Colin Christopher

Invariant algebraic curves and conditions for a centre

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.

Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in \Bbb C 2

AbstractIn this paper, we consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio 1 : −λ with λ ∈

Algorithmic Derivation Of Centre Conditions

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Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces

. We address the questions of orbital normalizability, normalizability (i.e., convergence of the normalizing transformation), integrability (i.e., orbital linearizability), and linearizability of the system. As for the experimental part of our study, we specialize to quadratic systems and study the values of λ for which these notions are distinct. For this purpose we give several tools for demonstrating normalizability, integrability, and linearizability.Our main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which we have integrable or linearizable saddles as λ and the other parameters of the system vary. Many of the results are valid in the larger context of polynomial or analytic vector fields. We explain several features of the bifurcation diagram, namely, the existence of a continuous skeleton of integrable (linearizable) systems with sequences of holes filled with orbitally normalizable (normalizable) systems and strata finishing at a particular value of λ. In particular, we introduce the Ecalle-Voronin invariants of analytic classifcation of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.

POLYNOMIAL SYSTEMS: A LOWER BOUND FOR THE HILBERT NUMBERS

Two algorithms are given for finding conditions for a critical point to be an isochronous center. The first is based on a systematic search for a transformation to the simple harmonic oscillator and as an example is used to find conditions for an isochronous center in the Kukles system; the second algorithm is specific to systems with homogeneous nonlinearities and is based on a connection with an Abel differential equation. General properties of systems with isochronous centers are also considered and results on Lienard and Hamiltonian systems are deduced; a close connection is demonstrated between isochronous centers and complex centers.

Darboux integrability and invariant algebraic curves for planar polynomial systems

The conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications. We demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres. We survey recent developments and illustrate our approach by means of examples.

Algebraic invariant curves and the integrability of polynomial systems

We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

Estimating Limit Cycle Bifurcations from Centers

We consider the second-order equation where f and g are polynomials with deg f,deg g n. Our interest is in the maximum number of isolated periodic solutions which can bifurcate from the steady state solution x = 0. Alternatively, this is equivalent to seeking the maximum number of limit cycles which can bifurcate from the origin for the Li?nard system, Assuming the origin is not a centre, we show that if either f or g are quadratic, then this number is . If f or g are cubic we show that this number is , for all 1n50. The results also hold for generalized Li?nard systems.