Narrow Results

We investigate stable manifolds of hyperbolic holomorphic automorphisms. Our main result is that these can always be embedded as open subsets of complex Euclidean space.

In this paper, we study the parameter space of the quadratic polynomial family f_λ,μ(z, w) = (λz + w², μw + z²), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family f_λ,μ, we prove that the sum of the Lyapunov exponents is continuous.

Let M be a positive integer and let f be a holomorphic mapping from a ball Δⁿ = {x ∈ ℂⁿ;|x| < δ} into ℂⁿ such that the origin 0 is an isolated fixed point of both f and the M-th iteration f^M of f. Then one can define the number , which can be interpreted to be the number of periodic orbits of f with period M hidden at the fixed point 0.

For a 3 × 3 matrix A, of which the eigenvalues are all distinct primitive M-th roots of unity, we will give a sufficient and necessary condition for A such that for any holomorphic mapping f: Δ ³ → ℂ³ with f(0) = 0 and Df(0) = A, if 0 is an isolated fixed point of the M-th iteration f^M, then .

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.]. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.] also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. In [The reconstruction theorem for endomorphisms, Bull. Braz. Math. Soc. (N.S.) 33(2) (2002) 231–262.] Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the (2m + 1)th image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.

We study the global dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, $f$ admits two canonical probability measures ${\mu }^{+}$ and ${\mu }^{-}$ which are invariant by $f^{\ast }$ and $f_{\ast },$ respectively. If the critical values of $f$ (respectively, $f^{-1}$) are not periodic, the backward (respectively, forward) orbit of any point $a\in X$ equidistributes towards ${\mu }^{+}$ (respectively, ${\mu }^{-}$), uniformly in $a$ and exponentially fast.

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: “Given a generic polynomial$P$ of degree at least$2,$ there exist positive integers$N,M$ such that$\text{Re}(P^k(z))=\text{Re}(P^k(w))$ for all$0\le k\le N$ implies$P^M(z)=P^M(w)$”.

One aim of synthetic biology is to construct increasingly complex genetic networks from interconnected simpler ones to address challenges in medicine and biotechnology. However, as systems increase in size and complexity, emergent properties lead to unexpected and complex dynamics due to nonlinear and nonequilibrium properties from component interactions. We focus on four different studies of biological systems which exhibit complex and unexpected dynamics. Using simple synthetic genetic networks, small and large populations of phase-coupled quorum sensing repressilators, Goodwin oscillators, and bistable switches, we review how coupled and stochastic components can result in clustering, chaos, noise-induced coherence and speed-dependent decision making. A system of repressilators exhibits oscillations, limit cycles, steady states or chaos depending on the nature and strength of the coupling mechanism. In large repressilator networks, rich dynamics can also be exhibited, such as clustering and chaos. In populations of Goodwin oscillators, noise can induce coherent oscillations. In bistable systems, the speed with which incoming external signals reach steady state can bias the network towards particular attractors. These studies showcase the range of dynamical behavior that simple synthetic genetic networks can exhibit. In addition, they demonstrate the ability of mathematical modeling to analyze nonlinearity and inhomogeneity within these systems.

This paper describes a full-custom mixed-signal chip that embeds digitally programmable analog parallel processing and distributed image memory on a common silicon substrate. The chip was designed and fabricated in a standard 0.5 μm CMOS technology and contains approximately 500 000 transistors. It consists of 1024 processing units arranged into a 32×32 grid. Each processing element contains two coupled CNN cores, thus, constituting two parallel layers of 32×32 nodes. The functional features of the chip are in accordance with the 2nd Order Complex Cell CNN-UM architecture. It is composed of two CNN layers with programmable inter- and intra-layer connections between cells. Other features are: cellular, spatial-invariant array architecture; randomly selectable memory of instructions; random storage and retrieval of intermediate images. The chip is capable of completing algorithmic image processing tasks controlled by the user-selected stored instructions. The internal analog circuitry is designed to operate with 7-bits equivalent accuracy. The physical implementation of a CNN containing second order cells allows real-time experiments of complex dynamics and active wave phenomena. Such well-known phenomena from the reaction–diffusion equations are traveling waves, autowaves, and spiral-waves. All of these active waves are demonstrated on-chip. Moreover this chip was specifically designed to be suitable for the computation of biologically inspired retina models. These computational experiments have been carried out in a developmental environment designed for testing and programming the analogic (analog-and-logic) programmable array processors.

The paper introduces a class of third-order nonsymmetric Cellular Neural Networks (CNNs), and shows through computer simulations that they undergo a cascade of period doubling bifurcations which leads to the birth of a large-size complex attractor. A major point is that these bifurcations and complex dynamics happen in a small neighborhood of a particular CNN with a symmetric interconnection matrix.

We study a family of quadratic rational maps and discuss dynamical and ergodic theoretic properties associated to various parameters. We prove results about the location of ergodic parameters in parameter space and show computer output illustrating the results. We show that the computer algorithm used to produce Julia sets in this setting is valid for all rational maps.

We give a complete description of the dynamics of the mapping f_ε(z)=z²+(ε/z) for positive real values of ε. We then consider two generalizations: the case of complex ε and the mapping z→zⁿ+(ε/z^m), where ε is positive and real. In both cases we provide a full characterization of the map for a certain set of parameters, and give observations based on numerical evidence for all other parameter values. The dynamics of all maps that we consider bears striking resemblance to that of complex quadratic maps.

This paper focuses on the dynamics of coupled Chua's circuits driven by two sinusoidal signals. In particular, it is shown that the application of signals with slightly different frequencies enables the new phenomenon of chaotic beats to be generated. Finally, the application of signals with equal frequencies is discussed, with the aim of understanding the formation of beats in nonlinear circuits.

In this paper, the possible presence of complex dynamics in nearly-symmetric standard Cellular Neural Networks (CNNs), is investigated. A one-parameter family of fourth-order CNNs is presented, which exhibits a cascade of period-doubling bifurcations leading to the birth of a complex attractor, close to some nominal symmetric CNN. Different from previous work on this topic, the bifurcations and complex dynamics are obtained for small relative errors with respect to the nominal interconnections. The fourth-order CNNs have negative (inhibitory) interconnections between distinct neurons, and are designed by a variant of a technique proposed by Smale to embed a given dynamical system within a competitive dynamical system of larger order.

The present paper focuses on the two time scale dynamics generated by 2D polynomial noninvertible maps T of (Z₀ - Z₂) and (Z₁ - Z₃ - Z₁) types. This symbolism, specific to noninvertible maps, means that the phase plane is partitioned into zones Z_k, where each point possesses the k real rank-one preimages. Of special interest here is the structure of slow and fast motion sets of such maps. The formation mechanism of a stable invariant close curve through the interaction of fast and slow dynamics, as well as its transformation into a canard are studied. A few among the plethora of chaotic attractors and chaotic transients produced by such maps are described as well.

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpinski Julia sets for certain elliptic functions. We give examples satisfying these conditions. We show this results in naturally occurring Sierpinski curves in the plane, sphere and torus as well.

We study the family of complex maps given by F_λ(z) = zⁿ + λ/zⁿ + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

In this paper, we consider the family of rational maps given by

In a quasi-1D thermal convective system consisting of a large array of nonlinearly coupled oscillators, clustering is the way to achieve a regime of mostly antiphase synchronized oscillators. This regime is characterized by a spatiotemporal doubling of traveling modes. As the dynamics is explored beyond a spatiotemporal chaos regime (STC) with weak coupling, new interacting modes emerge through a supercritical bifurcation. In this new regime, the system exhibits coherent subsystems of antiphase synchronized oscillators, which are stationary clusters following a spatiotemporal beating phenomena (ZZ regime). This regime is the result of a stronger coupling. We show from a phase mismatch model applied to each oscillator, that these phase coherent domains undergo a global phase instability, meanwhile the interactions between oscillators become nonlocal. For each value of the control parameter we find out the time-varying topology (link matrix) from the contact interactions between oscillators. The new characteristic spatiotemporal scales are extracted from the antiphase correlations at the time intervals defined by the link matrix. The interpretation of these experimental results contributes to widen the understanding of other complex systems exhibiting similar phase chaotic dynamics in 2D and 3D.

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g₂, g₃) = (g₂, 0) to show that in parameter space (g₂-space) one sees both quadratic-like attracting orbit behavior and prepole dynamics. In the case of prepole parameters all critical orbits terminate at poles and the Julia set of ℘_L is the entire sphere. We show that both the Mandelbrot-like dynamics and the prepole parameters accumulate on prepole parameters of lower order providing results on the dynamics occurring in parameter space "between Mandelbrot sets".

Gene-regulatory networks are potentially capable of more complex behavior than convergence to a stationary state, or even cycling through a simple sequence of expression patterns. The analysis of qualitative dynamics for these networks is facilitated by using piecewise-linear equations and its state transition diagram (an n-dimensional hypercube, in the case of n genes with a single effective threshold for the protein product of each). Our previous work has dealt with cycles of states in the state transition diagram that allow periodic solutions. Here, we study a particular kind of figure-8 pattern in the state transition diagram and determine conditions that allow complex behavior. Previous studies of complex behavior, such as chaos, in such networks have dealt only with specific examples. Our approach allows an appreciation of the design principles that give rise to complex dynamics, which may have application in assessing the dynamical possibilities of gene networks with poorly known parameters, or for synthesis or control of gene networks with complex behavior.