Cycles,chaos, and noise in predator–prey dynamics

In contrast to the single species models that were extensively studied in the 1970s and 1980s, predator–prey models give rise to long-period oscillations, and even systems with stable equilibria can display oscillatory transients with a regular frequency. Many fluctuating populations appear to be governed by such interactions. However, predator–prey models have been poorly studied with respect to the interaction of nonlinear dynamics, noise, and system identification. I use simulated data from a simple host–parasitoid model to investigate these issues. The addition of even a modest amount of noise to a stable equilibrium produces enough structured variation to allow reasonably accurate parameter estimation. Despite the fact that more-or-less regular cycles are generated by adding noise to any of the classes of deterministic attractor (stable equilibrium, periodic and quasiperiodic orbits, and chaos), the underlying dynamics can usually be distinguished, especially with the aid of the mechanistic model. However, many of the time series can also be fit quite well by a wrong model, and the fitted wrong model usually misidentifies the underlying attractor. Only the chaotic time series convincingly rejected the wrong model in favor of the true one. Thus chaotic population dynamics offer the best chance for successfully identifying underlying regulatory mechanisms and attractors.     

Triple product L-functions and quantum chaos on SL(2, ℂ)

Let Y be an arithmetic hyperbolic 3-manifold. We establish a link between quantum unique ergodicity for sections of automorphic vector bundles on Y and subconvexity for the triple product L-function, which extends a result of Watson in the case of hyperbolic 2-manifolds. The proof combines the representation theoretic microlocal lift for bundles developed by Bunke and Olbrich with the triple product formula of Ichino. A key step is determining the asymptotic behaviour of the local integrals at complex places that appear in Ichino’s formula.     

Stein’s method on Wiener chaos

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.     

DC3 and Li–Yorke chaos

There are three versions of distributional chaos, namely DC1, DC2 and DC3. By using an example of constant-length substitution system, we show that DC3 need not imply Li–Yorke chaos. (In this paper, chaos means the existence of an uncountable scrambled set of the corresponding type, while the existing example only deals with a single pair of points.)     

A note on Schweizer–Smital chaos

In this paper, we consider a continuous map f:X→X$f:X\to X$, where X$X$ is a compact metric space, and prove that for any positive integer N$N$, f$f$ is Schweizer–Smital chaotic if and only if f^N$f^N$ is too.     

Hyperchaotic set in continuous chaos–hyperchaos transition

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.     

Several transitive properties and Devaney’s chaos

The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system (each point is non-wandering), indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, Z-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney's chaos are given by weak transitivity. Moreover, it is proved that both d-shadowing property and d-shadowing property imply weak transitivity.     

Is there chaos in the Spanish labour market?

One could argue that there is a resurgence of the non-linear modelling in economics. Some instruments have been developed to measure the complexity or instability of the analysed systems. At the present work some of these developed techniques are applied to verify the non-linearity present in the time series of Spanish unemployment, as well as to quantify the degree of complexity of the system that has generated the series. Using these techniques we find evidence of chaos in Spanish unemployment time series.     

The Fibonacci sequence modulo π, chaos and some rational recursive equations

In this paper, we provide the closed form solution to the inter-related equations

     

Li–Yorke chaos translation set for linear operators

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).     

Distributional chaos for operators with full scrambled sets

In this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator (we even prove that the mixing property does not suffice). Moreover, we show that an extremal situation is possible: There are (hypercyclic and non-hypercyclic) operators such that the whole space consists, except zero, of distributionally irregular vectors.     

Li–Yorke chaos in higher dimensions: a review

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ? ⁿ are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (?  ⁿ , D) of fuzzy sets on the base space ? ⁿ .     

Evolving to the edge of chaos: Chance or necessity?

We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Low-dimensional chaos in high-dimensional phase space: how does it occur?

A fundamental observation in nonlinear dynamics is that the asymptotic chaotic invariant sets in many high-dimensional systems are low-dimensional. We argue that such a behavior is typically associated with chaos synchronism. Numerical support using coupled chaotic systems including a class derived from a nonlinear partial differential equation is provided.