Related papers: On Planar Holomorphic Systems

Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide…

In this paper we obtain the global dynamics and phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous systems. We first prove that all planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial…

In this paper we provide a new method to study global dynamics of planar quasi--homogeneous differential systems. We first prove that all planar quasi--homogeneous polynomial differential systems can be translated into homogeneous…

This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the…

We classify the global dynamics of the five-parameter family of planar Kolmogorov systems \begin{equation*} \begin{split} \dot{y}&=y \left( b_0+ b_1 y z + b_2 y + b_3 z\right), \dot{z}&=z\left( c_0 + b_1 y z + b_2 y + b_3 z\right),…

If f: C -> P^n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation t(z)=z+c, then f is periodic with period c. This result,…

This paper deals with random dynamical systems of polynomial automorphisms (complex generalized H\'{e}non maps and their conjugate maps) of $\Bbb{C}^{2}.$ We show that a generic random dynamical system of polynomial automorphisms has ``mean…

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…

We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…

Generalized Heisenberg algebras $\H(f)$ for any polynomial $f(h)\in\C[h]$ have been used to explain various physical systems and many physical phenomena for the last 20 years. In this paper, we first obtain the center of $\H(f)$, and the…

This paper considers some work done by the author and Catlin [CD1,CD2,CD3] concerning positivity conditions for bihomogeneous polynomials and metrics on bundles over certain complex manifolds. It presents a simpler proof of a special case…

The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and…

This is a full study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a homogeneous polynomial of arbitrary degree $n>1$. It extends previous work by other…

For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting…

In polarization optics, various topological constructs, namely Poincar\'e spheres of different orders, are used to represent uniform and structured polarization distributions. Similarly, there are also structured polarization optical…

We prove that the graph of a continuous function $f$, defined on a domain of ${\mathbb C}^n$, is pluripolar if and only if $f$ is holomorphic.

We study piecewise-smooth systems with three zones, $\dot{z} = f_i(z)$, $i = 1,2,3,$ whose discontinuity set $\Sigma$ consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each…

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite…

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…