Evolution Of Chaos Research Articles

Chaotic band-gap modulation mechanism for nonlinear vibration isolation systems based on time-delay feedback control

Abstract Systems designed for nonlinear vibration isolation that incorporate chaotic states demonstrate superior capabilities in vibration attenuation, adeptly modulating the spectral constituents of vibrational noise. Yet, the challenge of eliciting low-amplitude chaotic dynamics and perpetuating these states across a diverse array of parameters remains formidable. This study proposes a pioneering strategy and technique for modulating the chaos band by incorporating a time-delayed feedback control mechanism within the framework of nonlinear vibration isolation systems.The investigation commences with an exhaustive analysis of the nonlinear dynamics, shedding light on the principles dictating the evolution of chaos. The study then advances to scrutinize the dynamics of systems with delays to elucidate the chaos-inducing processes engendered by feedback with temporal lags. Building upon the system’s responses, the chaotic performance and the effectiveness of the vibration isolation are crafted. Consequently, the time-delayed feedback control parameters are identified as pivotal design variables, which are then employed to dissect the control mechanisms influenced by the time-delayed feedback on the chaos band. Utilizing the delineated control mechanism, the nonlinear vibration isolation system is precipitously transitioned from a state of stable periodicity to one of chaos, fostering low-amplitude chaotic dynamics across an expansive parameter space, and in turn, resolving the previously stated challenge. Perhaps most significantly, the mechanism for attaining low-amplitude chaos introduced here paves the way for innovative methodologies in the active vibration isolation design of similar systems. Furthermore, it is anticipated to yield theoretical guidance for the manipulation of chaos bands and the formulation of active vibration isolation strategies within the domain of nonlinear vibration isolation systems.

Chaotic systems based on higher-order oscillatory equations

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.

Analytical studies on third-order chaotic systems with Sprott type nonlinearities and their microcontroller implementation

The evolution of chaos in a generic third-order autonomous mathematical model with nonlinearities described by simple mathematical functions is reported in this paper. The nonlinearities termed as Sprott type nonlinear functions are used in the design of a class of third-order systems exhibiting chaotic behavior. The evolution and confirmation of chaos in their system dynamics is observed through numerical simulation studies of one-parameter bifurcation diagrams and Lyapunov exponents. Analytical solutions are developed for systems with piecewise-linear nonlinear functions. Finally, the microcontroller implementation of the third-order system equations with different nonlinearities and analog circuit simulation results are presented to confirm the numerical and analytical results. Chaos in generic third-order systems studied through numerical, analytical and microcontroller results has been reported in the literature for the first time.

Grid-scroll memristive chaotic system with application to image encryption

The construction of multi-scroll chaotic systems has been a hot research topic in recent years. In this paper, a four-dimensional multi-scroll chaotic system is established on the basis of a simple three-dimensional chaotic system by adding a flux-controlled non-volatile memristor. After that, a novel system is extended to generate grid-scroll chaotic attractors by replacing the linear term in the multi-scroll chaotic system with a triangular wave function. The evolution of chaos is studied by means of bifurcation diagrams and Lyapunov exponent diagrams. The existence of coexisting attractors is observed, and the attraction basin of coexisting attractors is drawn. In addition, the partial amplitude of the proposed system can be controlled by adjusting the parameters. The results of circuit implementation are consistent with the numerical simulation, verifying the physical existence of the system. Finally, according to the proposed grid-scroll memristive chaotic system, a new image encryption algorithm is designed. It adopts a novel confusion-diffusion structure to scramble and diffuse all pixels of the plain-image. And the security analysis experiments are performed to verify the good encryption performance of the proposed scheme such as information entropy analysis, correlation coefficient analysis, differential attack analysis and robustness analysis.

Warlords, the United States, and the state of anarchy in Afghanistan

As part of its “War on Terror”, the United States (US) provided immense sums of money and advanced equipment to Afghan warlords in order to defeat and dismantle the Taliban and al-Qaeda in Afghanistan. Nearly two decades after the 2001 US-led intervention in Afghanistan that toppled the Taliban regime, the US continues supporting the warlords in various ways. As the intervention was also aimed at establishing a functioning state and reconstruction of the war-torn country, the US needed the support of local warlords to achieve its goals. However, over time, warlords and warlordism became a major challenge to the post-Taliban state-building project and in many ways undermined the overall security and the state monopoly on violence. These warlords, who had been mostly expelled and defeated by the Taliban regime, returned under the aegis of the B52 bombers, recaptured parts of the country and reestablished their fiefdoms with US support and resources. They not only resist giving up the power and prestige they have accumulated over the past few years, but also hamper the effort to improve governance and enact necessary reforms in the country. In addition, many of them run their private militias and have been accused of serious human rights abuses as well as drug trafficking, arms smuggling, illegal mining and extortion in the areas under their control or influence. In many ways, they challenge the government authority and have become a major hurdle to the country’s emerging from lawlessness and anarchy. This paper explores the emergence and re-emergence of warlords in Afghanistan as well as the evolution of chaos and anarchy in the country, especially after the US-led intervention of late 2001. It also analyzes the impact of the post-9/11 US support to Afghan warlords and its negative consequences for the overall stability and the US-led state-building process in Afghanistan.

The concept of “chaos measure” in the aspect of social synergetics

We understand chaos as a bifurcation process, the quantitative characteristic of which is reduced to the number of possibilities for the further evolution of the system, which in real time is directly expressed by self-oscillations of chaos and order (bifurcation cascade). The questions are being investigated: the scientific correctness of such formulations as ―chaos measure‖, ―evolution of chaos‖, correlation of the concepts ―measure of chaos‖ and ―magnitude of entropy‖; does the concept of ―measure of chaos‖ correlate with the concept of entropy; how the concepts of ―measure of chaos‖ and ―measure of order‖ relate, and is it possible to ―control‖ chaos? Models that study the evolution of open systems provide a bifurcation picture of development opportunities in at least two directions: complication and simplification, progress and regression, hierarchization of systems and their de-hierarchization, where a quantitative measure of chaos corresponds to the number of possible outcomes. The conclusion drawn from this is that human influence on the vector of the evolutionary process is reduced to a deliberate impact on the structure, which is commonly called the norm: an individual is able to comprehend the "measure of chaos" of the system of which he is an element, correlate it with the "measure of order" of the system and, setting the norm, thereby determine the direction to the superattractor.

The Image of the Change: From the I Ching to the Evolution of Chaos

This article is dedicated to artistic explorations of change, with the I Ching (The Book of Changes) as a stepping stone. The author uses modern mathematics to identify the basic types of change in the I Ching codes and to build a bottom-up I Ching systemization with an associated aesthetic principle. Moreover, the author introduces other (chaotic) types of change to sparsely fill the gap between the basic I Ching orders and the ultimate Change, allowing artistic speculation reflecting the evolution of many types of change by means of digital simulations, 3D volumetric display, etc.

Observation and evolution of chaos for a cantilever plate in supersonic flow

For a wing-like plate in supersonic flow cantilevered at its root, chaotic motions are studied in this paper. Prior literature has mainly focused on a simply supported plate or the limit cycle oscillations (LCOs) of a cantilever plate. The governing equations are constructed using von Karman plate theory and first-order piston theory. The Rayleigh–Ritz approach is adopted to discretize (and reduce the order of) the partial differential equations of the plate, and the resulting ordinary differential equations (ODEs) are solved numerically by the fourth-order Runge–Kutta (RK4) method. Numerical simulations demonstrate that the evolution of chaos is very complex, and the route to chaos depends on the panel׳s length-to-width ratio a/b. For a/b=1, a period-doubling of periodic motion occurs before transition to chaos. Another route to chaos is via quasi-periodic response directly for a/b=2. The most complicated prechaos and postchaos pattern is for a/b=0.5, which shows the presence of chaos regions with periodicity windows. Additionally, bifurcation diagrams show that certain features of the aeroelastic system such as quasi-periodic motions may be missed with too few Rayleigh–Ritz modes. Time histories, phase plane portraits, Poincaré maps and frequency spectra are used for identifying periodic and chaotic motions.

Artifacts & Illuminations: Critical Essays on Loren Eiseley by Susan N. Maher (review)

Reviewed by: Artifacts & Illuminations: Critical Essays on Loren Eiseley by Susan N. Maher Andrew Angyal Artifacts & Illuminations: Critical Essays on Loren Eiseley. Edited by Tom Lynch and Susan N. Maher. Lincoln: University of Nebraska Press, 2012. 358 pages, $40.00. Artifacts & Illuminations, the first comprehensive collection of new critical essays about Loren Eiseley’s work, will transform Eiseley scholarship and encourage fresh interpretations. Tom Lynch and Susan N. Maher have assembled a wide variety of critical perspectives, ranging from ecocriticism to structural, rhetorical, thematic, and place-oriented approaches. These essays testify to the scope and diversity of contemporary Eiseley scholarship. In their introduction, the editors lament the dearth of scholarship for a writer of Eiseley’s stature, suggesting that he might be a victim of the same “two cultures” split that he decried. [End Page 413] Or it may be that given his wide range of interests, Eiseley demands a unique interdisciplinary scholarship. Whatever the causes, this long overdue volume should remedy that neglect. The growing importance of Eiseley’s work reflects a new interest in science and nature writing as well as in the genre of creative nonfiction. The fourteen critical essays in this volume highlight the diversity of new interdisciplinary approaches to Eiseley criticism. Several contributors discuss the significance of rural and urban places in shaping Eiseley’s literary cartography. One of the most noteworthy essays, Jacqueline Cason’s “The Spirit of Synecdoche” examines Eiseley’s use of metaphor to depict ecological complexity in his use of the “concealed essay” form in exploring the conflict of order and chaos in evolution (187, 208). A reexamination of Eiseley’s poetry in terms of ecopoetics finds a unique expression of “evolutionary consciousness” (136–40). Another study, Stephen Mercier’s “Epic Narratives of Evolution,” notes the similarities between Eiseley’s and John Burroughs’s use of the tropes of the journey and the road to present the complexities of evolution in nontechnical terms. Pamela Gossin’s “Lessons of an Interdisciplinary Life” examines rhetorical strategies for teaching Eiseley in the interdisciplinary classroom and shows how his work can be used to model new ways of thinking and writing. Kathleen Boardman’s “Anthropomorphizing the Essay” uses new insights from cognitive ethnology to refute charges of anthropocentricism in Eiseley’s thought. Perhaps the most unusual essay, Dimitri N. Breschinsky’s “From the American Great Plains to the Steppes of Russia,” concerns efforts to translate Eiseley’s work into Russian and its mixed reception in Russia. A number of essays explore Eiseley’s personal spirituality in terms of Emerson, Thoreau, Dante, Jung, Whitehead, and Burroughs. New approaches to “The Star Thrower” find an “ecology of redemption” in his persona’s quest for meaning (214–15). Anthony Lioi’s “In a Dark Wood” compares Eiseley’s “dark night of the soul” to Dante’s spiritual journey in the Comedia, contrasting Eiseley’s ecological redemption with Dante’s Christian redemption. Jonathan Weidenbaum’s “Emerson and Eiseley: Two Religious Visions” shows how both writers contributed to the formation of a new “American Religion” of interior spirituality (239). Eiseley’s use of Jungian depth psychology is illustrated in concepts such as sacred time and the collective unconscious and in the archetypal symbol of the pyramid, especially in his 1998 collection of essays The Invisible Pyramid. Eiseley’s synthesis of science and personal reflection is more important than ever as a model of a new awareness of humanity’s place in the pageant of evolution. The essays in Artifacts & Illuminations will introduce Eiseley’s work to a new generation of readers beyond the two cultures’ rift. [End Page 414] Andrew Angyal Elon University, Elon, North Carolina Copyright © 2013 Western Literature Association

The evolution of chaos in active galaxy models with an oblate or a prolate dark halo component

AbstractThe evolution of chaotic motion in a galactic dynamical model with a disk, a dense nucleus and a flat biaxial dark halo component is investigated. Two cases are studied: (i) the case where the halo component is oblate and (ii) the case where a prolate halo is present. In both cases, numerical calculations show that the extent of the chaotic regions decreases exponentially as the scale‐length of the dark halo increases. On the other hand, a linear relationship exists between the extent of the chaotic regions and the flatness parameter of the halo component. A linear relationship between the critical value of the angular momentum and the flatness parameter is also found. Some theoretical arguments to support the numerical outcomes are presented. An estimation of the degree of chaos is made by computing the Lyapunov Characteristic Exponents. Comparison with earlier work is also made (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Chaotic dynamics in small inert gas clusters: The influence of potential energy saddles

We have investigated the effect of potential energy saddles on the evolution of chaos in small (three- to seven-atom) inert gas clusters by comparing the local Kolmogorov entropy (K entropy) accumulated near a saddle with that accumulated in a potential well. We find that saddles which are relatively flat along the isomerization coordinate are most effective at reducing the local K entropy in the saddle region, indicating that these saddles tend to regularize the dynamics close to the saddle point. In three- and four-atom clusters, we also find that flat saddles decouple some or all of the cluster’s vibrational modes from one another; this leads to approximately quasiperiodic behavior in some degrees of freedom near the saddle regions of these clusters.

Invariant torus bifurcation series and evolution of chaos exhibited by a forced non-linear vibration system

A forced non-linear vibration system with two degrees of freedom is investigated numerically. It is found that there is an invariant torus bifurcation series that precedes the onset of chaos. The evolution of chaos is also shown by the Poincaré maps.

Numerical simulations of periodic and chaotic responses in a stable duffing system

A stable Duffing system is examined by numerical simulations in order to obtain a better understanding of the behavior of periodic and chaotic responses to sinusoidal excitations. It is found that beside the multiplicity of responses, there is a duality for both periodic and chaotic responses. Period doubling does exist and this process may originate from different basic responses even with the same forcing frequency. The evolution of chaos is shown by a sequence of Poincaré maps. Finally a possible pattern for transition to chaos is suggested.

Chaos and cooperation in nonlinear pictorial feedback systems 3: Quantized feedback systems

We investigate the behavior of pictorial feedback systems with quantized intensity. These systems are used for numerical experiments to investigate the evolution of chaos and order in pictures. It turns out that chaos experiments with quantized feedback systems may show completely different results compared with those of nonquantized systems. However, the essential properties of the nonquantized system can be restored, in quantized experiments, by addition of noise. We demonstrate a use of the quasi-chaotic behavior of quantized systems, that is, cryptography of pictures.

Mechanism of onset and evolution of chaos in stratified positive column of a gas discharge

Mechanism of onset and evolution of chaos in stratified positive column of a gas discharge

Observation and Evolution of Chaos for an Autonomous System

Time histories, phase plane portraits, power spectra, and Poincare maps are used as descriptors to observe the evolution of chaos in an autonomous system. Although the motions of such a system can be quite complex, these descriptors prove helpful in detecting the essential structure of the motion. Here the principal interest is in phase plane portraits and Poincare maps, their methods of construction, and physical interpretation. The system chosen for study has been previously discussed in the literature, i.e., the flutter of a buckled elastic plate in a flowing fluid.