Related papers: Random local complex dynamics
We study robust properties of zero sets of continuous maps $f:X\to\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The…
We study corank one $A$-finite germs $f:(\mathbb{R}^n,0)\rightarrow (\mathbb{R}^{n+1},0)$ and their complexifications. More precisely, we study when these germs provide good real pictures of the complex germs, i.e., when there is a real…
Understanding which system structure can sustain stable dynamics is a fundamental step in the design and analysis of large scale dynamical systems. Towards this goal, we investigate here the structural stability of systems with a random…
Relationship for dynamical properties in the vicinity of fixed points between two-dimensional continuous and its positivity-preserving discretized dynamical systems is studied. Based on linear stability analysis, we reveal the conditions…
The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map $f$ of the unit disc $\mathbb{D}$. Far less well understood are…
We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.
We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ at a given parameter can be followed holomorphically with the…
The image of a holomorphic map germ is not necessarily locally open, and it is not always well-defined as a set germ. We find the structure of what becomes the image of a map germ when the target is a surface. We encode it as a decorated…
Given any positive sequence (\{c_n\}_{n \in {\Bbb N}}), we construct orientation preserving homeomorphisms (f:{\Bbb R}^3 \to {\Bbb R}^3) such that (Fix(f)=Per(f)=\{0\}), (0) is Lyapunov stable and (\limsup \frac{|i(f^m, 0)|}{c_m}= \infty).…
The local dynamics around a fixed point has been extensively studied for germs of one and several complex variables. In one dimension, there exist a complete picture of the trajectory of the orbits on a whole neighborhood of the fixed…
Jamming is a phenomenon shared by a wide variety of systems, such as granular materials, foams, and glasses in their high density regime. This has motivated the development of a theoretical framework capable of explaining many of their…
We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects the…
We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we…
We study here the dynamics (and stability) of Probabilistic Population Protocols, via the differential equations approach. We provide a quite general model and we show that it includes the model of Angluin et. al. in the case of very large…
We study discrete-time random dynamical systems where each fibre map is an orientation-preserving homeomorphism of the circle. We prove that the existence of a random periodic cycle with period at least two implies that the random rotation…
The environment in which a population evolves can have a crucial impact on selection. We study evolutionary dynamics in finite populations of fixed size in a changing environment. The population dynamics are driven by birth and death…
A coupled computational approach to simultaneously learn a vector field and the region of attraction of an equilibrium point from generated trajectories of the system is proposed. The nonlinear identification leverages the local stability…
We investigate stability of a solution of a hybrid system in the sense that the graphs of solutions from nearby initial conditions remain close and tend towards the graph of the given solution. In this manner, a small continuous-time…
We study the spectral properties of sparse random graphs with different topologies and type of interactions, and their implications on the stability of complex systems, with particular attention to ecosystems. Specifically, we focus on the…
A particular type of random dynamical processes is considered, in which the stochasticity is introduced through randomly fluctuating parameters. A method of local multipliers is developed for treating the local stability of such dynamical…