Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.     

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.     

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.     

Modal exact linearization of a class of second-order switched nonlinear systems

This paper considers the problem of stabilizing single-input affine switched nonlinear systems. The main idea is to transform a switched nonlinear system to an equivalent controllable switched linear system. First, we define the notion of modal state feedback linearization. Then, we develop a set of conditions for modal state feedback linearizability of a certain class of second order switched nonlinear systems. Considering two special structures, easily verifiable conditions are proposed for the existence of suitable state transformations for modal feedback linearization. The results are constructive. Finally, the method is illustrated with two examples, including a Continuous Stirred Tank Reactor (CSTR) to demonstrate the applicability of the proposed approach.     

<Emphasis Type="Italic">A</Emphasis>-Orbital Linearization of Affine Systems

The problem of transformation of an affine system into a linear controllable system is considered. For affine systems with a single control, the notion of A-orbital linearizability is introduced, which generalizes the notion (well known for affine systems) of orbital linearizability to the case where the control-dependent changes of independent variable are used. A necessary and sufficient condition for the A-orbital linearizability is proved, and an algorithm for determining linearizable transformations is proposed based on the construction of the derived series of the codistribution associated with the original system.     

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

Dynamical systems of translation groups on a torus

The criterion for linearizability is obtained by a property of denominator of rational numbers that are given by continued fractions. We can regard this criterion as a condition given by behaviors of orbits of dynamical systems of translation groups on a one-dimensional real torus. By using continued fractions, we obtain a criterion that is an analogous phenomenon in several complex variables that is determined by behaviors of dynamical systems of translation groups on a one-dimensional complex torus.     

On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

Generalized isochronous centers for complex systems

In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained.     

Approximately linear tracking control of nonlinear systems

An elegant approach to control nonlinear state-space systems is the exact input-to-state linearization, where a nonlinear change of coordinates combined with a nonlinear feedback law yields a linear controllable system. In this contribution, we treat the single-input case, where input-to-state linearizability is equivalent to flatness. Sufficient and necessary existence conditions are well-known, but quite restrictive. We propose the design of a tracking controller for flat single-input systems, where the explicit knowledge of the flat output is not required. Our approach is based on a series expansion of the tracking error along a given reference trajectory. The controller gain can even be computed for non-flat systems with regular controllability matrix. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometry of cascade feedback linearizable control systems

Cascade feedback linearization provides geometric insights on explicit integrability of nonlinear control systems with symmetry. A central piece of the theory requires that the partial contact curve reduction of the contact sub-connection be static feedback linearizable. This work establishes new necessary conditions on the equations of Lie type - in the abelian case - that arise in a contact sub-connection with the desired static feedback linearizability property via families of codimension one partial contact curves. Furthermore, an explicit class of contact sub-connections admitting static feedback linearizable contact curve reductions is presented, hinting at a possible classification of all such contact sub-connections. Key tools in proving, and stating, the main results of this paper are truncated versions of the total derivative and Euler operators. Additionally, the Battilotti-Califano system with three inputs is used as a clarifying example of both cascade feedback linearization and the new necessary conditions.     

Linearization of two-dimensional systems of ODEs without conditions on small denominators

We study the problem of linearizability for two-dimensional systems of ODEs in a neighborhood of the saddle type singular point with rationally incommensurable eigenvalues. It is shown that if the linearizing transformation is convergent in one of the variables, then it is absolutely convergent.     

Integrability and linearizability of the Lotka-Volterra systems

Integrability and linearizability of the Lotka-Volterra systems are studied. We prove sufficient conditions for integrable but not linearizable systems for any rational resonance ratio. We give new sufficient conditions for linearizable Lotka-Volterra systems. Sufficient conditions for integrable Lotka-Volterra systems with 3:−q resonance are given. In the particular cases of 3:−5 and 3:−4 resonances, necessary and sufficient conditions for integrable systems are given.     

Computer Algebra Algorithms for Control Related Tests of Implicit Dynamic Systems

This contribution is focused on control related tests for implicit dynamic systems, like accessibility, observability or input to output, input to state linearizability. Since the performance of these tests needs tedious symbolic calculations, computer algebra systems are the ideal tool to cope with this problem. Accessibility and observability are exemplarily used to present a new approach based on Lie groups. It is shown that non accessible or non observable systems admit Lie‐groups acting on their solutions such that distinguished parts remain unchanged. This fact allows us to apply this technique, as well as its realization by computer algebra algorithm, to several fundamental problems in control.     

Classification of discrete equations linearizable by point transformation on a square lattice

We provide a complete set of linearizability conditions for nonlinear partial difference equations defined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Möbious transformation.     

Integrability conditions for complex systems with homogeneous

In this paper we obtain&nbsp;conditions for the existence&nbsp; of a local analytic first integral for four eight-parameter families of&nbsp;quintic complex system. We also discuss computational difficulties arising in the study of the problem of integrability for these systems.     

Semi-Fredholm problems on periodic solutions of functional-differential equations of neutral type

We consider the problem on periodic solutions for linear systems of functionaldifferential equations of neutral type with periodic coefficients and periodic deviations of the argument. By reduction to associated functional equations, we derive necessary and sufficient conditions under which the problem on periodic solutions for such a system is Fredholm or semi-Fredholm.     

Linearizability conditions of time-reversible cubic systems

In this paper we present the necessary and sufficient conditions for linearizability of the planar time-reversible cubic complex system , . From these conditions, the necessary and sufficient conditions for the origin to be an isochronous center of the time-reversible cubic real system , can be obtained. Thus, the isochronous center problem of time-reversible cubic systems is solved completely.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.