Polynomial Differential Equations Research Articles

Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations $${\dot{x} = y+{\rm sgn}(g_m(x, y))F(x)}$$ , $${\dot{y} = -x}$$ , where the zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m, and sgn(z) denotes the sign function.

ON THE EQUIVALENCE OF DIFFERENTIAL EQUATIONS

In this article we use the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Riccati equation and some polynomial differential equations. The results are applied to discussion of the qualitative behavior of periodic solutions of these complicated differential equations.

Algorithmic Construction of Lyapunov Functions for Power System Stability Analysis

We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition, which have been powerful tools for the analysis of systems with polynomial vector fields. In order to apply these techniques to power grid systems described by trigonometric nonlinearities we use an algebraic reformulation technique to recast the system's dynamics into a set of polynomial differential algebraic equations. We demonstrate the application of these techniques to the transient stability analysis of power systems by estimating the region of attraction of the stable operating point. An algorithm to compute the local stability Lyapunov function is described together with an optimization algorithm designed to improve this estimate.

Features in chemical kinetics. I. Signatures of self-emerging dimensional reduction from a general format of the evolution law

Simplification of chemical kinetics description through dimensional reduction is particularly important to achieve an accurate numerical treatment of complex reacting systems, especially when stiff kinetics are considered and a comprehensive picture of the evolving system is required. To this aim several tools have been proposed in the past decades, such as sensitivity analysis, lumping approaches, and exploitation of time scales separation. In addition, there are methods based on the existence of the so-called slow manifolds, which are hyper-surfaces of lower dimension than the one of the whole phase-space and in whose neighborhood the slow evolution occurs after an initial fast transient. On the other hand, all tools contain to some extent a degree of subjectivity which seems to be irremovable. With reference to macroscopic and spatially homogeneous reacting systems under isothermal conditions, in this work we shall adopt a phenomenological approach to let self-emerge the dimensional reduction from the mathematical structure of the evolution law. By transforming the original system of polynomial differential equations, which describes the chemical evolution, into a universal quadratic format, and making a direct inspection of the high-order time-derivatives of the new dynamic variables, we then formulate a conjecture which leads to the concept of an "attractiveness" region in the phase-space where a well-defined state-dependent rate function ω has the simple evolution ω[over dot]=-ω(2) along any trajectory up to the stationary state. This constitutes, by itself, a drastic dimensional reduction from a system of N-dimensional equations (being N the number of chemical species) to a one-dimensional and universal evolution law for such a characteristic rate. Step-by-step numerical inspections on model kinetic schemes are presented. In the companion paper [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234102 (2013)] this outcome will be naturally related to the appearance (and hence, to the definition) of the slow manifolds.

Features in chemical kinetics. II. A self-emerging definition of slow manifolds

In the preceding paper of this series (Part I [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013)]) we have unveiled some ubiquitous features encoded in the systems of polynomial differential equations normally applied in the description of homogeneous and isothermal chemical kinetics (mass-action law). Here we proceed by investigating a deeply related feature: the appearance of so-called slow manifolds (SMs) which are low-dimensional hyper-surfaces in the neighborhood of which the slow evolution of the reacting system occurs after an initial fast transient. Indeed a geometrical definition of SM, devoid of subjectivity, "naturally" follows in terms of a specific sub-dimensional domain embedded in the peculiar region of the concentrations phase-space that in Part I we termed as "attractiveness region." Numerical inspections on simple low-dimensional model cases are presented, including the benchmark case of Davis and Skodje [J. Chem. Phys. 111, 859 (1999)] and the preliminary analysis of a simplified model mechanism of hydrogen combustion.

On Groebner Bases and Their Use in Solving Some Practical Problems

Groebner basis are an important theoretical building block of modern (polynomial) ring theory. The origin of Groebner basis theory goes back to solving some theoretical problems concerning the ideals in polynomial rings, as well as solving polynomial systems of equations. In this article four practical applications of Groebner basis theory are considered; we use Groebner basis to solve the systems of nonlinear polynomial equations, to solve an integer programming problem, to solve the problem of chromatic number of a graph, and finally we consider an original example from the theory of systems of ordinary (polynomial) differential equations. For practical computations we use systems »MATHEMATICA« and »SINGULAR«.

Limit cycles of polynomial differential equations with quintic homogeneous nonlinearities

In this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers ẋ=−y,ẏ=x;ẋ=−y(1−(x2+y2)2),ẏ=x(1−(x2+y2)2); when they are perturbed inside the class of all polynomial differential systems with quintic homogeneous nonlinearities. We do this study using the averaging theory of first, second and third orders.

An explicit bound of the number of vanishing double moments forcing composition

We give two new characterizations of pairs of polynomials or trigonometric polynomials that form a composition pair. One of them proves that the cancellation of a given number of double moments implies that they form a composition pair. This number only depends on the maximum degree of both polynomials. This is the first time that composition is characterized in terms of the cancellation of an explicit number of double moments. Our results allow to recognize the composition centers for polynomial and trigonometric Abel differential equations.

Differential algebra for control systems design: Constructive computation of canonical forms

Many systems can be represented using polynomial differential equations, particularly in process control, biotechnology, and systems biology [1], [2]. For example, models of chemical and biochemical reaction networks derived using the law of mass action have the form ẋ = Sv(k,x), (1) where x is a vector of concentrations, S is the stoichiometric matrix, and v is a vector of rate expressions formed by multivariate polynomials with real coefficients k . Furthermore, a model containing nonpolynomial nonlinearities can be approximated by such polynomial models as explained in "Model Approximation". The primary aims of differential algebra (DALG) are to study, compute, and structurally describe the solution of a system of polynomial differential equations,f (x,ẋ, ...,x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k)</sup> ) =0, (2) where f is a polynomial [3]-[6]. Although, in many instances, it may be impossible to symbolically compute the solutions, or these solutions may be difficult to handle due to their size, it is still useful to be able to study and structurally describe the solutions. Often, understanding properties of the solution space and consequently of the equations is all that is required for analysis and control design.

A Transformation Method to Construct Family of Exactly Solvable Potentials in Quantum Mechanics

A transformation method is applied to the second order ordinary differential equation satisfied by orthogonal polynomials to construct a family of exactly solvable quantum systems in any arbitrary dimensional space. Using the properties of orthogonal polynomials, the method transforms polynomial differential equation to D-dimensional radial Schrodinger equation which facilitates construction of exactly solvable quantum systems. The method is also applied using associated Laguerre and Hypergeometric polynomials. The quantum systems generated from other polynomials are also briefly highlighted.

Robust nonlinear stability and performance analysis of an F/A‐18 aircraft model using sum of squares programming

SUMMARYIn this paper, we develop algorithms for the computational analysis of stability and robust performance properties of nonlinear models governed by polynomial differential equations in quasi‐Linear Parameter Varying form subject to parametric uncertainty. The methods presented use the sum of squares decomposition and ideas from real algebraic geometry to represent polynomial non‐negativity over closed sets to compute various system properties such as gain, regions of attraction, reachable sets and nonlinear Hankel norm approximations. The methods we present are then illustrated on a nonlinear model of an F/A‐18 aircraft short‐period dynamics subject to uncertainty in the aerodynamic coefficients. Copyright © 2012 John Wiley &amp; Sons, Ltd.

A note on the topological classification of complex polynomial differential equations with only centre singularities

We give the exact number of topologically different separatrix configurations on the Poincaré disc for complex polynomial differential equations with only centre singularities.

A suspension of the Hénon map by periodic orbits

We create polynomial differential equations for a suspension of the Hénon map embedded in R3. By globalizing the local tangent vectors to suspended periodic orbits of the Hénon map, we are able to find approximate autonomous differential equations for that geometric suspension. Using as few as two suspended periodic orbits, we can generate a robust three dimensional attractor whose Poincaré map has very nearly the dynamics of the original Hénon map on the attractor.

Hybrid models of the cell cycle molecular machinery

Piecewise smooth hybrid systems, involving continuous and discrete variables, are suitable models for describing the multiscale regulatory machinery of the biological cells. In hybrid models, the discrete variables can switch on and off some molecular interactions, simulating cell progression through a series of functioning modes. The advancement through the cell cycle is the archetype of such an organized sequence of events. We present an approach, inspired from tropical geometry ideas, allowing to reduce, hybridize and analyse cell cycle models consisting of polynomial or rational ordinary differential equations.

Robustness-based Model Validation of an Apoptosis Signalling Network Model

Models of intracellular biochemical reaction networks are difficult to parameterise due to the low number of quantitative time series experimental values. Therefore, model validation or invalidation plays an important role, as it allows to check qualitatively whether a model structure is suited or not to reproduce qualitatively the experimental findings.This paper analyses the robustness of an experimentally validated polynomial differential equation model of TNF-induced pro-and anti-apoptotic signalling. The bistability of the median model is shown robust to large single parameter variations. Only two parameters (XIAP and Procaspase-3 production rates) are shown to be fragile, in particular when changed simultaneously. Therefore, the model seems valid from the point of view of robustness analysis of the bistability.Many biological experiments quantify average concentrations or the percentage of viable cells, while other methods such as microscopy-based experiments observe single cells. The integration of single cell and cell population behaviour of TNF-induced pro- and anti-apoptotic signalling has been achieved via a cell ensemble model, whose robustness is also analysed here. We show that within the cell population there are cells with not only quantitative differences, but also qualitative ones. In particular, all cells are not bistable. The degree of robustness applicable for the median cell is expanded to combine mono- and bistable models. This measure, applied solely to the two-dimensional subspace of fragile parameters, is shown to correlate well with the time of death. While robustness of bistability can serve for model validation of the median cell model, it cannot for the model of the cell population.

Stability analysis of networked control systems: A sum of squares approach

This paper presents a sum of squares (SOS) approach to the stability analysis of networked control systems (NCSs) incorporating bounded time-varying delays, bounded time-varying transmission intervals and a shared communication medium. A shared communication medium imposes that per transmission only one node, which consists of several actuators or sensors, can access the network and transmit its corresponding data. Which node obtains access is determined by a network protocol. We will provide mathematical models that describe these NCSs and transform them into suitable hybrid systems formulations. Based on these hybrid systems formulations we construct Lyapunov functions using SOS techniques that can be solved using LMI-based computations. This leads to several beneficial features: (i) we can deal with plants and controllers which are described by nonlinear (piecewise) polynomial differential equations, (ii) we can allow for non-zero lower bounds on the delays and transmission intervals in contrast with various existing approaches, (iii) we allow more flexibility in the Lyapunov functions thereby obtaining less conservative estimates of the maximal allowable transmission intervals (MATI) and maximal allowable delay (MAD), and finally (iv) it provides an automated method to address stability analysis problems in NCS. Several numerical examples illustrate the strengths of our approach.

On Liouvillian integrability of the first–order polynomial ordinary differential equations

Recently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Giné and Llibre (2012) [8]. In this note, we prove that, if a complex differential equation of the form y′=a0(x)+a1(x)y+⋯+an(x)yn, with ai(x) polynomials for i=0,1,…,n, an(x)≠0, and n≥2, has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n=1, i.e., for linear polynomial differential equations.

A differential evolutionary method for solving a class of differential equations numerically

A method for solving differential equations using the differential evolutionary algorithm (DE) is presented in this contribution. In the case of differential equation is a polynomial function of its variables, the algorithm makes possible solution vectors evolve toward the best solution of the differential equation based on certain criteria, such as least squares or another error measure. In the case of non polynomial differential equations, the DE is applied in the context of variational methods. Numerical results for solving a segment of the Lorenz and van der Pol differential equations are provided.

Periodic orbits for perturbed non-autonomous differential equations

We consider non-autonomous differential equations, on the cylinder ( t , r ) ∈ S 1 × R d , given by d r / d t = f ( t , r , ε ) and having an open continuum of periodic solutions when ε = 0 . From the study of the variational equations of low order we obtain successive functions such that the simple zeroes of the first one that is not identically zero control the periodic orbits that persist for the unperturbed equation. We apply these results to several families of differential equations with d = 1 , 2 , 3 . They include some autonomous polynomial differential equations and some Abel type non-autonomous differential equations.

Inverse Problems in Darboux’ Theory of Integrability

The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A number of relevant examples and applications is included.