Related papers: Using Symbolic Computation to Analyze Zero-Hopf Bi…
The double Hamiltonian Hopf bifurcation is studied, i.e. a generic two-parametric unfolding of a smooth Hamiltonian system with four degrees of freedom which has at the critical value of parameters the equilibrium with two pairs of double…
We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with independent coefficients. This problem reduces, by way of classical…
We discuss the role and merits of symmetry methods for the analysis of biological systems. In particular, we consider systems of first order ordinary differential equations and provide a comprehensive review of the geometrical foundations…
Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…
We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic…
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in time polynomial in both the encoding size of the system of equations and in log(1/\epsilon), where…
In a previous paper, the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with…
We analyze rate-dependent tipping in a fast/slow system with an equilibrium near the fold of a critical manifiold. We find a Hopf bifurcation as the rate parameter increases in the reduced co-moving system. This implies the growth of a…
In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit. In the linear case, we apply basic "symbolic" methods from linear…
We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D…
In this paper we analyze a generic dynamical system with $\mathbb{D}_2$ constructed via a Cayley graph. We study the Hopf bifurcation and find conditions for obtaining a unique branch of periodic solutions. Our main result comes from…
A bilevel program is an optimization problem whose constraints involve another optimization problem. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as…
We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari for interpolating polynomials over rings with characteristic zero,…
The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection…
We study multifractal decompositions based on Birkhoff averages for sequences of functions belonging to certain classes of symbolically continuous functions. We do this for an expanding interval map with countably many branches, which we…
Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial…
The orbits of the reversible differential system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, with $x,y \in R$ and $z\in R^d$, are periodic with the exception of the equilibrium points $(0,0, z)$. We compute the maximum number of limit cycles…
Zernike circular polynomials (ZCP) play a significant role in optics engineering. The symbolic expressions for ZCP are valuable for theoretic analysis and engineering designs. However, there are still two problems which remain open:…
We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the…
In this paper we describe how to improve the performance of the symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case. By…