are a class of practical shape descriptors, and that the QU distance between images can accurately measure their similarity.

Solving polynomial systems using no-root elimination blending schemes

Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its Bernstein–Bézier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surface–surface–surface intersection (SSSI) and surface–curve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks.

Solving parametric piecewise polynomial systems

We deal with C r smooth continuity conditions for piecewise polynomial functions on Δ , where Δ is an algebraic hypersurface partition of a domain Ω in R n . Piecewise polynomial functions of degree, at most, k on Δ that are continuously differentiable of order r form a spline space C k r . We present a method for solving parametric systems of piecewise polynomial equations of the form Z ( f 1 , … , f n ) = { X ∈ Ω ∣ f 1 ( V , X ) = 0 , … , f n ( V , X ) = 0 } , where f ω ∈ C k ω r ω ( Δ ) , and f ω ∣ σ i ∈ Q [ V ] [ X ] for each n -cell σ i in Δ , V = ( u 1 , u 2 , … , u τ ) is the set of parameters and X = ( x 1 , x 2 , … , x n ) is the set of variables; σ 1 , σ 2 , … , σ m are all the n -dimensional cells in Δ and Ω = ⋃ i = 1 m σ i . Based on the discriminant variety method presented by Lazard and Rouillier, we show that solving a parametric piecewise polynomial system Z ( f 1 , … , f n ) is reduced to the computation of discriminant variety of Z . The variety can then be used to solve the parametric piecewise polynomial system. We also propose a general method to classify the parameters of Z ( f 1 , … , f n ) . This method allows us to say that if there exist an open set of the parameters’ space where the system admits exactly a given number of distinct torsion-free real zeros in every n -cells in Δ .

Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests

We present an algorithm which robustly computes the intersection curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n + 1 unknowns. The solution of such a system is typically a curve in R n + 1 . This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in R n + 1 . Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution curve consists of just one single component. For such domains, a numerical curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting curve is guaranteed. Several possible applications of the solver, namely solving the surface–surface intersection problem, computing 3D trisector curves, flecnodal curves or kinematic simulations in 3D are also discussed.

Continuous Shearlet Tight Frames

Based on the shearlet transform we present a general construction of continuous tight frames for L 2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform.

Direction-Specific Gradient Scaling for Interactive Multicriterion Optimization Using an Abstract Mass Concept

In the Method of Abstract Forces for the direction finding (or tradeoff cut) subproblems of interactive multicriterion optimization, it is necessary to scale criterion gradients. Previously, an ad hoc, analyst assisted, but nevertheless effective scaling method was used. This paper introduces a concept of abstract mass. It is shown that the previous ad hoc scaling method may be considered an approximation to the present direction-specific approach. Also, the abstract mass approach provides a strengthening of the original Newton's Second Law of Motion analogy motivation for the Method of Abstract Forces. A method for automatic and direction-specific scaling is proposed which depends on the solution of a continuous piecewise polynomial system of equations. The method is illustrated on an example with three criterion functions. In this example, Newton's method for solving simultaneous nonlinear systems converges to a solution. More generally, restart homotopy methods may be required.