Planar cubic polynomial differential systems with the maximum number of invariant straight lines
نویسنده
چکیده
We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every class we provide the configuration of its invariant straight lines in the Poincaré disc. Moreover, every class is characterized by a set of affine invariant conditions.
منابع مشابه
Quadratic Systems with Invariant Straight Lines of Total Multiplicity Two Having Darboux Invariants
In this paper we present the global phase portraits in the Poincaré disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants.
متن کاملOn the Number of Limit Cycles in Discontinuous Piecewise Linear Differential Systems with Two Pieces Separated by a Straight Line
In this paper we study the maximum number N of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies 2 ≤ N ≤ 4 if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity.
متن کاملPlanar quadratic differential systems with invariant straight lines of the total multiplicity 4 Dana
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrica...
متن کاملLarge amplitude oscillations for a class of symmetric polynomial differential systems in R 3
In this paper we study a class of symmetric polynomial differential systems in R3, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n ∈ N there is εn > 0 such that for 0 < ε < εn the system has at...
متن کاملIntegrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity
In this article we prove that all real quadratic differential systems dx dt = p(x, y), dy dt = q(x, y) with gcd(p, q) = 1, having invariant lines of total multiplicity at least five and a finite set of singularities at infinity, are Darboux integrable having integrating factors whose inverses are polynomials over R. We also classify these systems under two equivalence relations: 1) topological ...
متن کامل