Singularity Structure of Third-Order Dynamical Systems. II

The singularity structure of the solutions of a general third-order system, with polynomial right-hand sides of degree less than or equal to two, is studied about a movable singular point. An algorithm for transforming the given third-order system to a third-order Briot–Bouquet system is presented. The dominant behavior of a solution of the given system near a movable singularity is used to construct a transformation that changes the given system directly to a third-order Briot–Bouquet system. The results of Horn for the third-order Briot–Bouquet system are exploited to give the complete form of the series solutions of the given third-order system; convergence of these series in a deleted neighborhood of the singularity is ensured. This algorithm is used to study the singularity structure of the solutions of the Lorenz system, the Rikitake system, the three-wave interaction problem, the Rabinovich system, the Lotka–Volterra system, and the May–Leonard system for different sets of parameter values. The proposed approach goes far beyond the ARS algorithm.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

Nonlocal representation of the sl(2, R) algebra for the Chazy equation

A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.     

Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painlevé-type differential equations of the second order and second degree that are built upon, in a natural well-defined sense, the "sn-log" equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling "exotic" Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé-type equations contains three, and only three, equations, which we denote SD-326-I, SD-326-II, and SD-326-III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn-log equation. The second and third are new, the third being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second-degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.     

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.     

A simple adaptive control for full and reduced-order synchronization of uncertain time-varying chaotic systems

In this paper, a simple adaptive feedback control is proposed for full and reduced-order synchronization of time-varying and strictly uncertain chaotic systems. Our method uses only one feedback gain with parameter adaptation law and converges very fast even in the presence of noise. For full synchronization, a drive-response system consisting of two second-order identical parametrically excited oscillators achieve global synchronization; while for reduced-order synchronization, the dynamical evolution of a second-order parametrically driven oscillator is synchronized with the projection of a third-order time-varying chaotic system. The effectiveness of our approach is demonstrated using numerical simulations.     

Localization of invariant compact sets of nonautonomous systems

We generalize the localization method for invariant compact sets of an autonomous dynamical system to the case of a nonautonomous system of differential equations. By using this method, we solve the localization problem for the Vallis third-order dynamical system governing some processes in atmosphere dynamics over the Pacific Ocean. For this system, we construct a one-parameter family of localizing sets bounded by second-order surfaces and find the intersection of all sets of the family.     

Stationary Schr?dinger equation in nonrelativistic quantum mechanics and the functional integral

We formulate a method for representing solutions of homogeneous second-order equations in the form of a functional integral or path integral. As an example, we derive solutions of second-order equations with constant coefficients and a linear potential. The method can be used to find general solutions of the stationary Schr?dinger equation. We show how to find the spectrum and eigenfunctions of the quantum oscillator equation. We obtain a solution of the stationary Schr?dinger equation in the semiclassical approximation, without a singularity at the turning point. In that approximation, we find the coefficient of transmission through a potential barrier. We obtain a representation for the elastic potential scattering amplitude in the form of a functional integral.     

Dynamics of a two-layer fluid sloshing problem

A second-order ordinary differential equation, which is a reducedform of the periodically forced extended Korteweg–de Vries(eKdV) equation, is derived in the physical context of sloshinga two-layer fluid tank. In the limit of small dispersion, numericalevidence is given of multiple periodic solutions displayingfast oscillations superimposed on slow periodic waves and ahigher-order Melnikov method is then used to verify the existenceof such solutions. The dynamical behaviour of a similar equationwith more general coefficients is also examined, demonstratingthe existence of periodic and chaotic behaviour. We highlightnew aspects which arise due to the presence of mixed nonlinearity.     

Periodic bouncing solutions of the Lazer–Solimini equation with weak repulsive singularity

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem.     

Algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom

We study the Zeeman-Stark effect for the hydrogen atom in crossed homogeneous electric and magnetic fields. A nonhomogeneous perturbing potential can also be present. If the crossed fields satisfy some resonance relation, then the degeneration in the resonance spectral cluster is removed only in the second-order term of the perturbation theory. The averaged Hamiltonian in this cluster is expressed in terms of generators of some dynamical algebra with polynomial commutation relations; the structure of these relations is determined by a pair of coprime integers contained in the resonance ratio. We construct the irreducible hypergeometric representations of this algebra. The averaged spectral problem in the irreducible representation is reduced to a second-or third-order ordinary differential equation whose solutions are model polynomials. The asymptotic behavior of the solution of the original problem concerning the Zeeman-Stark effect in the resonance cluster is constructed using the coherent states of the dynamical algebra. We also describe the asymptotic behavior of the spectrum in nonresonance clusters, where the degeneration is already removed in the first-order term of the perturbation theory.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 530–555, March, 2005.     

Classification and refined singularity of positive solutions for nonlinear Maxwell equations arising in mesoscopic electromagnetism

This paper is a continuation of [13], where we studied the existence and other analytic properties of positive radial solutions for a system of nonlinear Maxwell equations in the plane R²${\mathbb{R}}^2$, which arises in the modelling of mesoscopic scale electromagnetic phenomena. In this paper we derive local estimates of singular positive solutions, based on which a classification theorem of general positive solutions is established. The refined singularity of general positive solutions is also investigated by employing the theory of infinite dimensional dynamical systems.     

Tendril Perversion in Intrinsically Curved Rods

Summary. A straight elastic rod with intrinsic curvature under varying tension can undergo an instability and bifurcate to a filament made out of two helices with opposite handedness. This inversion of handedness, known as perversion, appears in a wide range of biological and physical systems and is investigated here within the framework of thin elastic rods described by the static Kirchhoff equations. In this context, a perversion is represented by a heteroclinic orbit joining asymptotically two fixed points representing helices with opposite torsion. A center manifold reduction and a normal form transformation for a triple zero eigenvalue reduce the dynamics to a third-order reversible dynamical system. The analysis of this reduced system reveals that the heteroclinic connection representing the physical solution results from the collapse of pairs of symmetric homoclinic orbits. Results of the normal form calculation are compared with numerical solutions obtained by continuation methods. The possibility of self-contact and the elastic characteristics of the perverted rod are also studied.     

On a generalization of the Chazy equation with a movable singular line

We generalize a third-order Chazy equation with a movable singular line, which has only negative resonances. For differential equations of order 2n+1 with resonances −1,−2, …, −(2n + 1), we study the convergence of the series representing their solutions, the existence of rational solutions, the invariance of these equations under certain transformations, and the existence of three-parameter solutions with a movable singular line.     

Oscillatory Singularities in Bianchi Models with Magnetic Fields

An idea which has been around in general relativity for more than 40  years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time, progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to cases where the Einstein equations are coupled to matter fields, focussing on the example of a dynamical system arising from the Einstein–Maxwell equations with symmetry of Bianchi type VI₀. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration than in the vacuum case. The difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.     

Positive solutions for third-order variable-coefficient nonlinear equation with weak and strong singularities

By application of Green's function and a fixed-point theorem, i.e. Leray–Schauder alternative principle, we establish some new existence results of positive periodic solutions for nonlinear third-order singular equation with variable-coefficient, these results can be applied to study the case of a strong singularity as well as the case of a weak singularity.     

A Landesman-Lazer type condition for second-order differential equations with a singularity at resonance

In this paper, we are concerned with the existence of positive periodic solutions for second-order differential equations with a singularity at resonance. A Landesman-Lazer type condition is given to obtain the existence of positive periodic solutions using phase-plane analysis method and topological degree method.     

Solitary wave solution for inhomogeneous nonlinear Schrödinger system with loss/gain

This paper deals with the propagation of solitons in real fibres, governed by the system of inhomogeneous nonlinear Schrödinger (INLS) equations. The Painlevé singularity structure analysis is utilized to check for the integrability of the system and from the analysis, the system is found to admit soliton-type lossless wave propagation. The system is transformed to its homogeneous counterpart using a suitable variable transformation and the soliton solutions are obtained through Bäcklund transformation after constructing the explicit Lax pair for the system. The one-soliton solutions are plotted for different choices of inhomogeneity parameters and the evolutionary characteristics of the solutions are analyzed.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.