Applying the Poincaré-Birkhoff theorem to antiperiodic problems

Multiple periodic solutions of scalar second order differential equations

For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solved

We define the concept of discrete weighted pseudo-S-asymptotically periodic function and prove some basic results including composition theorem. We investigate the existence, and uniqueness of discrete weighted pseudo-S-asymptotically periodic solution to nonautonomous semilinear difference equations. Furthermore, an application to scalar second order difference equations is given. The working tools are based on the exponential dichotomy theory and fixed point theorem.

The paper deals with classical polynomial Liénard equations, i.e. planar vector fields associated to scalar second order differential equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x plus f left-parenthesis x right-parenthesis x Superscript prime Baseline plus x equals 0"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x+f(x)x’+ x=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial. We prove that for a well-chosen polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6 comma"> <mml:semantics> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">6,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the equation exhibits <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> limit cycles. It induces that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exist polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the related equations exhibit more than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Liénard equations as above, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n comma"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2n,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the maximum number of limit cycles is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n period"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classical polynomial Liénard equations. More precisely we find our example inside a family of second order differential equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon x plus f Subscript mu Baseline left-parenthesis x right-parenthesis x Superscript prime Baseline plus x equals 0 period"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi> μ </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon x+f_\mu (x)x’+x=0.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript mu"> <mml:semantics> <mml:msub>

In this chapter we are concerned with the oscillation of nonlinear two-dimensional differential systems and second order vector-matrix differential equations. In Section 7.1 we shall present criteria for the oscillation of nonlinear two-dimensional differential systems. This includes the superlinear, linear, and sublinear cases. Section 7.2 deals with the oscillation of linear second order differential systems. Here, first the system considered will be reduced to a certain scalar Riccati inequality, so that the known results from the literature can be applied to obtain oscillation criteria. Then, we shall employ the notation and definitions of Section 6.1 to present some general results. Finally, we shall use Riccati and variational techniques which involve assumptions on the behavior of the eigenvalues of the coefficient matrix (or of its integral) to present a number of sufficient conditions which guarantee the oscillation of linear second order systems. In Section 7.3 we shall discuss the oscillation of nonlinear second order differential systems with functionally commutative matrix coefficients. Here, we shall show that the oscillation theory of such systems can be effectively reduced to the study of diagonal systems of scalar second order differential equations. In Section 7.4 we shall prove some comparison theorems of Hille-Wintner type for second order operator-valued linear differential equations. In Section 7.5 some oscillation results for second order differential systems with a forcing term are given.

In this paper, Maxwell's equations are transformed into a set of uncoupled, scalar first order differential equations. The spatial derivative operator in the transformed differential equations is a fractional Laplacian. Numerical method for solving the equations are investigated

We prove the existence of periodic solutions for perturbations of some autonomous second order nonlinear differential equations by the use of the Poincaré- Birkhoff fixed point theorem.

Two-point boundary value problems for planar systems: A lower and upper solutions approach

Using a recent modified version of the Poincaré-Birkhoff fixed point theorem [19], we study the existence of one-signed T-periodic solutions and sign-changing subharmonic solutions to the second order scalar ODE u′′ + f (t, u) = 0, being f : ℝ × ℝ → ℝ a continuous function T-periodic in the first variable and such that f (t, 0) ≡ 0. Partial extensions of the results to a general planar Hamiltonian systems are given, as well.

A technique is developed for explicitly eliminating the constraint torques from a canonical system of n vector equations for the attitude dynamics of a satellite consisting of n arbitrarily interconnected rigid bodies. This elimination reduces the number of scalar second order differential equations from 3n to r, the number of degrees of rotational freedom of the satellite. At the same time, the number of dependent variables in these equations is reduced from the full set of 3n angular velocity components to just 3 such components for one body, together with r — 3 relative angular rates. This elimination and reduction saves computer time when the equations are integrated, and also avoids a possible build-up of numerical errors violating the constraints. The final equations resemble those obtained from a Lagrangian approach, but are simpler to derive and to modify to account for additional effects.

A technique is developed for explicitly eliminating the constraint torques from a canonical system of n vector equations for the attitude dynamics of a satellite consisting of n arbitrarily interconnected rigid bodies. This elimination reduces the number of scalar second order differential equations from 3n to r, the number of degrees of rotational freedom of the satellite. At the same time, the number of dependent variables in these equations is reduced from the full set of 3n angular velocity components to just 3 such components for one body, together with r — 3 relative angular rates. This elimination and reduction saves computer time when the equations are integrated, and also avoids a possible build-up of numerical errors violating the constraints. The final equations resemble those obtained from a Lagrangian approach, but are simpler to derive and to modify to account for additional effects.

Adaptive algorithms based on exact difference schemes for nonlinear BVPs on the half-axis

Assuming only asymptotic conditions on the potential function, we prove the existence of periodic solutions for equations whose nonlinearity stays below the first curve of Fučik’s spectrum.

Assuming only asymptotic conditions on the potential function, we prove the existence of periodic solutions for equations whose nonlinearity stays below the first curve of Fučik’s spectrum.

Radially symmetric systems with a singularity and asymptotically linear growth