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An episode of the podcast produced by The Last Alliance: University of Alberta Tolkien Society discussing the works of J.R.R. Tolkien.

In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov's convex regularization and via differential inclusions, the underlying discontinuous initial value problem is first recast in terms of a set-valued problem and hence it is continuously approximated by using Cellina's Theorem for differential inclusions. For chaos control, an active control technique is implemented so that the unstable equilibria become stable. As example, Shimizu--Morioka's system is considered. Numerical simulations are obtained by means of the Adams-Bashforth-Moulton method for differential equations of fractional-order.

In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov's convex regularization and via differential inclusions, the underlying discontinuous initial value problem is first recast in terms of a set-valued problem and hence it is continuously approximated by using Cellina's Theorem for differential inclusions. For chaos control, an active control technique is implemented so that the unstable equilibria become stable. As example, Shimizu--Morioka's system is considered. Numerical simulations are obtained by means of the Adams-Bashforth-Moulton method for differential equations of fractional-order.

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We study the regular or chaotic nature of orbits in a 3D potential describing a triaxial galaxy surrounded by a spherical dark halo component. Our numerical calculations show, that the percentage of chaotic orbits decreases exponentially, as the mass of the dark halo increases. A linear increase of the percentage of the chaotic orbits was observed, as the scale length of the dark halo component increases. In order to distinguish between regular and chaotic character of orbits, we use the total angular momentum Ltot, as a new indicator. Comparison of this new dynamical parameter, with other, previously used chaos indicators, shows that the Ltot gives very fast and reliable results in order to detect the character of orbits in galactic potentials.

The dynamics of glacial cycles is studied in terms of the dynamical systems theory. We explore the dependence of the climate state on the phase of astronomical forcing by examining five conceptual models of glacial cycles proposed in the literature. The models can be expressed as quasiperiodically forced dynamical systems. It is shown that four of them exhibit a strange nonchaotic attractor (SNA), which is an intermediate regime between quasiperiodicity and chaos. Then, the dependence of the climate state on the phase of astronomical forcing is not given by smooth relations, but constitutes a geometrically strange set. Our result suggests that the dynamics of SNA is a candidate for the dynamics of glacial cycles, in addition to the well-known dynamics of quasiperiodicity and chaos.

The dynamics of glacial cycles is studied in terms of the dynamical systems theory. We explore the dependence of the climate state on the phase of astronomical forcing by examining five conceptual models of glacial cycles proposed in the literature. The models can be expressed as quasiperiodically forced dynamical systems. It is shown that four of them exhibit a strange nonchaotic attractor (SNA), which is an intermediate regime between quasiperiodicity and chaos. Then, the dependence of the climate state on the phase of astronomical forcing is not given by smooth relations, but constitutes a geometrically strange set. Our result suggests that the dynamics of SNA is a candidate for the dynamics of glacial cycles, in addition to the well-known dynamics of quasiperiodicity and chaos.

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Includes bibliographical references and index

A number of observations are made on Hofstadter's integer sequence defined by Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2)), for n > 2, and Q(1)=Q(2)=1. On short scales the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k-th generation has 2**k members which have ``parents'' mostly in generation k-1, and a few from generation k-2. In this sense the series becomes Fibonacci type on a logarithmic scale. The mean square size of S(n)=Q(n)-n/2, averaged over generations is like 2**(alpha*k), with exponent alpha = 0.88(1). The probability distribution p^*(x) of x = R(n)= S(n)/n**alpha, n >> 1, is well defined and is strongly non-Gaussian. The probability distribution of x_m = R(n)-R(n-m) is given by p_m(x_m)= lambda_m * p^*(x_m/lambda_m). It is conjectured that lambda_m goes to sqrt(2) for large m.

We study the phenomena at the overlap of quantum chaos and nonclassical statistics for the time-dependent model of nonlinear oscillator. It is shown in the framework of Mandel Q-parameter and Wigner function that the statistics of oscillatory excitation number is drastically changed in order-to chaos transition. The essential improvement of sub-Poissonian statistics in comparison with an analogous one for the standard model of driven anharmonic oscillator is observed for the regular operational regime. It is shown that in the chaotic regime the system exhibits the range of sub- and super-Poissonian statistics which alternate one to other depending on time intervals. Unusual dependence of the variance of oscillatory number on the external noise level for the chaotic dynamics is observed.

Alan Watt CTTM LIVE on RBN 794 World In Chaos Terrifying To Some Radiation And Wars Order To Come Mar 22, 2011 March 22, 2011 Alan Watt "Cutting Through The Matrix" LIVE on RBN: Poem Copyright Alan Watt March 22, 2011: World in Chaos, Terrifying to Some, Radiation and Wars, Order to Come: "World's Elite are Gobbling Up Nations, Stomachs Ravenous, Growling Sensations, Salivating Wolves, Running as a Pack, Their New World Order is on Fast Track, Each Citizen Suspect, Total Info Collected In Case Some Awake, Watched, Suspected, For Who Knows Where Truth May Lead? Pollination of Minds Can Produce Seed, Those Dead in Spirit Think All is Well, Deaf to the Roaring of Masters of Hell" © Alan Watt March 22, 2011

Alan Watt CTTM LIVE on RBN 794 World In Chaos Terrifying To Some Radiation And Wars Order To Come Mar 22, 2011 March 22, 2011 Alan Watt "Cutting Through The Matrix" LIVE on RBN: Poem Copyright Alan Watt March 22, 2011: World in Chaos, Terrifying to Some, Radiation and Wars, Order to Come: "World's Elite are Gobbling Up Nations, Stomachs Ravenous, Growling Sensations, Salivating Wolves, Running as a Pack, Their New World Order is on Fast Track, Each Citizen Suspect, Total Info Collected In Case Some Awake, Watched, Suspected, For Who Knows Where Truth May Lead? Pollination of Minds Can Produce Seed, Those Dead in Spirit Think All is Well, Deaf to the Roaring of Masters of Hell" © Alan Watt March 22, 2011

The energies and wave functions of stationary many-body states are analyzed to look for the signatures of quantum chaos. Shell model calculations with the Wildenthal interaction are performed in the $J-T$ scheme for 12 particles in the $sd$-shell. The local level statistics are in perfect agreement with the GOE predictions. The analysis of the amplitudes of the eigenvectors in the shell model basis with the aid of the informational entropy and moments of the distribution function shows evidence for local chaos with a localization length reaching 90\% of the total dimension in the middle of the spectrum. The degree of chaoticity is sensitive to the the strength of the residual interaction as compared to the single particle energy spacing.

Transient chaos is an ubiquitous phenomenon characterizing the dynamics of phase space trajectories evolving towards a steady state attractor in physical systems as diverse as fluids, chemical reactions and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems or even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability ($k$-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate $\kappa$, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution.We show that for a given density of constraints and fixed number of Boolean variables $N$, the hardness of formulas in random $k$-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints $\alpha$, hardness appears through a second-order phase transition at $\alpha_{\chi}$ in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter $\kappa \sim N^{B|\alpha - \alpha_{\chi}|^{1-\gamma}}$ with $0 < \gamma < 1$. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints $\alpha$ is increased.

All inhabitants of this universe, from galaxies to people, are finite. Yet the universe itself is often assumed to be infinite. If instead the universe is topologically finite, then light and matter can take chaotic paths around the compact geometry. Chaos may lead to ordered features in the distribution of matter throughout space.

All inhabitants of this universe, from galaxies to people, are finite. Yet the universe itself is often assumed to be infinite. If instead the universe is topologically finite, then light and matter can take chaotic paths around the compact geometry. Chaos may lead to ordered features in the distribution of matter throughout space.

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This paper is devoted to a discussion of the Discrete Fourier Transform (DFT) representation of a chaotic finite-duration sequence. This representation has the advantage that is itself a finite-duration sequence corresponding to samples equally spaced in the frequency domain. The Fast Fourier Transform (FFT) algoritm allows us an effective computation, and it can be applied to a relatively short time series. DFT representation requirements were analized and applied for determining the order-chaos transition in a nonlinear system described by the equation $x[n+1]=rx[n](1-x[n])$. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent for the time series set, obtained from the logistic equation.

We use the Smaller Alignment Index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behavior of the SALI for the two cases: the index fluctuates around non--zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behavior for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents $\sigma_1$, $\sigma_2$ i.e. $SALI \propto e^{-(\sigma_1-\sigma_2)t}$. Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of 2 and 3 degrees of freedom.

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Tracklist: 1. Sideways 2. All I Want From You Is Love 3. Icicles 4. It's As Clear 5. Too Many Stars 6. We Get Along 7. Better Off 8. This Rope Is Long 9. Heart Condition 10. This Much 11. Come Home Safely

18 September 2023 - David Clements live 6PM EST - The evening Apollo & David discuss the rampant crime that has overtaken our country and the judicial system's reward for said crime. Recently, Illinois became the first state to remove cash bail, which could enable criminals to more easily receive a \"get out of jail free\" card. In addition, the pair discuss the distressing, disappointing, and quite frankly ridiculous situation of the missing F-35. Apollo and David urge the people to not lose hope and to continue fighting for the things and mechanisms that will make our country great again! Happy Monday, let's make this a great week! Follow us on Social Media: https://libertylinks.io/ConservativeDailyhttps://libertylinks.io/JoeOltmannhttps://libertylinks.io/Apollo HEY! SAY NO to VOTING MACHINES! Send this Message to Congress and let them know We MUST BAN the Voting Machines! https://conservative-daily.com/election/ban-the-voting-machines-act-now-congress-must-ban-the-voting-machines-before-the-2024-election 

Chris Downs - Order in Chaos

Chris Downs - Order in Chaos

Chris Downs - Order in Chaos

239 pages 26 cm

The quantitative determination of the order-chaos transition in a nonlinear dynamical system described by H\'enon mapping defined as $x[n+1]=1.0-A*x[n]^2+B*y[n], y[n+1]=B*x[n]$, where $B=0.3$, and $A$ is an adjustable control parameter, was made. This was achieved by applying the Inverse Average Magnitude-Squared Coherence Index (IAMSCI). This method is based on the Welch average periodogram technique and it has the advantage respect to nonlinear dynamical methods that it may be applied to any stationary signal by using discrete Fourier transform (DFT) representation which allows to operate on a short discrete-time series. Its effectiveness was demonstrated by comparing the results obtained by applying IAMSCI method with those obtained by calculating the largest Lyapounov exponent (LLE), both applied to the same discrete-time series set derived from the H\'enon mapping.

Coexistence of various ordered chaotic states in a Hamiltonian system is studied with the use of a symplectic coupled map lattice. Besides the clustered states for the attractive interaction, a novel chaotic ordered state is found for a system with repulsive interaction, characterized by a dispersed state of particles. The dispersed and clustered states form an onion-like structure in phase space. The degree of order increases towards the center of the onion, while chaos is enhanced at the edge between ordered and random chaotic states. For a longer time scale, orbits itinerate over ordered and random states. The existence of these ordered states leads to anomalous long-time correlation for many quantifiers such as the global diffusion.

Bars in galaxies are mainly supported by particles trapped around closed periodic orbits. These orbits respond to the bar's forcing frequency only and lack free oscillations. We show that a similar situation takes place in double bars: particles get trapped around orbits which only respond to the forcing from the two bars and lack free oscillations. We find that writing the successive positions of a particle on such an orbit every time the bars align generates a closed curve, which we call a loop. Loops allow us to verify consistency of the potential. As maps of doubly periodic orbits, loops can be used to search the phase-space in double bars in order to determine the fraction occupied by ordered motions.

Bars in galaxies are mainly supported by particles trapped around closed periodic orbits. These orbits respond to the bar's forcing frequency only and lack free oscillations. We show that a similar situation takes place in double bars: particles get trapped around orbits which only respond to the forcing from the two bars and lack free oscillations. We find that writing the successive positions of a particle on such an orbit every time the bars align generates a closed curve, which we call a loop. Loops allow us to verify consistency of the potential. As maps of doubly periodic orbits, loops can be used to search the phase-space in double bars in order to determine the fraction occupied by ordered motions.

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We investigate numerically conditions for order and chaos in the dynamics of an interacting Bose- Einstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size.

Book Source: Digital Library of India Item 2015.46952 dc.contributor.author: Cole, G. D. H. dc.date.accessioned: 2015-06-26T12:35:39Z dc.date.available: 2015-06-26T12:35:39Z dc.date.digitalpublicationdate: 2006-05-15 dc.date.citation: 1920 dc.identifier.barcode: 4990010212865 dc.identifier.origpath: /data3/upload/0067/017 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/46952 dc.description.scanningcentre: C-DAK, Kolkata dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 310 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Methuen And Co., London dc.rights: In Public Domain dc.source.library: State Central Library, Kolkata dc.subject.classification: Technology dc.subject.classification: Various Industries, Trades And Crafts dc.subject.classification: Industries Based On Various Processable Materials dc.subject.keywords: Coal Industry dc.subject.keywords: Industry - Motives dc.title: Chaos And Order In Industry

We revisit the equilibrium one-dimensional $\phi^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These "computationally stable" sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles' coordinates and momenta are either identical or differ only in sign. "Positive Lyapunov exponents" can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

We revisit the equilibrium one-dimensional $\phi^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These "computationally stable" sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles' coordinates and momenta are either identical or differ only in sign. "Positive Lyapunov exponents" can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

Alan Watt~Cutting Through the Matrix~1748~Nov 24, 2019 Nov. 24, 2019 "Cutting Through the Matrix" with Alan Watt (Blurb, i.e. Educational Talk) Global Order Out of Chaos: "Stuck in a Time Loop of Bolshevist Replays, Rent-a-Mob Anarchists, Civilization to Raze." © Alan Watt Nov. 24, 2019

We have proven using renormalization theory that coupled nonlinear systems exhibit universal behavior as the strength parameter and coupling parameters are varied. This qualitative as well as quantitative universality has a wide range of applicability in particular the limits of parameter ranges where free electron lasers can produce coherent waves (no chaos in electron motion) has also been studied and the results published.

This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of 'order' less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of 'order' less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

This report studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. Varying the total system order incrementally from 2.6 to 3.7 demonstrates that systems of 'order' less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable.

We investigate the transition from regular to chaotic motion in a composite galaxy model with a disk-halo, a massive dense nucleus and a dark halo component. We obtain relationships connecting the critical value of the mass of the nucleus or the critical value of the angular momentum, with the mass of the dark halo, where the transition from regular motion to chaos occurs. We also present 3D diagrams connecting the mass of nucleus the energy and the percentage of stars that can show chaotic motion. The fraction of the chaotic orbits observed in the phase plane, as a function of the mass of the dark halo is also computed. We use a semi-numerical method, that is a combination of theoretical and numerical procedure. The theoretical results obtained using the version 8.0 of the Mathematica package, while all the numerical calculations were made using a Bulirsch-Stoer FORTRAN routine in double precision. The results can be obtained in semi-numerical or numerical form and give good description for the connection of the physical quantities entering the model and the transition between regular and chaotic motion. We observe that the mass of the dark halo, the mass of the dense nucleus and the L_z component of the angular momentum, are important physical quantities, as they are linked to the regular or chaotic character of orbits in disk galaxies described by the model. Our numerical experiments suggest, that the amount of the dark matter plays an important role in disk galaxies represented by the model, as the mass of the halo affects, not only the regular or chaotic nature of motion but it is also connected with the existence of the different families of regular orbits. Comparison of the present results with earlier work is also presented.

The theory of the inverse problem is used in order to find a two dimensional galactic potential generating a mono-parametric family of elliptic periodic orbits. The potential is made up of a two-dimensional harmonic oscillator with perturbing terms of third and fourth degree and can be considered to describe local motion in the central parts of a barred galaxy. Our numerical calculations indicate that the potential produces also other families of orbits and there are cases, where distinct chaotic components are also observed. The numerical experiments suggest that there are regular as well as chaotic orbits supporting the barred structure. We use a variety of dynamical indicators, in order to determine the character of motion. Of significant interest, is the S(g) dynamical spectrum, which is used in order to identify the islandic motion of resonant orbits and also proves to be a very useful and fast indicator, in order to distinguish between regular and chaotic motion. Comparison with other methods for detecting chaos is also discussed. The present results are compared with outcomes from earlier work.

Alan Watt CTTM LIVE on RBN 794 World In Chaos Terrifying To Some Radiation And Wars Order To Come Mar 22, 2011 March 22, 2011 Alan Watt "Cutting Through The Matrix" LIVE on RBN: Poem Copyright Alan Watt March 22, 2011: World in Chaos, Terrifying to Some, Radiation and Wars, Order to Come: "World's Elite are Gobbling Up Nations, Stomachs Ravenous, Growling Sensations, Salivating Wolves, Running as a Pack, Their New World Order is on Fast Track, Each Citizen Suspect, Total Info Collected In Case Some Awake, Watched, Suspected, For Who Knows Where Truth May Lead? Pollination of Minds Can Produce Seed, Those Dead in Spirit Think All is Well, Deaf to the Roaring of Masters of Hell" © Alan Watt March 22, 2011

We investigate the properties of motion in a map model derived from a galactic Hamiltonian made up of perturbed elliptic oscillators. The phase space portrait is obtained in all three different cases using the map and numerical integration of the equations of motion. Our numerical calculations suggest that the map describes very well the properties of regular motion in all three cases. There are cases where the map fails to describe chaotic motion and cases where the map describes satisfactorily the chaotic phase plane. We also compare the Lyapunov characteristic number and the spectra of orbits derived by the map and numerical integration in each case. The agreement of the outcomes is satisfactory.

We investigate the properties of motion in a map model derived from a galactic Hamiltonian made up of perturbed elliptic oscillators. The phase space portrait is obtained in all three different cases using the map and numerical integration of the equations of motion. Our numerical calculations suggest that the map describes very well the properties of regular motion in all three cases. There are cases where the map fails to describe chaotic motion and cases where the map describes satisfactorily the chaotic phase plane. We also compare the Lyapunov characteristic number and the spectra of orbits derived by the map and numerical integration in each case. The agreement of the outcomes is satisfactory.

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