THE NON-DICRITICAL ORDER AND ATTRACTING DOMAINS OF HOLOMORPHIC MAPS TANGENT TO THE IDENTITY

2014 ◽  
Vol 25 (01) ◽  
pp. 1450003 ◽  
Author(s):  
FENG RONG
Keyword(s):  
Fixed Point ◽  
Duke Math ◽  
Holomorphic Maps ◽  
Characteristic Direction ◽  
Local Dynamics ◽  
Analytic Transformation ◽  
Tangent To The Identity

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp, 0) tangent to the identity, Duke Math. J.92 (1998) 403–428], n - 1 invariants αj, 1 ≤ j ≤ n - 1, called the directors, were associated to [v] and it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Re αj = 0 for some j. With the help of a new invariant μ called the non-dicritical order, we show that f has an attracting domain at p tangent to [v] if μ ≥ 1. We also study the "spiral domains" when μ = 0. For n = 2, we show that f has an attracting domain at p tangent to [v] if and only if either the director α > 0 or μ ≥ 1.

scholarly journals Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Q. Khan ◽  
M. B. Javaid
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Chaotic Behavior ◽  
Discrete Analogue ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Time Model ◽  
Local Dynamics

AbstractThe local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.

scholarly journals Random local complex dynamics

2018 ◽  
Vol 40 (8) ◽  
pp. 2156-2182
Author(s):  
LORENZO GUERINI ◽  
HAN PETERS
Keyword(s):  
Fixed Point ◽  
Probability Measure ◽  
Complex Dynamics ◽  
Linear Part ◽  
Interesting Case ◽  
Holomorphic Maps ◽  
Deterministic Case ◽  
Local Complex ◽  
The Family ◽  
The Stability

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper, we will consider the corresponding random setting: given a probability measure $\unicode[STIX]{x1D708}$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_{n}\circ \cdots \circ f_{1}$, where each $f_{i}$ is chosen independently with probability $\unicode[STIX]{x1D708}$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov exponents vanish, in which case stability implies simultaneous linearizability of all germs in $\text{supp}(\unicode[STIX]{x1D708})$.

Towards a semi-local study of parabolic invariant curves for fibered holomorphic maps

2011 ◽  
Vol 32 (6) ◽  
pp. 2056-2070 ◽  
Author(s):  
MARIO PONCE
Keyword(s):  
Rotation Number ◽  
Invariant Curve ◽  
Invariant Curves ◽  
Holomorphic Maps ◽  
Main Ingredient ◽  
Local Study ◽  
Local Dynamics

AbstractWe introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibered holomorphic maps. As in the classical non-fibered case, we show that petals are the main ingredient. Nevertheless, one expects that the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypotheses on the arithmetic and smoothness of the map, we present a theorem that allows a characterization of the local dynamics around a parabolic invariant curve.

scholarly journals On Ecalle-Hakim 's theorems in holomorphic dynamics

2014 ◽  
Author(s):  
Marco Arizzi ◽  
Jasmin Raissy
Keyword(s):  
Fixed Point ◽  
Simple Form ◽  
Linear Term ◽  
Holomorphic Dynamics ◽  
Holomorphic Map ◽  
Tangent To The Identity

This chapter provides detailed proofs for the results by M. Hakim regarding the dynamics of germs of biholomorphisms tangent to the identity of order k + 1 ≥ 2 and fixing the origin. One of the main questions in the study of local discrete holomorphic dynamics, i.e., in the study of the iterates of a germ of a holomorphic map of ℂᵖ at a fixed point, which can be assumed to be the origin, is when it is possible to holomorphically conjugate it to a “simple” form, possibly its linear term. It turns out that the answer to this question strongly depends on the arithmetical properties of the eigenvalues of the linear term of the germ.

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