Two-dimensional Autonomous System Research Articles

Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial

In this work we study isochronous centers of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We first found necessary conditions for such isochronous center in polar coordinates. Finally we give a proof of the isochronicity of these systems using different methods.

Liénard systems, limit cycles, and Melnikov theory

Li\'enard systems constitute a general class of two-dimensional autonomous systems, among which the van der Pol equation is found. Recently Giacomini and Neukirch [Phys. Rev. E 57, 3809 (1997)] introduced a sequence of polynomials whose roots are related to the number and location of limit cycles of Li\'enard systems. We show that in the limit of these sequences, the same information is given by a polynomial which Melnikov theory associates with a given Li\'enard system, and discuss the relationship existing among them.

Integrability of linear center perturbed by a fifth degree homogeneous polynomial

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrability of a linear center perturbed by fourth degree homogeneous polynomial

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

A class of integrable polynomial vector fields

We study the integrability of two-dimensional autonomous systems in the plane of the form $\dotx=-y+X_s(x,y)$, $\doty=x+Y_s(x,y)$, where X_s(x,y) and Y_s(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial

First integrals of autonomous systems of differential equations and the Prelle-Singer procedure

The Prelle-Singer procedure for determining elementary first integrals of two-dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed. The application of this procedure in several dynamical systems is reported.

Integrable systems in the plane with center type linear part

We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for thes

Stochastic resonance without external periodic force.

A model of a two-dimensional autonomous system subject to external noise is investigated. Without noise the system has a stable limit cycle in a certain region of control parameter. Various noise-induced effects have been found numerically, such as a noise-induced frequency shift in the presence of the deterministic limit cycle, and noise-induced coherent oscillations in the absence of the deterministic limit cycle. An interesting result is that the stochastic resonance phenomenon appears in a system without an external signal and when the asymptotic state of the deterministic system is stationary.

Limit cycles of periodically forced oscillations

The existence of limit cycles of two-dimensional autonomous integrable systems periodically perturbed by delta-peaks is discussed. For that purpose, Poincaré maps are introduced considering analytical solutions between successive pulses.

Some properties of two-dimensional iterative systems

No finite algorithm exists which determines whether a random two-dimensional autonomous regular iterative system can be separated.

Bounded quadratic systems in the plane

This paper contains a study of those two-dimensional autonomous systems with quadratic polynomial right-hand sides which have all of their trajectories bounded for t > 0. Such systems will be referred to as bounded quadratic systems. It follows from the PoincarbBendixson Theorem that any such system will have a rest point in the plane and by translating the origin to that rest point it will have the form

Bounds for the lengths and periods of closed orbits of two-dimensional autonomous systems of differential equations

Bounds for the lengths and periods of closed orbits of two-dimensional autonomous systems of differential equations

Dynamic snap-through of a simple viscoelastic truss

In order to understand the behavior of shallow structures in dynamic snap-through or buckling, a detailed study has been made for a plane, viscoelastic, three-hinged truss with a concentrated mass at the central hinge, and with a normal load of constant magnitude applied suddenly at this hinge. The dynamic buckling criterion is found to correspond to values of the parameters for which the solution goes into the saddle point of a two-dimensional autonomous system. It is shown that another dynamic buckling criterion, based upon the asymptotic behavior of solutions in time, can give incorrect results in certain cases. Two methods to compute buckling loads are investigated with the aid of a phase plane diagram and potential curves. Approximations to the buckling load, including an upper bound, are computed by means of an energy integral method. The exact buckling loads are computed by numerical integration of the governing differential equation.

Analogy of nonlinear systems to classical dynamics

The two-dimensional autonomous system in the general form is considered. The solution is in terms of the phase trajectories in the phase plane. In this paper, a force field over the phase plane is formulated such that the path of a particle in this force field coincides with the path of the trajectories, provided the initial condition is properly chosen. Hence, the trajectories can be obtained by solving the path of the particle subject to the formulated force field. It is further developed that, for any given system, it is theoretically possible to formulate a conservative force field to give the same property. Consequently, it is found out that the trajectories can be solved either analytically by a simple substitution or graphically by means of the family of trajectories which are orthogonal to the trajectories of the original system. Various examples including the generalized Van der Pol equation are given gs illustrations.

On the behavior of paths of the analytic two-dimensional autonomous system in a neighborhood of an isolated critical point (continued)

On the behavior of paths of the analytic two-dimensional autonomous system in a neighborhood of an isolated critical point (continued)

On the Behavior of Paths of the Analytic Two-dimensional Autonomous System in a Neighborhood of an Isolated Critical Point

On the Behavior of Paths of the Analytic Two-dimensional Autonomous System in a Neighborhood of an Isolated Critical Point