Chaotic phenomena
type: Item
instance of: scholarly article
title: Chaotic phenomena
zbMATH Open document ID: 0586.92002
DOI: 10.1007/BF02469299
published in: Bulletin of Mathematical Biology
publication date: 1985-01-01T00:00:00Z
review text: The essential phenomena of chaotic systems are reviewed with several references to biological applications. Difference and differential equations are discussed separately, followed by a shorter chapter on measures, dimensions, and entropy. Several well-known model systems (like the logistic map, the Baker's transformation, or the horseshoe map for difference equations, the Hénon equation, the Lorentz equations, or the periodically forced van der Pol oscillator for differential systems) are presented in order to illustrate the general features of chaos.
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 39Axx
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 34D99
Mathematics Subject Classification ID: 35B99
zbMATH DE Number: 3941312
zbMATH Keywords: chaos
zbMATH Keywords: chaotic systems
zbMATH Keywords: measures
zbMATH Keywords: dimensions
zbMATH Keywords: entropy
zbMATH Keywords: logistic map
zbMATH Keywords: Baker's transformation
zbMATH Keywords: horseshoe map
zbMATH Keywords: Hénon equation
zbMATH Keywords: Lorentz equations
zbMATH Keywords: van der Pol oscillator
author: Peter E. Kloeden
author: Alistair Mees
reviewed by: Michael Baake
MaRDI profile type: Publication
Wikidata QID: Q52678698
cites work: Hypergraphic Functions and Bifurcations in Recurrence Relations
cites work: Attractors of partial differential evolution equations and estimates of their dimension
cites work: Chaotic motion in nonlinear feedback systems
cites work: Q4081516
cites work: Homoclinic Points of Mappings of the Interval
cites work: Stability of Periodic Orbits in the Theorem of Sarkovskii
cites work: Q4745028
cites work: Q3966384
cites work: Notes on chaos in the cell population partial differential equation
cites work: Ergodicity and exactness of the shift on C[0,\(\infty)\) and the semiflow of a first-order partial differential equation
cites work: How chaotic is chaos? Chaotic and other ''noisy'' dynamics in the frequency domain
cites work: Bursting Phenomena in Excitable Membranes
cites work: Almost Periodic Flows and Solutions of Differential Equations
cites work: On Non-Linear Differential Equations of the Second Order: I. the Equation y¨ − k (1-y 2 )y˙ + y = b λk cos(λl + α), k Large
cites work: Q3908199
cites work: An Interesting Cantor Set
cites work: Q5847520
cites work: The Danziger-Elmergreen theory of periodic catatonic schizophrenia
cites work: Period-doubling bifurcations and associated universal properties including parameter dependence.
cites work: Homoclinic bifurcations and the area-conserving Henon mapping
cites work: Q3314128
cites work: Are chaotic functions really chaotic
cites work: Roads to turbulence in dissipative dynamical systems
cites work: Q5186444
cites work: Quantitative universality for a class of nonlinear transformations
cites work: Asymptotic analysis of the Navier-Stokes equations
cites work: Local and global behavior near homoclinic orbits
cites work: Measuring the strangeness of strange attractors
cites work: Approximate solution of a class of operator equations
cites work: Dimensions and entropies of strange attractors from a fluctuating dynamics approach
cites work: Analysis of Newton’s Method at Irregular Singularities
cites work: Sensitive dependence to initial conditions for one dimensional maps
cites work: Dimension estimates for attractors
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: The dynamics of density dependent population models
cites work: Q1167078
cites work: Recurrences and discrete dynamic systems
cites work: A two-dimensional mapping with a strange attractor
cites work: The dynamical systems approach to differential equations
cites work: Q4067388
cites work: Nonlinear double-diffusive convection
cites work: Q3894691
cites work: Spatiotemporal patterns in epileptic seizures
cites work: Chaotic difference equations are dense
cites work: On Sharkovsky's cycle coexistence ordering
cites work: Q3924618
cites work: A computer-assisted proof of the Feigenbaum conjectures
cites work: Stable and chaotic solutions of a first order partial differential equation
cites work: Qualitative analysis of the periodically forced relaxation oscillations
cites work: A second order differential equation with singular solutions

Chaotic phenomena (cites work)
cites work: Period Three Implies Chaos

Chaotic phenomena (cites work)
cites work: Deterministic Nonperiodic Flow

Chaotic phenomena (cites work)
cites work: The dynamics of recurrent inhibition

Chaotic phenomena (cites work)
cites work: Oscillation and Chaos in Physiological Control Systems

Chaotic phenomena (cites work)
cites work: Negatively invariant sets of compact maps and an extension of a theorem of Cartwright

Chaotic phenomena (cites work)
cites work: Q4154814

Chaotic phenomena (cites work)
cites work: Snap-back repellers imply chaos in \(\mathbb{R}^n\)

Chaotic phenomena (cites work)
cites work: Simple mathematical models with very complicated dynamics

Chaotic phenomena (cites work)
cites work: Asymptotic properties of sequences of iterates of nonlinear transformations

Chaotic phenomena (cites work)
cites work: Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Chaotic phenomena (cites work)
cites work: Homoclinic and heteroclinic orbits in the double scroll attractor

Chaotic phenomena (cites work)
cites work: Chaos and bifurcation phenomena in limiting central difference scheme

Chaotic phenomena (cites work)
cites work: Sur l'itération des polynômes réels quadratiques

Chaotic phenomena (cites work)
cites work: Q5658860

Chaotic phenomena (cites work)
cites work: Cayley's problem and Julia sets

Chaotic phenomena (cites work)
cites work: Q3900299

Chaotic phenomena (cites work)
cites work: Iterates of maps on an interval

Chaotic phenomena (cites work)
cites work: Q4150377

Chaotic phenomena (cites work)
cites work: Q3715933

Chaotic phenomena (cites work)
cites work: Differentiable dynamical systems and the problem of turbulence

Chaotic phenomena (cites work)
cites work: On the nature of turbulence

Chaotic phenomena (cites work)
cites work: Newton's Method, Circle Maps, and Chaotic Motion

Chaotic phenomena (cites work)
cites work: Chaos and Arnold diffusion in dynamical systems

Chaotic phenomena (cites work)
cites work: Q5553902

Chaotic phenomena (cites work)
cites work: Q5563285

Chaotic phenomena (cites work)
cites work: Q3733844

Chaotic phenomena (cites work)
cites work: A generalization of a theorem of marotto

Chaotic phenomena (cites work)
cites work: Chaotic Difference Equations: Generic Aspects

Chaotic phenomena (cites work)
cites work: Stable Orbits and Bifurcation of Maps of the Interval

Chaotic phenomena (cites work)
cites work: Differentiable dynamical systems

Chaotic phenomena (cites work)
cites work: Structural stability of nonchaotic difference equations

Chaotic phenomena (cites work)
cites work: Chaos in a three-dimensional single loop feedback system with a piecewise linear feedback function

Chaotic phenomena (cites work)
cites work: The Lorenz equations: bifurcations, chaos, and strange attractors

Chaotic phenomena (cites work)
cites work: A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line

Chaotic phenomena (cites work)
cites work: Q5518885

Chaotic phenomena (cites work)
cites work: Periodic Points of Continuous Functions

Chaotic phenomena (cites work)
cites work: Recurrent motions in certain continuum dynamical systems

Chaotic phenomena (cites work)
cites work: The topological conjugacy problem for generalized Hénon mappings: some negative results

Chaotic phenomena (cites work)
cites work: On the appearance of chaos in a model of the Belousov reaction

Chaotic phenomena (cites work)
cites work: The unstable separatrix of Feigenbaum's fixed-point

Chaotic phenomena (cites work)
cites work: Homoclinic solution and chaos in

Chaotic phenomena (cites work)
cites work: Discrete Dynamical Systems in Dimensions One and Two

Chaotic phenomena (cites work)
cites work: Determining Lyapunov exponents from a time series

Chaotic phenomena (cites work)
cites work: Newton's Method and Symbolic Dynamics

Chaotic phenomena (cites work)
cites work: Q3869633

Chaotic phenomena (cites work)
cites work: Entropy, Lyapunov exponents, and Hausdorff dimension in differentiable dynamical systems

Chaotic phenomena (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02469299

Chaotic phenomena (OpenAlex ID)
OpenAlex ID: W3190201959