Order Autonomous Ordinary Differential Equation Research Articles

Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x_0,y_0) in mathbb {C}^2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x_0)=y_0.

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Probabilistic evolution approach is a newly developed theory which may be utilized for the solution of ordinary differential equations. The approach may directly be applied for initial value problems of explicit first order autonomous ordinary differential equation sets with analytic right hand side functions. Analyticity plays an important role since it facilitates the expansion into direct power series which is the key element of the approach. Direct power series appear not only in all applications of probabilistic evolution but also show themselves as a promising tool for novel approximation methods. In this work, similarities and differences between Taylor series and direct power series are rigorously studied. Arbitrariness in transposed vector coefficients of direct power series is detailed. Equipartition theorem of direct power series is conjectured and proven in order to obtain unique transposed vector coefficients.

On the similarity solutions for a steady MHD equation

In this paper, we investigate the similarity solutions for the steady laminar incompressible boundary layer equations governing the magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies. This leads to the study of a boundary value problem involving a third order autonomous ordinary differential equation. Our main results are the existence, uniqueness and non-existence for concave or convex solutions.

Combined nonstandard numerical methods for ODEs with polynomial right-hand sides

In this paper we formulate conditions that guarantee second-order accuracy of the one-step nonstandard finite difference methods for general first order autonomous ordinary differential equations. We also develop a new class of nonstandard finite-difference methods for differential equations of the form d x / d t = polynomial ( x ) , based on earlier results obtained in reference [H.V. Kojouharov, B.M. Chen, Nonstandard Eulerian–Lagrangian methods for multi-dimensional reactive transport problems, Appl. Numer. Math. 49 (2) (2004) 225–243]. Error analysis and numerical examples that demonstrate the performance of the proposed new method are also provided.

Analytic solitary waves of nonintegrable equations

A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ordinary differential equation (ODE) satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg–Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

On Resonant Differential Equations with Unbounded Non-Linearities

We present a method to study asymptotically linear degenerate problems with sublinear unbounded non-linearities. The method is based on the uniform convergence to zero of projections of non-linearity increments onto some finite-dimensional spaces. Such convergence was used for the analysis of resonant equations with bounded non-linearities by many authors. The unboundedness of nonlinear terms complicates essentially the analysis of most problems: existence results, approximate methods, systems with parameters, stability, dissipativity, etc. In this paper we present statements on projection convergence for unbounded non-linearities and apply them to various resonant asymptotically linear problems: existence of forced periodic oscillations and unbounded sequences of such oscillations, existence of unbounded solutions, sharp analysis of integral equations with simple degeneration of the linear part (a scalar two-point boundary value problem is considered as an example), existence of non-trivial cycles for higher order autonomous ordinary differential equations, and Hopf bifurcations at infinity.

A parameter study of epitaxial crystal growth

A third order autonomous ordinary differential equation is studied that is derived from a mathematical model of epitaxial crystal growth on misoriented crystal substrates. The solutions of the ODE correspond to the traveling wave solutions of a nonlinear partial differential equation which is related to the Kuramoto–Sivashinsky equation. The fixed points, the periodic solutions, and the heteroclinic orbits of the ODE are analysed, and stability results are given. A variety of nonlinear phenomena are observed, including Gavrilov–Guckenheimer bifurcations, homoclinic bifurcations, and a cascade of period doublings.

Pretzel Orbits in Simulations of Epitaxial Crystal Growth

A third order autonomous ordinary differential equation is studied that describes stationary solutions of a nonlinear partial differential equation. The PDE models the growth of an epitaxial film on misoriented crystal substrates and is similar to the Kuramoto–Sivashinsky equation, but contains an additional nonlinear term. The equilibria, the periodic solutions, and the heteroclinic orbits of the ODE are analyzed, and stability results are given. Parameter regions are identified where the equilibria and the periodic solutions are unstable, but other bounded solutions exist. Their phase portrait is a double focus ("pretzel") that connects the stable and the unstable manifolds of the equilibria.

Periodic solution of a second order, autonomous, nonlinear system

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.

Cluster formation in a symmetrical network: a dynamical system for the description of the suppression among non-immune T lymphocytes and its application to the effects of immunization

A mathematical model has been developed for the description of the suppressive regulation between polyclonally activated normal and immune T cells. The model assumes reversible cell-cell interactions to interpret results from limiting dilution experiments performed to determine the frequencies of precursor cells for antigen-specific T effector lymphocytes and to analyse mechanisms regulating the maturation of precursor into effector T cells. In particular, the model deals with the changes induced in the T lymphocytes population following immunization with antigens. In these limiting dilution experiments, T cells are placed in cultures at varying cell numbers with all other essential culture constituents kept in excess. After polyclonal activation of the T cells in culture they are supplied with growth and maturation factors so that they form daughter clones of functionally active T effector cells. The typical result observed was that effector T cells develop in cultures at low cell input but that this development is totally suppressed at high cell numbers. This result suggested that, at high cell numbers, the effector T cells are exposed to a sufficient number of other T cells of appropriate specificity to permit suppressive interactions. Whereas this is the case for non-immune T cells, T cells after immunization develop into effector cells both at high as well as at low cell concentrations, though with efficiencies less than proportional to their number of precursors. Our mathematical model is made up of a set of first order autonomous ordinary differential equations in many variables permitting the calculations of numbers of free cells and of cells engaged in cellular clusters of varying sizes. Free cells can develop into effector cells whereas cells engaged in clusters cannot. We calculate the consequences of several reasonable hypotheses concerning the effects of immunization. We consider the possibility that immunization modifies the growth behavior of the antigen-specific cells to permit an increased or accelerated clonal expansion in culture. Alternatively, we consider the possibility that immunization changes the interaction strength between cells specific for the immunizing antigen and other cells. Thirdly, we have connected both behaviors by calculating the case of an inverse relationship between growth rates and intensities of interaction between cells. Our model has been inspired by the symmetrical network model and can be interpreted in this framework. It proposes that immune regulation is a consequence of idiotype-anti-idiotype interactions.(ABSTRACT TRUNCATED AT 400 WORDS)

Qualitative dynamics of a network model of regulation of the immune system: A rationale for the IgM to IgG switch

A mathematical model has been developed that simulates some of the main features of a network theory of regulation of the immune system. According to the network viewpoint, the V regions (idiotypes) on antibodies and lymphocytes are self-antigens, to which other lymphocytes of the system can respond specifically, just as they respond to foreign antigens. The resultant couplings between the lymphocytes are considered to be basic for the regulation of the system. The present mathematical model simulates the interactions between cells that recognize the antigen (“positive cells”), and “negative cells” that have receptors that specifically recognize the V regions of the positive cells. The mathematical model incorporates only the interactions that are postulated to be important in the four steady states of the theory, and includes neither the antigen nor any accessory (“A”) cells. The effects of both antigen-specific and anti-idiotypic T and B cells are included, as well as antigen-specific and anti-idiotypic T cell factors, and the two main classes of antibodies. The model is a first order autonomous ordinary differential equation in two variables. We describe a geometric technique that gives strong information on the model, without explicitly solving the ordinary differential equation. This technique proves to be powerful in permitting us to systematically scan the parameter space of the model. The detailed analysis leads to support for the idea that the model provides a rationale for the switch observed in the immune system from the production of one major class of antibody (IgM) to the other major class (IgG). The analysis also leads to a new, previously unsuspected possibility for the nature of the suppressed state within the context of the postulates of the symmetrical network theory.