DOCSLIB.ORG

Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points Hindawi Complexity Volume 2018, Article ID 9430637, 8 pages https://doi.org/10.1155/2018/9430637 Research Article Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points Aiguo Wu,1 Shijian Cang ,1,2 Ruiye Zhang,1 Zenghui Wang ,3 and Zengqiang Chen 4 1 School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China 2Department of Product Design, Tianjin University of Science and Technology, Tianjin 300457, China 3Department of Electrical and Mining Engineering, University of South Africa, Florida 1710, South Africa 4College of Computer and Control Engineering, Nankai University, Tianjin 300071, China Correspondence should be addressed to Shijian Cang; [email protected] Received 8 November 2017; Revised 21 February 2018; Accepted 12 March 2018; Published 22 April 2018 Academic Editor: Jose´ Angel´ Acosta Copyright © 2018 Aiguo Wu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in diferent disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. Tis paper proposes a fve-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fxed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents. 1. Introduction white-noise-like, the conservative chaos has no chaotic attractor, which means that the conservative chaos when As a research hotspot in the feld of nonlinear science applied in information security has distinct advantages over over the last fve decades, chaos theory has achieved great dissipative chaos. development since the Ueda attractor [1], the Lorenz attractor It is well known that the nonlinear dynamical systems [2], and Li-Yoke chaos [3] were discovered. So far, chaos theory has been successfully applied in many felds, such can produce various dynamical behaviors, such as periodic as electronic engineering [4], computer science [5], com- motion, quasiperiodic motion, chaos, and hyperchaos. Gen- munication systems [6, 7], complex networks [8], chemical erally, hyperchaos is defned as its dynamical behavior with at engineering [9], and economic models [10]. In the recent least two positive Lyapunov exponents (LEs), and the mini- decade, hundreds of physical chaotic models [11–14] and mal dimension for an autonomous continuous hyperchaotic artifcial chaotic systems [15–19] have been investigated in system is four. In comparison with chaos, hyperchaos, espe- theory and by numerical simulations due to the poten- cially conservative hyperchaos, is preferable for those appli- tial applications of chaotic system in various chaos-based cations that require the chaotic systems showing complicated technologies [20, 21]. By now, numerous dissipative-chaos- dynamics, such as network security and data encryption. Te based encryption algorithms have been developed to ensure frst hyperchaotic system was proposed in 1979 by Rossler¨ the safety of information, but these algorithms are not [22]. Since then, various hyperchaotic systems have been strong enough because the dissipative chaotic attractors can found [23–26], but almost all of these hyperchaotic systems be reconstructed by delay embedding method based on are dissipative and it is difcult to fnd odd-dimensional the sampled data. Besides the general properties of chaos systems that can generate conservative hyperchaotic fows. such as ergodicity, aperiodic, uncorrelated, broadband, and In 1994, Sprott found a 3D nonequilibrium non-Hamiltonian 2 Complexity system with conservative chaos, lately known as the Sprott A �=�̇ V system [15], which is described as V̇=��−�� �=�̇ �=−�̇ V −��, �=−�+��̇ (1) (2) �=−�̇ 2 +1, where �, �, �, V,and� are the system variables and � and �∈ R+ � � � are the system’s constant parameters. Obviously, system where , ,and are the system states. It has been proven that (2) does not meet the requirements for classic Hamiltonian system (1) is a special case of the Nose-Hoover´ system [27]. systems. Although system (2) is not Hamiltonian, it does have Moreover, there exist coexisting quasiperiodic and conserva- a conserved quantity, which will be discussed in the following tive chaotic fows under diferent initial conditions. In view of sections. thefactthatthesesystemswithconservativehyperchaosare rare, thus, it is very interesting to fnd hyperchaotic systems 2.1. Invariance and Time-Reversibility. Many of chaotic tra- with conservative fows. jectories are symmetric; if �→+∞,itiseasytoget Trough sorting out the published papers about chaotic the invariance of the system (2) under the coordinate systems of recent years, it is found that hidden attractors and transformation (�,�,�,V,�) → (�,�,−�,−V,−�),which perpetual points become two important new research topics. persists for all values of � and �.However,theinvari- Te properties of hidden attractor are diferent from that of ance of the solution of system (2) in the time-reversible self-excited chaotic attractor [28, 29]. It has been shown that direction can also be achieved under the transformation (�,�,�,�,V, �) → (−�, −�, �, �, V,−�) the dynamical systems having hidden attractors include these . Generally, a dynami- systems without equilibrium, with no unstable equilibrium, cal system is reversible if there is an involution in phase space with one stable equilibrium, and with lines, curves, planes, which reverses the direction of time. It is a very common and surfaces of equilibria [30–35]. Recently, perpetual points thing to fnd the time-reversibility for conservative systems, were introduced as another structural feature of nonlinear but there are exceptions [38, 39]. systems [36]. In some cases, the perpetual points are useful to locate hidden attractors and to fnd coexisting attractors in 2.2. Conservation multistable systems [36, 37]. Defnition 1. Dynamical systems, whose Hamiltonian (i.e., Te main contribution of this paper is that a rare fve- conserved quantity) does not vary in time, are called conser- dimensional (5D) autonomous dynamical system is proposed vative systems; otherwise they are called dissipative systems and investigated. In comparison with the known chaotic [40]. systems, the system has the following four characteristics: (i) Te system with nonzero initial values has two non- Remark 2. Tere is diference between the conservative sys- hyperbolic fxed points. temandtheconservativemotion.Generally,theconservative motion is most ofen encountered in conservative systems, (ii) Te system is conservative, which can be theoretically especiallyinHamiltoniansystems,butitcanbealsofound verifed by the existence of a conserved quantity, in nonconservative systems, such as the Sprott A system; but cannot be confrmed by the trace of its Jacobian in other words, the conservative motion exists not only in matrix, which means that the fows generated by the conservative systems but also in nonconservative systems, system are compressible. which can be verifed by numerical techniques, such as LE (iii) Numerically, the symmetric LEs spectrum shows that spectra. the system has conservative chaos. By the defnition, if we can fnd a Hamiltonian for system (iv) Hyperchaotic motion can be found under specifc (2)andprovethetimederivativeoftheconservedquantity conditions in this conservative system. equals zero, system (2) is conservative. Te rest of this paper is organized as follows. In Section 2, we introduce a new 5D dynamical system and analyze its Teorem 3. One of the Hamiltonians of system (2) is �(�, 2 2 2 2 2 ̇ basic dynamics. In Section 3, the existence of perpetual point �, �, V, �) = (1/2)(� +� +� + V +� ),whichsatisfes�(�, of the new system is investigated. Section 4 illustrates the �, �, V,�)=0. dynamicalbehaviorsoftheproposedsystemwiththehelpof LEs and phase portraits. Te conclusion is presented in the Proof. Te linear frst-order PDE of system (2) can be last section. expressed as [41, 42] �� �� �� �� �� �̇ + �̇ + �̇ + V̇ + �̇ =0. (3) 2. System Model �� �� �� �V �� In order to get the general solutions of (3), we choose (4) as Consider the following 5D continuous dynamical system: the associated system. 2 �=��+�̇ �� �� �� �V �� = = = = . (4) �=−��̇ �� + �2 −�� �V �� − �� −�V −�� Complexity 3 According to (4), for simplicity, we choose a particular Obviously, all FPs of system (2) are center FPs (nonhyperbolic solution of the following smooth positive-defnite standard FPs) since there exist one real root �1 =0and two pairs of quadratic form: purely imaginary roots �2,3 =±��and �4,5 =±��,which 1 can efectively exhibit that there are no asymptotically stable �(�,�,�,V,�)= (�2 +�2 +�2 + V2 +�2). 2 (5) equilibria or limit cycles in conservative systems in phase space. BasedontheTeorem3,wecanconcludethatsystem(2)is conservative. Teorem 5. Forsystem(2),therearetwocenterFPsunderany initial conditions (�0,�0,�0, V0,�0) with �0 =0̸ . In addition, we can also discuss
Load more

Recommended publications
  • Role of Nonlinear Dynamics and Chaos in Applied Sciences
    v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings.
    [Show full text]
  • Arxiv:0705.0033V3 [Math.DS]
    ERGODIC THEORY: RECURRENCE NIKOS FRANTZIKINAKIS AND RANDALL MCCUTCHEON Contents 1. Definition of the Subject and its Importance 3 2. Introduction 3 3. Quantitative Poincaré Recurrence 5 4. Subsequence Recurrence 7 5. Multiple Recurrence 11 6. Connections with Combinatorics and Number Theory 14 7. Future Directions 17 References 19 Almost every, essentially: Given a Lebesgue measure space (X, ,µ), a property P (x) predicated of elements of X is said to hold for almostB every x X, if the set X x: P (x) holds has zero measure. Two sets A, B are∈ essentially disjoint\ { if µ(A B) =} 0. ∈B Conservative system: Is an∩ infinite measure preserving system such that for no set A with positive measure are A, T −1A, T −2A, . pairwise essentially disjoint.∈ B (cn)-conservative system: If (cn)n∈N is a decreasing sequence of posi- tive real numbers, a conservative ergodic measure preserving transforma- 1 tion T is (cn)-conservative if for some non-negative function f L (µ), ∞ n ∈ n=1 cnf(T x)= a.e. arXiv:0705.0033v3 [math.DS] 4 Nov 2019 ∞ PDoubling map: If T is the interval [0, 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T : T T, defined by Tx = 2x mod 1, preserves Lebesgue measure, hence induces→ a measure preserving system on T. Ergodic system: Is a measure preserving system (X, ,µ,T ) (finite or infinite) such that every A that is T -invariant (i.e. T −B1A = A) satisfies ∈B either µ(A) = 0 or µ(X A) = 0. (One can check that the rotation Rα is ergodic if and only if α is\ irrational, and that the doubling map is ergodic.) Ergodic decomposition: Every measure preserving system (X, ,µ,T ) can be expressed as an integral of ergodic systems; for example,X one can 2000 Mathematics Subject Classification.
    [Show full text]
  • Dissipative Dynamical Systems and Their Attractors
    Dissipative dynamical systems and their attractors Grzegorz ukaszewicz, University of Warsaw MIM Qolloquium, 05.11.2020 Plan of the talk Context: conservative and dissipative systems 3 Basic notions 10 Important problems of the theory. G.ukaszewicz Dissipative dynamical systems and their attractors Qolloquium 2 / 14 Newtonian mechanics ∼ conservative system of ODEs in Rn system reversible in time ∼ groups ∼ deterministic chaos From P. S. de Laplace to H. Poincaré, and ... Evolution of a conservative system Example 1. Motivation: Is our solar system stable? (physical system) G.ukaszewicz Dissipative dynamical systems and their attractors Qolloquium 3 / 14 Evolution of a conservative system Example 1. Motivation: Is our solar system stable? (physical system) Newtonian mechanics ∼ conservative system of ODEs in Rn system reversible in time ∼ groups ∼ deterministic chaos From P. S. de Laplace to H. Poincaré, and ... G.ukaszewicz Dissipative dynamical systems and their attractors Qolloquium 3 / 14 Mech. of continuous media ∼ dissipative system of PDEs in a Hilbert phase space system irreversible in time ∼ semigroups ∼ innite dimensional dynamical systems ∼ deterministic chaos Since O. Ladyzhenskaya's papers on the NSEs (∼ 1970) Evolution of a dissipative system Example 2. Motivation: How does turbulence in uids develop? G.ukaszewicz Dissipative dynamical systems and their attractors Qolloquium 4 / 14 Evolution of a dissipative system Example 2. Motivation: How does turbulence in uids develop? Mech. of continuous media ∼ dissipative system of PDEs in a Hilbert phase space system irreversible in time ∼ semigroups ∼ innite dimensional dynamical systems ∼ deterministic chaos Since O. Ladyzhenskaya's papers on the NSEs (∼ 1970) G.ukaszewicz Dissipative dynamical systems and their attractors Qolloquium 4 / 14 Let us compare the above problems Example 1.
    [Show full text]
  • Energy Cycle for the Lorenz Attractor
    Energy cycle for the Lorenz attractor Vinicio Pelino, Filippo Maimone Italian Air Force, CNMCA Aeroporto “De Bernardi”, Via di Pratica Di Mare, I-00040 Pratica di Mare (Roma) Italy In this note we study energetics of Lorenz-63 system through its Lie-Poisson structure. I. INTRODUCTION In 1955 E. Lorenz [1] introduced the concept of energy cycle as a powerful instrument to understand the nature of atmospheric circulation. In that context conversions between potential, kinetic and internal energy of a fluid were studied using atmospheric equations of motion under the action of an external radiative forcing and internal dissipative processes. Following these ideas, in this paper we will illustrate that chaotic dynamics governing Lorenz-63 model can be described introducing an appropriate energy cycle whose components are kinetic, potential energy and Casimir function derived from Lie-Poisson structure hidden in the system; Casimir functions, like enstrophy or potential vorticity in fluid dynamical context, are very useful in analysing stability conditions and global description of a dynamical system. A typical equation describing dissipative-forced dynamical systems can be written in Einstein notation as: x&ii=−Λ+{xH, } ijji x f i = 1,2...n (1) Equations (1) have been written by Kolmogorov, as reported in [2], in a fluid dynamical context, but they are very common in simulating natural processes as useful in chaos synchronization [3]. Here, antisymmetric brackets represent the algebraic structure of Hamiltonian part of a system described by function H , and a cosymplectic matrix J [4], {}F,G = J ik ∂ i F∂ k G . (2) Positive definite diagonal matrix Λ represents dissipation and the last term f represents external forcing.
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]
  • Strong Chaos Without Butter Y E Ect in Dynamical Systems with Feedback
    Strong Chaos without Buttery Eect in Dynamical Systems with Feedback 1 2 3 Guido Boetta Giovanni Paladin and Angelo Vulpiani 1 Istituto di Fisica Generale Universita di Torino Via PGiuria I Torino Italy 2 Dipartimento di Fisica Universita del lAquila Via Vetoio I Coppito LAquila Italy 3 Dipartimento di Fisica Universita di Roma La Sapienza Ple AMoro I Roma Italy ABSTRACT We discuss the predictability of a conservative system that drives a chaotic system with p ositive maximum Lyapunov exp onent such as the erratic motion of an asteroid 0 in the gravitational eld of two b o dies of much larger mass We consider the case where in absence of feedback restricted mo del the driving system is regular and completely predictable A small feedback of strength still allows a go o d forecasting in the driving system up to a very long time T where dep ends on the details of the system p The most interesting situation happ ens when the Lyapunov exp onent of the total system is strongly chaotic with practically indep endent of Therefore an exp onential tot 0 amplication of a small incertitude on the initial conditions in the driving system for any co exists with very long predictability times The paradox stems from saturation eects in the evolution for the growth of the incertitude as illustrated in a simple mo del of coupled maps and in a system of three p oint vortices in a disk PACS NUMBERS b It is commonly b elieved that a sensible dep endence on initial condition makes foreca sting imp ossible even in systems with few degrees
    [Show full text]
  • Frequently Asked Questions About Nonlinear Science J.D
    Frequently Asked Questions about Nonlinear Science J.D. Meiss Sept 2003 [1] About Sci.nonlinear FAQ This is version 2.0 (Sept. 2003) of the Frequently Asked Questions document for the newsgroup s ci.nonlinear. This document can also be found in Html format from: http://amath.colorado.edu/faculty/jdm/faq.html Colorado, http://www-chaos.engr.utk.edu/faq.html Tennessee, http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html England, http://www.sci.usq.edu.au/mirror/sci.nonlinear.faq/faq.html Australia, http://www.faqs.org/faqs/sci/nonlinear-faq/ Hypertext FAQ Archive Or in other formats: http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.pdf PDF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.rtf RTF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.tex old version in TeX, http://www.faqs.org/ftp/faqs/sci/nonlinear-faq the FAQ's site, text version ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/nonlinear-faq text format. This FAQ is maintained by Jim Meiss [email protected]. Copyright (c) 1995-2003 by James D. Meiss, all rights reserved. This FAQ may be posted to any USENET newsgroup, on-line service, or BBS as long as it is posted in its entirety and includes this copyright statement. This FAQ may not be distributed for financial gain. This FAQ may not be in cluded in commercial collections or compilations without express permission from the author. [1.1] What's New? Fixed lots of broken and outdated links. A few sites seem to be gone, and some new sites appeare d.
    [Show full text]
  • Some Examples in Ergodic Theory
    SOME EXAMPLES IN ERGODIC THEORY By YAEL NAIM DOWKER a& PBUL ERDBS [Received 22 February 1958.-Read 20 March 19581 Introduction THE purpose of this paper is to give some counter examples in ergodic theory. Some of the examples answer questions raised previously by some authors, some are counter examples to published theorems later withdrawn and some answer questions entertained privat,ely by the authors of this note. Let Y be an abstract space, y a a-field of subsets of Y and m a measure defined on y. We shall assume throughout that (Y, y, m) is a o-finite, non- atomic measure space. Let T be a 1: 1, ergodic measure preserving trans- formation of Y on to itself. Examples 1 and 2 deal with the case where (Y, y, a) is a finite measure space (in fact it is t,he ordinary line segment [O, 1). Example 1 shows that one cannot generalize theorems in Diophantine approximation t,heory like Theorem 2 (12, 76) to arbitrary ergodic conservative systems on [0, 1). Example 2 answers completely the questions raised by Halmos (7) and more recently by St’andish (13) concerning non-homogeneous ergodic theorems. We put and where fa(y) is the characteristic function of A. Examples 3, 4, and 5 and Theorems 1 and 2 deal with the case where m(Y) is infinite and study the relative behaviour of two ergodic measure preserving transformations T1 and T2. Example 3 shows that contrary to some expectations (1,2) the sequence -K(A, % $9 Y) = JfAG 4 Y)/~,(~, 4 Y) need not converge p.p.
    [Show full text]
  • Activated Random Walks Arxiv:1507.04341V1 [Math.PR] 15 Jul 2015
    Activated Random Walks∗ Leonardo T. Rolla Contents 1 Overview 2 1.1 The Activated Random Walk reaction-diffusion model . .2 1.2 Infinite conservative system, fixation, phase transition . .3 1.3 Physical motivation: “self-organized criticality” . .3 1.4 Predictions . .4 1.5 Difficulties . .5 1.6 Results . .5 2 Definitions and Diaconis-Fulton construction 8 2.1 Notation . .8 2.2 Diaconis-Fulton construction . .9 3 One-dimensional counting arguments 13 4 Exploring the instructions in advance 16 5 Higher-dimensional arguments 19 6 A multi-scale argument 24 7 Arguments using particle-wise constructions 27 arXiv:1507.04341v1 [math.PR] 15 Jul 2015 7.1 Preliminaries . 27 7.2 Results . 29 References 32 ∗Minicourse given at the workshop “Activated Random Walks, DLA, and related topics” at IMéRA- Marseille, March 2015. Preliminary version. July 15, 2015. 1 1 Overview 1.1 The Activated Random Walk reaction-diffusion model The Activated Random Walk model is defined as follows. Particles sitting on the graph Zd can be in state A for active or S for passive. Each active particle, that is, each particle in the A state, performs a continuous-time random walk with jump rate DA = 1 and with translation-invariant jump distribution given by a probability p(·) on Zd. Several active particles can be at the same site, and they do not interact among themselves. When a particle is alone, it may become passive, a transition denoted by A → S, which occurs at a sleeping rate 0 < λ 6 ∞. In other words, each particle carries two clocks, one for jumping and one for sleeping.
    [Show full text]
  • Spontaneous Breakdown of the Time Reversal Symmetry
    Article Spontaneous Breakdown of the Time Reversal Symmetry Janos Polonyi Department of Physics and Engineering, Strasbourg University, CNRS-IPHC, 23 rue du Loess, BP28 67037 Strasbourg Cedex 2, France; [email protected]; Tel.:+33-(0)3-88-10-62-91 Academic Editor: Martin-Isbjörn Trappe Received: 23 December 2015; Accepted: 13 April 2016; Published: 20 April 2016 Abstract: The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized e-prescription where the non-uniform convergence of the limit e ! 0 leaves behind a spontaneous breakdown of the time reversal symmetry. Keywords: symmetry breaking; non-uniform convergence; irreversibility; causality 1. Introduction One usually observes a system in interaction with its environment. The laws, discovered in such a manner are highly complex; they describe an open, effective rather than conserved, closed dynamics. Some simplifications may take place when we bring the environment into the thermodynamical limit and the complicated, open features of the dynamics are reduced to irreversibility and dissipative forces. The emergence of the ensuing dynamical breakdown of the time reversal invariance is the subject of this work. The number of the dynamical variables of a large environment is controlled by an artificially introduced quantity, the cutoff, and its removal defines the infinitely large environment.
    [Show full text]
  • Touch Here to Download Free
    The Vajra Sequence A PSYCHEDELIC JOURNEY INTO THE HEART OF RELIGION _____________________ Richard Merrick ___________________________________________________ For Sherolyn Copyright © 2016 by Richard Merrick. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by an information storage and retrieval system—except by a reviewer who may quote brief passages in a review to be printed in a magazine or newspaper—without permission in writing from the publisher. ISBN: 978-1-5323-2678-3 Printed in the United States of America Contents Chapter 1: A Cabinet of Curiosities .............................................. 1 Chapter 2: The Vegetation Gods ................................................ 14 Chapter 3: The Radcliffe Camera ............................................... 24 Chapter 4: The Egersis ............................................................... 35 Chapter 5: What is Real? ............................................................ 47 Chapter 6: Of Seven Heavens ..................................................... 57 Chapter 7: Coding the Sequence ................................................. 73 Chapter 8: Enter The Vegetalismo .............................................. 83 Chapter 9: The Entheotech Theory ............................................. 93 Chapter 10: Council of the Twelve Apostles ............................. 104 Chapter 11: Receiving the Vajra-kīla .......................................
    [Show full text]
  • Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System
    entropy Article Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System Ze Wang and Guoyuan Qi * Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, Tiangong University, Tianjin 300387, China; [email protected] * Correspondence: [email protected] Abstract: In this paper, a three-terminal memristor is constructed and studied through changing dual-port output instead of one-port. A new conservative memristor-based chaotic system is built by embedding this three-terminal memristor into a newly proposed four-dimensional (4D) Euler equation. The generalized Hamiltonian energy function has been given, and it is composed of conservative and non-conservative parts of the Hamiltonian. The Hamiltonian of the Euler equation remains constant, while the three-terminal memristor’s Hamiltonian is mutative, causing non- conservation in energy. Through proof, only centers or saddles equilibria exist, which meets the definition of the conservative system. A non-Hamiltonian conservative chaotic system is proposed. The Hamiltonian of the conservative part determines whether the system can produce chaos or not. The non-conservative part affects the dynamic of the system based on the conservative part. The chaotic and quasiperiodic orbits are generated when the system has different Hamiltonian levels. Lyapunov exponent (LE), Poincaré map, bifurcation and Hamiltonian diagrams are used to analyze the dynamical behavior of the non-Hamiltonian conservative chaotic system. The frequency and initial values of the system have an extensive variable range. Through the mechanism adjustment, instead of trial-and-error, the maximum LE of the system can even reach an incredible value of 963. An analog circuit is implemented to verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos.
    [Show full text]