What Could Be Worse than the Butterfly Effect?

2008 ◽  
Vol 38 (4) ◽  
pp. 519-547 ◽  
Author(s):  
Robert C. Bishop
Keyword(s):  
Nonlinear Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Classical Mechanics ◽  
Physical Systems ◽  
Physics Community ◽  
Sensitive Dependence ◽  
Classical Models ◽  
In The Beginning

Our understanding of classical mechanics (CM) has undergone significant growth in the latter half of the twentieth century and in the beginning of the twenty-first. This growth has much to do with the explosion of interest in the study of nonlinear systems in contrast with the focus on linear systems that had colored much work in CM from its inception. For example, although Maxwell and Poincaré arguably were some of the first to think about chaotic behavior, the modern study of chaotic dynamics traces its beginning to the pioneering work of Edward Lorenz (1963). This work has yielded a rich variety of behavior in relatively simple classical models that was previously unsuspected by the vast majority of the physics community (see Hilborn 2001). Chaos is a property of nonlinear systems that is usually characterized by sensitive dependence on initial conditions (SDIC). In CM the behavior of simple physical systems is described using models (such as the harmonic oscillator) that capture the main features of the systems in question (Giere 1988).

CHAOTIC BEHAVIOR AND DYNAMICS OF MAPS USED IN A METHOD OF SCRAMBLING SIGNALS

2010 ◽  
Vol 20 (12) ◽  
pp. 4097-4101
Author(s):  
REZA MAZROOEI-SEBDANI ◽  
MEHDI DEHGHAN
Keyword(s):  
Fixed Points ◽  
Limit Cycles ◽  
Secure Communication ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Chaotic Encryption ◽  
Input And Output ◽  
Sensitive Dependence ◽  
Close Relationship ◽  
Existence And Stability

The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. In this manuscript, we prove that a class of maps that have been proposed as suitable for scrambling signals possess the property of sensitive dependence on initial conditions (s.d.i.c.) necessary for chaos and cryptography. Our result can also be used for generating other maps with s.d.i.c., through a suitable semiconjugacy between their input and output parts. Using the condition of semiconjugacy we also establish for this class of maps rigorous criteria for the existence and stability of their fixed points and limit cycles.

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

2004 ◽  
Vol 14 (07) ◽  
pp. 2161-2186 ◽  
Author(s):  
GOONG CHEN ◽  
TINGWEN HUANG ◽  
YU HUANG
Keyword(s):  
Nonlinear Systems ◽  
Initial Data ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Oscillatory Behavior ◽  
Exponential Rate ◽  
Interval Maps ◽  
Interval Map ◽  
Sensitive Dependence

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

2018 ◽  
Vol 28 (14) ◽  
pp. 1850176 ◽  
Author(s):  
Hegui Zhu ◽  
Wentao Qi ◽  
Jiangxia Ge ◽  
Yuelin Liu
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Function System ◽  
Topological Transitivity ◽  
One Dimensional ◽  
Devaney’S Chaos ◽  
Chebyshev Map ◽  
Sensitive Dependence ◽  
Sine Map ◽  
The One

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

scholarly journals Chaotic Dynamics in Smart Grid and Suppression Scheme via Generalized Fuzzy Hyperbolic Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Qiuye Sun ◽  
Yaguang Wang ◽  
Jun Yang ◽  
Yue Qiu ◽  
Huaguang Zhang
Keyword(s):  
Smart Grid ◽  
Power System ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Linear Matrix ◽  
Hyperbolic Model ◽  
Distributed Generations ◽  
Generalized Fuzzy Hyperbolic Model ◽  
Suppression Strategy

This paper presents a method to control chaotic behavior of a typical Smart Grid based on generalized fuzzy hyperbolic model (GFHM). As more and more distributed generations (DG) are incorporated into the Smart Grid, the chaotic behavior occurs increasingly. To verify the behavior, a dynamic model which describes a power system with DG is presented firstly. Then, the simulation result shows that the power system can lead to chaos under certain initial conditions. Based on the universal approximation of GFHM, we confirm that the chaotic behavior could be suppressed by a new controller, which is designed by means of solving a linear matrix inequality (LMI). This approach could make a good application to suppress the chaos in Smart Grid. Finally, a numerical example is given to demonstrate the effectiveness of the proposed chaotic suppression strategy.

scholarly journals Spectral Features of Systems With Chaotic Dynamics

2020 ◽  
pp. 58-65
Author(s):  
Perevoznikov E. N.
Keyword(s):  
Nonlinear Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Traditional Approach ◽  
Short Wave ◽  
Spectral Features ◽  
Lorentz Model ◽  
Weakly Ionized ◽  
Weakly Ionized Plasma ◽  
Spectral Equation

Using the Lorentz model and Hamiltonian systems without dissipation as an example, spectral methods for analyzing the dynamics of systems with chaotic behavior are considered. The insufficiency of the traditional approach to the study of perturbation dynamics based on an analysis of the roots of the classical spectral equation is discussed. It is proposed to study nonlinear systems using the method of constructing spectral equations with different eigenvalues, which allows one to take into account the randomness and multiplicity of states. The spectral features of instability and chaos for systems without dissipation are shown by the example of short-wave perturbations of a flow of a weakly ionized plasma gas.

scholarly journals Chaos in a simple model of a delta network

2020 ◽  
Vol 117 (44) ◽  
pp. 27179-27187
Author(s):  
Gerard Salter ◽  
Vaughan R. Voller ◽  
Chris Paola
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Fundamental Limits ◽  
Experimental Systems ◽  
Flux Partitioning ◽  
Sensitive Dependence ◽  
Statistical Behavior ◽  
The Mean ◽  
Simple Numerical Model

The flux partitioning in delta networks controls how deltas build land and generate stratigraphy. Here, we study flux-partitioning dynamics in a delta network using a simple numerical model consisting of two orders of bifurcations. Previous work on single bifurcations has shown periodic behavior arising due to the interplay between channel deepening and downstream deposition. We find that coupling between upstream and downstream bifurcations can lead to chaos; despite its simplicity, our model generates surprisingly complex aperiodic yet bounded dynamics. Our model exhibits sensitive dependence on initial conditions, the hallmark signature of chaos, implying long-term unpredictability of delta networks. However, estimates of the predictability horizon suggest substantial room for improvement in delta-network modeling before fundamental limits on predictability are encountered. We also observe periodic windows, implying that a change in forcing (e.g., due to climate change) could cause a delta to switch from predictable to unpredictable or vice versa. We test our model by using it to generate stratigraphy; converting the temporal Lyapunov exponent to vertical distance using the mean sedimentation rate, we observe qualitatively realistic patterns such as upwards fining and scale-dependent compensation statistics, consistent with ancient and experimental systems. We suggest that chaotic behavior may be common in geomorphic systems and that it implies fundamental bounds on their predictability. We conclude that while delta “weather” (precise configuration) is unpredictable in the long-term, delta “climate” (statistical behavior) is predictable.

Oscillatory Tracking Control of a Class of Nonlinear Systems

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Zheng Wang ◽  
Yi Guo
Keyword(s):  
Nonlinear Systems ◽  
Tracking Control ◽  
State Feedback ◽  
Inverted Pendulum ◽  
Feedforward Control ◽  
Initial Conditions ◽  
Pendulum System ◽  
Physical Systems ◽  
Inverted Pendulum System ◽  
State Measurement

Vibration control is an effective alternative to conventional feedback and feedforward control. Motivated by its important application in physical systems and few results on general oscillatory tracking control, we consider tracking control of a class of nonlinear systems using oscillation in the paper. We propose a new oscillatory control design using general averaging analysis for the tracking problem. Based on the oscillation functions associated with accessible vibrating components of the system, oscillatory control is designed to track a desired trajectory. Comparing to existing oscillatory tracking control, our approach is robust to initial conditions. We show the effectiveness of the proposed method by two simulation examples, which include a second-order nonholonomic integrator and the inverted pendulum system. For the inverted pendulum system, we show that our designed oscillatory control does not need state feedback to track a desired trajectory, which is desirable for systems where state measurement is not feasible.

On Invariant Qualitative Chaos in Multi-Black-Hole Spacetimes

1997 ◽  
Vol 06 (06) ◽  
pp. 741-770 ◽  
Author(s):  
Marek Szydłowski
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Optical Model ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Cylindrical Symmetry ◽  
Geodesic Motion ◽  
Local Instability ◽  
Black Hole Spacetimes ◽  
Sensitive Dependence

It is analytically shown when chaos exists in the behavior of null and timelike geodesics in the general case of geodesic motion in static and diagonal fields of general relativity. We demonstrate the effectiveness of our method of investigating chaos in the behavior of geodesic motion in the multi-black-hole spacetimes. An optical model of chaotic behavior of geodesics in spacetimes with cylindrical symmetry is presented. The Lyapunov characteristic time is defined and estimated for geodesic motion of a test particle in the external fields of general relativity. We find that its value is positive in some compact regions of the configuration space. This means that the trajectories have the property of local instability which implies the sensitive dependence on initial conditions.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

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