Chaotic Dynamics

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 study an influence of nonlinear dissipation and external perturbations onto transition scenarious to chaos in Lorenz-Haken system. It will be show that varying in external potential parameters values leads to parameters domain formation of chaos realization. In the modified Lorenz-Haken system transitions from regular to chaotic dynamics can be of Ruelle-Takens scenario, Feigenbaum scenario, or through intermittency.

We propose a theory of deterministic chaos for discrete systems, based on their representations in binary state spaces $ \Omega $, homeomorphic to the space of symbolic dynamics. This formalism is applied to neural networks and cellular automata; it is found that such systems cannot be viewed as chaotic when one uses the Hamming distance as the metric for the

According to a previous conjecture, spatial and temporal Lyapunov exponents of chaotic extended systems can be obtained from derivatives of a suitable function, the entropy potential. The validity and the consequences of this hypothesis are explored in detail. The numerical investigation of a continuous-time model provides a further confirmation to the existence of the entropy potential. Furthermore, it is shown that the knowledge of the entropy potential allows determining also Lyapunov spectra in general reference frames where the time-like and space-like axes point along generic directions in the space-time plane. Finally, the existence of an entropy potential implies that the integrated density of positive exponents ͑Kolmogorov-Sinai entropy͒ is independent of the chosen reference frame. © 1997 American Institute of Physics. ͓S1054-1500͑97͒02704-3͔ Out-of-equilibrium systems give rise to a rich variety of phenomena that are often dealt with using ad hoc methods. Perhaps the main...

The drainage of ice-dammed lakes produces floods that can pose hazards, waste water resources and modulate ice flow. In this thesis I investigate several aspects of ice-dammed lake drainage through the development and analysis of mathematical models.
After an introduction in the first chapter and a description of the mathematical background to the thesis in the second, the third chapter investigates the mechanisms behind observed variability in the size and timing of subglacial floods from ice-dammed lakes. In particular, I examine how environmental controls like the weather and the shape of glaciers affect floods.

In the next chapter, I quantify how well simple models can predict the dates of floods from an ice-marginal lake in Kyrgyzstan. I find that incorporating environmental controls into models improves their prediction ability.

Next I investigate the coupling between subglacial drainage and glacier motion during ice-dammed lake drainage by developing and analysing a model which couples a marginal lake, glacier sliding, subglacial drainage through a channel and subglacial drainage through a distributed system of cavities. I show how changes in lake level cause the rate at which a glacier slides to increase during the first half of floods and decrease during the second half.

The next two chapters are concerned with two lake-drainage scenarios that involve water flowing as an open stream: firstly, the subglacial open-channel flow that occurs after a marginal lake drains completely during a flood, and secondly, the drainage of supraglacial lakes across the surface of ice sheets.

I end the thesis with a summary of my findings and some suggestions of theoretical and field-based investigations that are worthwhile pursing in the future.

We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of toy model used for climate studies. The system is subjected to both temporal and spatiotemporal perturbations. Through the analysis of the system's time evolution and its time correlations, we have obtained numerical evidence for two stochastic resonance-like behaviors.

The essence of the pendulum problem in analytical mechanics is a brilliant case of study for theoretical physicists and a base ground for other cases of Lagrangian problems. And by Lagrangians we can find the forces acting on the entire system. This paper has an approach for a generalized formula for an theoretical pendulum with n mass , n length and n angles .

"Patterns in War Dynamics. WARning 2020". In this study, complexity and network science are applied to the dynamics and development of the (International) System. This study shows that the System periodically becomes critical and produces "systemic wars". The typical dynamics that are revealed in this study are a consequence of laws of physics that apply also to the System. Systemic wars are instrumental in rebalancing the System, and in periodically producing "upgraded" international orders that allow for further growth and development. The patterns that can be observed in the war dynamics of the System, make it possible to predict these dynamics and the System's direction of development.
Data-analysis shows that the System will again become critical around 2020 (+/- two years).
This study shows that intensifying war dynamics pose an existential threat to human kind: Systemic wars will cause increasing damage and suffering and will put our climate system at (additional) risk: Fundamental change is a prerequisite for our survival.
This study was initially published in 2016. The publication "On the Thermodynamics of War and Social Evolution" (2019) provides a scientific explanation for the war dynamics that can be observed.

The study On the Thermodynamics of War and Social Evolution, shows that patterns can be identified in the war dynamics of the System, and that a relationship exists between these war dynamics and social evolution. The research suggests that Prigogine’s idea about non-equilibrium systems being able to attain highly ordered states in response to an increase of energy flux, can also be successfully applied to the social sciences. These new insights could have profound implications for our understanding of war (dynamics), and for our ability to better control and prevent war, in the future.I argue that the System can be considered a non-equilibrium system, and that the (relationship between) war dynamics - and the patterns they produce - and social evolution, can be explained from a (non-equilibrium) thermodynamic perspective: Interactions between components of the System (individual humans, communities, societies, states, etc.) are irreversible, and result in the production of entropy - tensions - in the System.
These tensions (entropy) serve as a source of order and are regulated by means of a dissipative structure that also puts kinetic activity (war) to use, to ensure the most efficient path to thermodynamic equilibrium of the System.

IT'S HERE NOW!!! 
IT’S GOING TO BE REAL STRANGE.  PEOPLE MUST NOT BE AFRAID, IN FACT, ALL PEOPLE NEED TO BE REAL HAPPY ABOUT THIS EVENT AND BE READY TO GET OUTSIDE. GROUNDED.  THIS IS MEANT TO CLEAN ALL OF THE POLLUTION AND PROBABLY EVEN THE RADIATION FROM THE PLANET.

SOLID EVIDENCE OF THE INCOMING WAVE

THE BIRTH OF CONSCIOUSNESS IS BEING OBSERVED:

ALL LIFE IS BORN OF CHAOTIC IDIOCY AS SHOULD BE OBVIOUS BY THE SHIT SHOW THAT HAS BEEN GOING ON FOR WAY TOO LONG.

SO, WHEN WE FINALLY BREAK OUT OF THE IDIOCY BORN IN DARKNESS - THE NEW UPGRADED IDIOCY WILL BE IN THE EUPHORIC PURPLISH GRAY BOUNDARY LAYER.

In the last decade, chaos has emerged as a new promising candidate for cryptography because many chaos fundamental characteristics such as a broadband spectrum, ergodicity, and high sensitivity to initial conditions are directly connected with two basic properties of good ciphers: confusion and diffusion. In this chapter we recount some of the saga undergone
by this field; we review the main achievements in the field of chaotic cryptography, starting with the definition of chaotic systems and their properties and the difficulties it has to outwit. According to their intrinsic dynamics, chaotic cryptosystems are classified depending on whether the system is discrete or continuous. Due to their simplicity and rapidity the discrete chaotic systems based on iterative maps have received a lot of attention. In spite of the significant achievements accomplished in this field, there are still many problems, basically speed, that restrict the application of existing encoding/decoding algorithms to real systems. The major advantages and drawbacks of the most popular chaotic map ciphers in terms of security and computational cost are analyzed. The most significant cryptanalytic techniques are considered and applied for testing the security of some chaotic algorithms. Finally, future trends in the development of this topic are discussed.

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

Modular algebra is widely used for synthesis of pseudo-random number generators and in data cryptography. However, electronic components to carry out the operations of this algebra remains quite rare on market. In order to facilitate the design of random number generators, we have intention in this work to design schematic electronic circuit able to calculate modulo 5 an 7 of positive integers encoded to  8 -bits (between 0-256). The approach presented  in this work has a particular interest in that it contrasts with the others algorithmic methods which present a problem of speed in terms of computations.

https://1drv.ms/w/s!Aik6CvaAE1b8jxYjJRcHw1WtKzOH?e=Giz5YK

I'd be a giddy 'jack-off',
but for the
social-cultural repressions
ANY GRATUITIES OF APPRECIATION FOR THE FOLLOWING
MAY BE LEFT AT THE QR-CODE BELOW



As I mused about my near-anarchist, chaos-organized, guardianee-grand daughter and her crew, I had that Godfather-II Final Scene of Michael Corleone sitting by himself outdoors with a look of retrospective musing of his life.

If his mood WAS, as I’m here wondering, also, ‘Where did it all go off track?’, as far as my disposition from being a happy jack-off, who goofed around with anyone…
To this more glum, sober-as-a-judge persona.
And I know what it was. It was from me learning that not only my cavalier-Devil doesn’t care..
-laissez faire-”Wherever you are ‘chaos’ reigns” mischievousness- was tempered first, by my profiling and signifying contemporaries who postured they were above and looked down upon anyone who didn’t take their “gravitas” of themselves seriously, or who wasn’t one whom they’d use as a proponent of their distorted and narcissistic egos and credos of gravitas.

Secondly from those slights upon my mannerisms, I saw first hand how others' more vulnerable states and statuses were taken advantage of by those of authority. Those more vulnerable were used indifferently and with cynical impunity. That corresponded to the third hand accounts about abuses, repressions, and oppressions on the micro-to-macro dimensions in other places.

I could not be indifferent to that, though I had the academic class opportunity to be that detached. But I participated in the second and third hand enabling of the systems and mechanism that enabled what I’d seen first hand and learned or heard about second-hand.

This took decades for me to realize then fully grasp, mentally: What was the  celebration of life in-the-moment for one’s free will was seen, contemporaneously as a materialism of privileged entitlement of formal or perceived status. Since I had forsworn any titular authority and was not associated with those who had such, I was one of the many who shared Franz Fanon’s appellation, the ‘Wretched of the Earth’. These wretched-deplorable and despicable-exist together in the social culture no better than a crowded crab-basket.

Just as those crabs jockey for position for their subjective vision of thriving survival, so are we doing a similar exercise in our metaphorical matrix of a crab basket.

Though I managed to detach myself from the hyper-jousting in this thrive-to-survive matrix for the faux goals of some ersatz, materialistic utopia of licensed indulgence and satiation, I was still affected by the vibes from the dramas of people I knew, people with whom I was aware, and the general social cultural environment of hyper-dramas.

That and (and even more important as a concurrent mea culpa) my own naive, economic knowledge for myself, other than following in someone else's shoes who was a personal or cultural icon-which itself was distracted by a #36 draft number in the December ‘69 Selective Service Draft-left me totally intellectually and emotionally unprepared for the skills I COULD HAVE used proactively to change the course of what would be a 45-plus year narrative after graduation from college. Just as the Hebrews wandered for forty years, I did my forty-five-plus years in similar ways in the banal to sublime, or mostly, misadventures.

Having emerged to this level, presently, I’m being more the rhetorical provocateur, as well as the irregular disrupter of decorum of institutions and social mores out on the streets (and here on line). These are the manic domains to which I may once again “celebrate” the ethos of life-itself in its revelation of its serendipities and synchronicities through my wandering renegade spirit of experiences.

Forced oscillation of a system composed of two pendulums coupled by a spring in the presence of damping is investigated. In the steady state and within the small angle approximation we solve the system equations of motion and obtain the amplitudes and phases of in terms of the frequency of the sinusoidal driving force. The resonance frequencies are obtained and the amplitude ratio is discussed in details. Contrary to a single oscillator, in this two-degree of freedom system four resonant frequencies, which are close to mode frequencies, appear. Within the pass-band interval the system is shown to exhibit a rich and complicated behaviour. It is shown that damping crucially affects the system properties. Under certain circumstances, the amplitude of the oscillator which is directly connected to the driving force becomes smaller than the one far from it. Particularly we show the existence of a driving frequency at which the connected oscillator's amplitude goes zero.

We provide a scheme for the synchronization of two chaotic mobile robots when a mismatch between the parameter values of the systems to be synchronized is present. We have shown how meta-heuristic optimization can be used to adapt the parameters in two coupled systems such that the two systems are synchronized, although their behavior is chaotic and they have started with different initial conditions and parameter settings. The controlled system synchronizes its dynamics with the control signal in the periodic as well as chaotic regimes. The method can be seen also as another way of controlling the chaotic behavior of a coupled system. In the case of coupled chaotic systems, under the interaction between them, their chaotic dynamics can be cooperatively self-organized. A synergistic approach to meta-heuristic optimization search algorithm is developed. To avoid being trapped into local optimum and to enrich the searching behavior, chaotic dynamics is incorporated into the proposed search algorithm. A chaotic Levy flight is firstly incorporated in the proposed search algorithm for efficiently generating new solutions. And secondly, chaotic sequence and a psychology factor of emotion are introduced for move acceptance in the search algorithm. We illustrate the application of the algorithm by estimating the complete parameter vector of a chaotic mobile robot.

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.

In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit

We show how the condition of isochronicity can be studied for two dimensional systems in the renormalization group (RG) context. We find a necessary condition for the isochronicity of the Cherkas and another class of cubic systems. Our conditions are satisfied by all the cases studied recently by Bardet et al \cite{bard} and Ghose Choudhury and Guha

This chapter surveys work on dynamic heterogeneous agent models (HAMs) in economics and finance. Emphasis is given to simple models that, at least to some extent, are tractable by analytic methods in combination with computational tools. Most of these models are behavioral models with boundedly rational agents using different heuristics or rule of thumb strategies that may not be perfect,

We study the dynamical and statistical behavior of the Hamiltonian mean field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of N fully coupled rotators which shows a second-order phase transition. The solution in the canonical ensemble is briefly recalled and its predictions are tested numerically at finite N. The Vlasov stationary solution is shown to give the same consistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical simulations, A link is established between the behavior of the maximal Lyapunov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the N → ∞ limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with N, at least near the critical point. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics.

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain , with elastic reflections on the boundary; here , and . After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example . Playing an important role in