Chaos (Physics) Research Papers - Academia.edu (original) (raw)

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up... more

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the
field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is
a three-dimensional phenomenon.

Outline: The reality of Catholicism; The question of the development of science; Historical outlook at some transitional moments; When dogma meets science; Contemporary physics and the worldview of Catholicism; Awaiting a 'Grand... more

Outline: The reality of Catholicism; The question of the development of science; Historical outlook at some transitional moments; When dogma meets science; Contemporary physics and the worldview of Catholicism; Awaiting a 'Grand Narrative' and the final vision of harmony.

In this 607 page book, in Spanish, are described in clear and complete way several problems of statics, mechanics, kinematics, dynamics and analytical dynamics. Includes non conventional subjects like perturbation theory, Kepler problem... more

In this 607 page book, in Spanish, are described in clear and complete way several problems of statics, mechanics, kinematics, dynamics and analytical dynamics. Includes non conventional subjects like perturbation theory, Kepler problem in parabolic coordinates, and connection with quantum mechanics.

İnsan bilgisizliğinden dolayı merak eder, en temel ihtiyaçlarından biridir merak; ayrıca zayıflığından dolayı da güçlenmek, çevresine hakim olmak, biçim vermek ister. Özellikle batı toplumlarındaki bu güçlenmeci, hakimolucu, kullanıcı ve... more

İnsan bilgisizliğinden dolayı merak eder, en temel ihtiyaçlarından biridir merak; ayrıca zayıflığından dolayı da güçlenmek, çevresine hakim olmak, biçim vermek ister. Özellikle batı toplumlarındaki bu güçlenmeci, hakimolucu, kullanıcı ve tüketici tutum; çevresiyle uyum içinde yaşayan pasif toplumları yutmuş ve dünyanın şu anki tablosunu ortaya çıkarmış görünüyor.

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... more

Essay review of Florin Diacu and Philip Holmes, Celestial Encounters: The Origins of Chaos and Stability.

In this work methods for performing time series prediction on complex real world time series are examined. In particular series exhibiting non-linear or chaotic behaviour are selected for analysis. A range of methodologies based on... more

In this work methods for performing time series prediction on complex real
world time series are examined. In particular series exhibiting non-linear or
chaotic behaviour are selected for analysis. A range of methodologies based
on Takens’ embedding theorem are considered and compared with more
conventional methods. A novel combination of methods for determining the
optimal embedding parameters are employed and tried out with multivariate
financial time series data and with a complex series derived from an
experiment in biotechnology. The results show that this combination of
techniques provide accurate results while improving dramatically the time
required to produce predictions and analyses, and eliminating a range of
parameters that had hitherto been fixed empirically. The architecture and
methodology of the prediction software developed is described along with
design decisions and their justification. Sensitivity analyses are employed to
justify the use of this combination of methods, and comparisons are made
with more conventional predictive techniques and trivial predictors showing
the superiority of the results generated by the work detailed in this thesis.

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... more

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.

The synchronization dynamics of two linearly coupled pendula is studied in this paper. Based on the Lyapunov stability theory and Linear matrix inequality (LMI); some necessary and sufficient conditions for global asymptotic... more

The synchronization dynamics of two linearly coupled pendula is studied in this paper. Based on the Lyapunov stability theory and Linear matrix inequality (LMI); some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling kth, for the on-set of full synchronization is obtained. The numerical value of kth determined from the average energies of the systems is in good agreement with theoretical analysis. Prior to the on-set of synchronization, the boundary crisis of the chaotic attractor is identified. In the bistable states, where two asymmetric periodic attractors co-exist, it is shown that the coupled pendula can attain multistable states via a new dynamical transition—the basin crisis that occur prior to the on-set of stable synchronization. The essential feature of basin crisis is that the two co-existing attractors are destroyed while new three or more co-existing attractors of the same or different periodicity are created. In addition, the linear perturbation technique and the Routh–Hurwitz criteria are employed to investigate the stability of steady states, and clearly identify the different types of bifurcations likely to be encountered. Finally, two-parameter phase plots, show various regions of chaos, hyperchaos and periodicity.

The behavior of some systems is noncomputable in a precise new sense. One infamous problem is that of the stability of the solar system: Given the initial positions and velocities of several mutually gravitating bodies, will any... more

The behavior of some systems is noncomputable in a precise new sense. One infamous problem is that of the stability of the solar system: Given the initial positions and velocities of several mutually gravitating bodies, will any eventually collide or be thrown off to infinity? Many have made vague suggestions that this and similar problems are undecidable: No finite procedure can reliably determine whether a given configuration will eventually prove unstable. But taken in the most natural way, this is trivial. The state of a system corresponds to a point in a continuous space, and virtually no set of points in space is strictly decidable. A new, more pragmatic concept is therefore introduced: A set is decidable up to measure zero (d.m.z.) if there is a procedure to decide whether a point is in that set and it only fails on some points that form a set of zero volume. This is motivated by the intuitive correspondence between volume and probability: We can ignore a zero-volume set of states because the state of an arbitrary system almost certainly will not fall in that set. D.m.z. is also closer to the intuition of decidability than other notions in the literature, which are either less strict or apply only to special sets, like closed sets. Certain complicated sets are not d.m.z., most remarkably including the set of know stable orbits for planetary systems (the KAM tori). This suggests that the stability problem is indeed undecidable in the precise sense of d.m.z. Carefully extending decidability concepts from idealized models to actual systems, we see that even deterministic aspects of physical behaviour can be undecidable in a clear and significant sense.

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are... more

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the
three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would greatly affect the field of fluid mechanics. In this paper, which had been published in an international journal in 2010, a reasoned, practical approach towards resolving the issue is adopted and a practical, statistical kind of mathematical solution is proposed.

We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of... more

We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation, still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that absolute validation, while highly desirable, is overvalued. Simulations also provide valuable insights that we cannot yet (if ever) prove.

—Passive dynamic walking (PDW) provides us better insight for understanding human walking, for developing prosthetic limbs and for designing superior bipedal robots. In this paper, we investigated the dynamics of a simple PDW, 2D... more

—Passive dynamic walking (PDW) provides us better insight for understanding human walking, for developing prosthetic limbs and for designing superior bipedal robots. In this paper, we investigated the dynamics of a simple PDW, 2D compass-gait biped model that loosely look like human legs (without knees), using time-series analysis based on nonlinear dynamics. Previously, this passive biped model has been explored using only bifurcation diagrams and phase-space plots, but we studied it using nonlinear time-series analysis. The experimental results from walking experiments of prototype passive compass-gait biped validated the simulation results. These results can be useful for designing efficient bipedal robots.

This paper presents the new Lorenz unlike chaotic attractor which is constructed by a three non linear first order differential equations. These equations are arranged in a three dimensional autonomous systems. The dynamic behavior of the... more

This paper presents the new Lorenz unlike chaotic attractor which is constructed by a three non linear first order differential equations. These equations are arranged in a three dimensional autonomous systems. The dynamic behavior of the new chaotic system is shown such as time series, strange attractors, and bifurcations. Numerical experience also shows that when the parameter 'd' is varied, the global non linear amplitude is also varying. The paper ends with some possible research and development recommendations.

This article proposes an approach to synchronize a class of unidimensional spatiotemporal chaotic systems using exponential nonlinear observers. The article focuses on generalized synchronization with parameter mismatching and a... more

This article proposes an approach to synchronize a class of unidimensional spatiotemporal chaotic systems using exponential nonlinear observers. The article focuses on generalized synchronization with parameter mismatching and a unidirectional master–slave topology. The approach involves the conception of two different nonlinear observers to estimate the unknown parameters of a master system, such that the estimated parameters can be injected into a controller to synchronize the slave with the master. To illustrate the proposed approach, an example based on the Gray–Scott equations that exposes the synchronization and the observer conception is presented.

A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can... more

A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of 𝔲(2). We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.

In this paper we report on how a commercial “Chaos pendulum system” was used to teach aspects of chaos theory to Greek upper secondary school students. The principal aim of this project was to investigate to what extent students can... more

In this paper we report on how a commercial “Chaos pendulum system” was used to teach aspects of chaos theory to Greek upper secondary school students.
The principal aim of this project was to investigate to what extent students can develop an understanding of the chaotic behaviour using representations in the phase space.
The didactical methodology used was that of the “Physics Workshop” where students follow a discovery type laboratory course with detailed lab worksheets and are guided to the study of chaotic motion starting from observations of the oscillation of a simple physical pendulum.
Interesting conclusions concerning students’ ability in using the technology associated with the microcomputer based laboratory (MBL), the development of a qualitative understanding of the chaotic behaviour using the representation in the phase space and in interpreting graphs can be drawn from the analysis of their recorded answers.

Many research groups developed bifurcation diagrams, Poincare maps and computed Feigenbaum constants for passive walkers. Very few attempts have been made for performing nonlinear time-series analyses of these complex dynamical systems.... more

Many research groups developed bifurcation diagrams, Poincare maps and computed Feigenbaum constants for passive walkers. Very few attempts have been made for performing nonlinear time-series analyses of these complex dynamical systems. Besides, Garcia’s et al.’s the simplest walking model, the compass-gait biped model is the most commonly used passive dynamic walking (PDW) biped robot by the biomechanists, robotics engineers and chaos theorists. We accomplished nonlinear analysis of a time-series generated by the compass-gait biped and aimed to examine its dynamical behavior. The walking gait time-series data presented chaotic dynamics as fractal dimensions and positive Lyapunov exponents were found.

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in a measure μ (or d-μ),... more

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in a measure μ (or d-μ), which is particularly appropriate for physics and in some ways more intuitive than Ko’s (1991) recursive approximability (r.a.). For Lebesgue measure λ, d-λ implies r.a. Sets with positive λ- measure that are sufficiently “riddled” with holes are never d-λ but are often r.a. This explicates Sommerer and Ott’s (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-λ.