scholarly journals Lorenz-63 Model as a Metaphor for Transient Complexity in Climate

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 951
Author(s):  
Sergey Kravtsov ◽  
Anastasios A. Tsonis
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Initial Conditions ◽  
External Forcing ◽  
Equilibrium Conditions ◽  
Natural Systems ◽  
Perturbed System ◽  
Climate Systems ◽  
The One ◽  
Sensitivity To Initial Conditions

Dynamical systems like the one described by the three-variable Lorenz-63 model may serve as metaphors for complex natural systems such as climate systems. When these systems are perturbed by external forcing factors, they tend to relax back to their equilibrium conditions after the forcing has shut off. Here we investigate the behavior of such transients in the Lorenz-63 model by studying its trajectories initialized far away from the asymptotic attractor. Counterintuitively, these transient trajectories exhibit complex routes and, in particular, the sensitivity to initial conditions is akin to that of the asymptotic behavior on the attractor. Thus, similar extreme events may lead to widely different variations before the perturbed system returns back to its statistical equilibrium.

scholarly journals Transient behavior in the Lorenz model

2014 ◽  
Vol 1 (2) ◽  
pp. 1905-1917 ◽  
Author(s):  
S. Kravtsov ◽  
N. Sugiyama ◽  
A. A. Tsonis
Keyword(s):  
Asymptotic Behavior ◽  
Initial Conditions ◽  
Transient Behavior ◽  
External Forcing ◽  
Lorenz Model ◽  
Equilibrium Conditions ◽  
Natural Systems ◽  
Perturbed System ◽  
The One ◽  
Sensitivity To Initial Conditions

Abstract. Dynamical systems like the one described by the three-variable Lorenz model may serve as metaphors for complexity in nature. When natural systems are perturbed by external forcing factors, they tend to relax back to their equilibrium conditions after the forcing has shut off. Here we investigate the behavior of such transients in the Lorenz model by studying its trajectories initialized far away from the asymptotic attractor. Perhaps somewhat surprisingly, these transient trajectories exhibit complex routes and, among other things, sensitivity to initial conditions akin to that of the asymptotic behavior on the attractor. Thus, similar extreme events may lead to widely different variations before the perturbed system returns back to its statistical equilibrium.

Boundedness and Other Theories for Complex Systems

2013 ◽  
pp. 108-125
Keyword(s):  
Initial Conditions ◽  
Traditional Approach ◽  
Deterministic Chaos ◽  
Mathematical Object ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Universal Properties ◽  
The One ◽  
Global Invariant ◽  
Sensitivity To Initial Conditions

A comparison between the concept of boundedness on the one hand, and the theory of self-organized criticality (SOC) and the deterministic chaos on the other hand, is made. The focus is put on the methodological importance of the general frame through which an enormous class of empirical observations is viewed. The major difference between the concept of boundedness and the theory of self organized criticality is that under boundedness, the response comprises both specific and universal part, and thus a system has well defined “identity,” while SOC assumes response as a global invariant which has only universal properties. Unlike the deterministic chaos, the boundedness is free to explain the sensitivity to initial conditions independently from the mathematical object that generates them. Alongside, it turns out that the traditional approach to the deterministic chaos has its ample understanding under the concept of boundedness.

Nonextensive Entropies and Sensitivity to Initial Conditions of Complex Systems

2004 ◽  
Author(s):  
Marcelo L. Lyra
Keyword(s):  
Initial Conditions ◽  
Scaling Properties ◽  
Tsallis Statistics ◽  
Natural Systems ◽  
Starting Point ◽  
Natural Frame ◽  
Low Dimensional ◽  
Main Ideas ◽  
Sensitivity To Initial Conditions ◽  
Entropic Index

Tsallis generalized statistics has been successfully applied to describe some relevant features of several natural systems exhibiting a nonextensive character. It is based on an extended form for the entropy, namely S<sub>q</sub> = (1 — Σ<sub>q</sub>p<sup>Q</sup><sub>q</sub>)/(Q —1), where q is a parameter that measures the degre of nonextensivity (q→1 for the traditional Boltzmann-Gibbs statistics). A series of recent works have shown that the power-law sensitivity to initial conditions in a complex state provides a natural link between the g-entropic parameter and the scaling properties of dynamical attractors. These results contribute to the growing set of theoretical and experimental evidences that Tsallis statistics can be a natural frame for studying systems with a fractal-like structure in phase space. Here, the main ideas underlying this relevant aspect are reviewed. The starting point is the weak sensitivity to initial conditions exhibited by low-dimensional dynamical systems at the onset of chaos. It is shown how general scaling arguments can provide a direct relation between the entropic index q and the scaling exponents associated with the multifractal critical attractor. These works shed light in the elusive problem concerning the connection between the g-entropic parameter of Tsallis statistics and the underlying microscopic dynamics of nonextensive systems.... Inspired on the scaling properties of multifractals, Tsallis introduced a generalized entropy with the aim of extending the usual statistical mechanics and thermodynamics [37].

scholarly journals Modelling chance and necessity in natural systems

2019 ◽  
Vol 77 (4) ◽  
pp. 1573-1588 ◽  
Author(s):  
Benjamin Planque ◽  
Christian Mullon
Keyword(s):  
Dynamic Systems ◽  
Paradigm Shift ◽  
Barents Sea ◽  
Initial Conditions ◽  
Deterministic Chaos ◽  
Participatory Management ◽  
Natural Systems ◽  
Emergence Of Chaos ◽  
Ecosystem Based Fisheries Management ◽  
Sensitivity To Initial Conditions

Abstract Nearly 30 years ago, emerged the concept of deterministic chaos. With it came sensitivity to initial conditions, nonlinearities, and strange attractors. This constituted a paradigm shift that profoundly altered how numerical modellers approached dynamic systems. It also provided an opportunity to resolve a situation of mutual misunderstanding between scientists and non-scientists about uncertainties and predictability in natural systems. Our proposition is that this issue can be addressed in an original way which involves modelling based on the principles of chance and necessity (CaN). We outline the conceptual and mathematical principles of CaN models and present an application of the model to the Barents Sea food-web. Because CaN models rely on concepts easily grasped by all actors, because they are explicit about knowns and unknowns and because the interpretation of their results is simple without being prescriptive, they can be used in a context of participatory management. We propose that, three decades after the emergence of chaos theories, CaN can be a practical step to reconcile scientists and non-scientists around the modelling of structurally and dynamically complex natural systems, and significantly contribute to ecosystem-based fisheries management.

A New Image Encryption Scheme Using Dual Chaotic Map Synchronization

2020 ◽  
Vol 18 (1) ◽  
Keyword(s):  
Image Encryption ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Chaotic Map ◽  
Security Applications ◽  
The Common ◽  
And Diffusion ◽  
Confusion And Diffusion ◽  
Sensitivity To Initial Conditions ◽  
Diffusion Properties

Chaotic systems behavior attracts many researchers in the field of image encryption. The major advantage of using chaos as the basis for developing a crypto-system is due to its sensitivity to initial conditions and parameter tunning as well as the random-like behavior which resembles the main ingredients of a good cipher namely the confusion and diffusion properties. In this article, we present a new scheme based on the synchronization of dual chaotic systems namely Lorenz and Chen chaotic systems and prove that those chaotic maps can be completely synchronized with other under suitable conditions and specific parameters that make a new addition to the chaotic based encryption systems. This addition provides a master-slave configuration that is utilized to construct the proposed dual synchronized chaos-based cipher scheme. The common security analyses are performed to validate the effectiveness of the proposed scheme. Based on all experiments and analyses, we can conclude that this scheme is secure, efficient, robust, reliable, and can be directly applied successfully for many practical security applications in insecure network channels such as the Internet

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

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.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

Numerical investigation of the propagation of shock waves in rigid porous materials: development of the computer code and comparison with experimental results

1996 ◽  
Vol 324 ◽  
pp. 163-179 ◽  
Author(s):  
A. Levy ◽  
G. Ben-Dor ◽  
S. Sorek
Keyword(s):  
Porous Materials ◽  
Initial Conditions ◽  
Computer Code ◽  
Experimental Results ◽  
Numerical Code ◽  
Materials Development ◽  
Numerical Predictions ◽  
Wide Range ◽  
Dimensional Version ◽  
The One

The governing equations of the flow field which is obtained when a thermoelastic rigid porous medium is struck head-one by a shock wave are developed using the multiphase approach. The one-dimensional version of these equations is solved numerically using a TVD-based numerical code. The numerical predictions are compared to experimental results and good to excellent agreements are obtained for different porous materials and a wide range of initial conditions.

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