Related papers: Higher bifurcations for polynomial skew-products
We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain $\Omega$ in $\mathbb R^m$ admit a parameterization by functions of bounded variation uniformly with respect to the…
We consider skew-products with concave interval fiber maps over a certain subshift obtained as the projection of orbits staying in a given region. It generates a new type of (essentially) coded shift. The fiber maps have expanding and…
Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known…
We present a family of birational transformations in $ CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of…
This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the…
We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than $1$ but…
This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of…
The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association…
By a nondegenerate $k$-parameterized family $K$ of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on $K$ system equals to $k.$ Bifurcation of asymptotically stable…
Stationary bifurcations in several nonlinear models of fluid conveying pipes fixed at both ends are analyzed with the use of Lyapunov-Schmidt reduction and singularity theory. Influence of gravitational force, curvature and vertical elastic…
Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding.…
We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…
We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves…
The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $\zeta_\lambda(z)=\lambda \frac{z}{z+1} e^{-z}$, $\lambda >0$, $z\in \mathbb{C}$ is investigated in the present paper. It is…
A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this…
We introduce a renormalization model which explains how the behavior of a discrete-time continuous dynamical system changes as the dimension of the system varies. The model applies to some two-dimensional systems, including H\'enon and Lozi…
We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values…
We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, \[ u_t+u u_x=\Lambda^{r}u-\varepsilon \Lambda^{s}u,\qquad x\in\mathbb T, \] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global…
This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic…
We present a new approach for estimating the set of bifurcation values at infinity. This yields a significant shrinking of the number of coefficients in the recent algorithm introduced by Jelonek and Kurdyka for reaching critical values at…