State Of Chaos Research Articles

Stacked generalization and simulated evolution

A stacked generalization strategy in which an evolutionary algorithm generates baselevel predictive models is described. The evolutionary algorithm incorporates model validation and an inductive ranking criterion which encourages diversity of prediction errors. Higher levels of the stack of generalizers yield predictors that work through memory-based correction and combination of predictions produced by populations of models obtained in lower levels of the stack. The strategy has been applied to the classic problem of predicting annual sunspots activity. Baselevel predictors are drawn from a class of recurrent neural networks. Normalized mean squared errors of 0.064 and 0.19 for the conventional test intervals of 1921–1955 and 1956–1979, respectively, improve upon previously published results. The strategy has also been used to accurately forecast the behaviour of a synthetic system which makes random transitions between two states of low-dimensional chaos.

Spatiotemporal chaos in a model of Rayleigh-Bénard convection.

We present a numerical investigation of mean-flow induced spiral turbulence in a generalized Swift-Hohenberg model in two dimensions. We show the existence of a spatiotemporal chaotic state comprised of a large number of rotating spirals in a large aspect ratio system. This state is not predicted by classical theory. We calculate the spatial correlation functions for the order parameter, vorticity, and mean flow field in order to characterize more quantitatively the spiral chaos state. Our simulations show that there is a power-law behavior in the temporal dynamics of spiral defect chaos. Our study suggests that this spiral defect state occurs for low Prandtl numbers and large aspect ratios. We use as global variables the spectra entropy and the magnitude of the vorticity to characterize the transition from the global parallel roll state to the localized spiral state. Unfortunately we cannot determine whether this transition is gradual or sharp.

Defect Dynamics for Spiral Chaos in Rayleigh-Bénard Convection

A theory of the novel spiral chaos state recently observed in Rayleigh-Benard convection is proposed in terms of the importance of invasive defects i.e defects that through their intrinsic dynamics expand to take over the system. The motion of the spiral defects is shown to be dominated by wave vector frustration, rather than a rotational motion driven by a vertical vorticity field. This leads to a continuum of spiral frequencies, and a spiral may rotate in either sense depending on the wave vector of its local environment. Results of extensive numerical work on equations modelling the convection system provide some confirmation of these ideas.

Order and disorder in fluid motion.

The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characterized statistically, and methods for doing so are evolving. Transport and mixing phenomena can also lead to spatial complexity, but the degree depends on the significance of molecular or thermal diffusion.

Characterization of dispersive chaos and related states of binary-fluid convection

Abstract We describe an extensive series of experiments on traveling-wave convection in an ethanol water mixture with separation ratio ψ = −0.020 in a long, narrow, annular container. As the stress parameter is quasi-statically increased through the range 0–2% above onset, this system evolves through a series of complex spatiotemporal states before settling into an extended pattern of steady, spatially-uniform convective rolls. Much of the dynamical behavior is influenced by strong nonlinear dispersion, which causes the repeated and often irregular appearance and sudden collapse of spatially-localized bursts of traveling waves. Small-amplitude states seen very close to onset are analyzed quantitatively using a complex Ginzburg-Landau equation. Complex dynamical states encountered further above onset are analyzed using a number of classical statistical techniques, focusing mainly on correlation functions and Fourier spectra. In addition, we use the technique of biorthogonal decomposition in an effort to extract the basic structures of the patterns and to characterize the complexty of their dynamics.

Effect of memory and dynamical chaos in long Josephson junctions

A long Josephson junction in a constant external magnetic field and in the presence of a dc bias current is investigated. It is shown that the system, simulated by the sine-Gordon equation, ``remembers'' a rapidly damping initial perturbation and final asymptotic states are determined exactly with this perturbation. Numerical solving of the boundary sine-Gordon problem and calculations of Lyapunov indices show that this system has a memory even when it is in a state of dynamical chaos, i.e, dynamical chaos does not destroy initial information having a character of rapidly damping perturbation.

Immune Network Behavior: Oscillations, Chaos and Stationary States

The authors report two types of behavior in models of immune networks. The typical behavior of simple models, which involve B cells only, consists of several coexisting steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While steady-state behavior leads to easy interpretations in terms of immune memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. The stability of the steady states, and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model`s behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e., limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the steady states.

The Need for the "Historical Jesus" A Response to Jacob Neusner's Review of Crossan and Meier

Abstract Jacob Neusner's review of John Dominic Crossan's The Historical Jesus and John P. Meier's A Marginal Jew is insightful and helpful at many points. Although Neusner is not himself a Jesus scholar, his work in rabbinica qualifies him for meaningful participation in what is a technical and difficult field of study. Neusner rightly criticizes Crossan's uncritical use of apocryphal gospels, especially with respect to Morton Smith's Secret Gospel of Mark. But Neusner infers too much from this particular controversy; Jesus research is not in a state of chaos, nor has the discipline been unable to defend itself from hucksters and sensationalists. Neusner claims too much when he accuses Crossan and Meier of defending their work on "blatantly theological grounds." Both discuss the implications that their research has for Christian faith, but the work itself is not defended on theological grounds. It is concluded that historical Jesus research is credible and necessary.

The Need for the "Historical Jesus" A Response to Jacob Neusner's Review of Crossan and Meier

Abstract Jacob Neusner's review of John Dominic Crossan's The Historical Jesus and John P. Meier's A Marginal Jew is insightful and helpful at many points. Although Neusner is not himself a Jesus scholar, his work in rabbinica qualifies him for meaningful participation in what is a technical and difficult field of study. Neusner rightly criticizes Crossan's uncritical use of apocryphal gospels, especially with respect to Morton Smith's Secret Gospel of Mark. But Neusner infers too much from this particular controversy; Jesus research is not in a state of chaos, nor has the discipline been unable to defend itself from hucksters and sensationalists. Neusner claims too much when he accuses Crossan and Meier of defending their work on "blatantly theological grounds." Both discuss the implications that their research has for Christian faith, but the work itself is not defended on theological grounds. It is concluded that historical Jesus research is credible and necessary.

Quadratic map modulated by additive periodic forcing.

We study the effects on the quadratic map ${\mathit{x}}_{\mathit{n}+1}$=\ensuremath{\lambda}-${\mathit{x}}_{\mathit{n}}^{2}$ of an additive periodic forcing term of period 2. The modulated system can possess two attractors which are noncomplementary. The system displays bistability and can exist in two different states of periodicity or chaos for a given \ensuremath{\lambda}. The two attractors coexist until, with increasing \ensuremath{\lambda}, the trajectory boundary of attractor I collides with the unstable fixed point created with attractor II and attractor I is destroyed. Thereafter, the system possesses only one attractor, except in periodic windows of the first attractor, where again the system has two attractors. For still larger \ensuremath{\lambda}, the trajectory boundary of attractor II also collides with the unstable fixed point that was created with it, resulting in a sudden enlargement of the chaotic regime. For even larger \ensuremath{\lambda}, in periodic windows, two attractors can again coexist. The range of \ensuremath{\lambda} for which the system is iteratively stable may not be continuous. We conclude by generalizing the broad features of period-2 forcing for the case of period-p forcing when the system can possess a maximum of p noncomplementary attractors.

Immune network behavior--II. From oscillations to chaos and stationary states.

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

Immune network behavior—II. From oscillations to chaos and stationary states

Immune network behavior—II. From oscillations to chaos and stationary states

Statistical properties of chaos demonstrated in a class of one-dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.

A tale of one city.

A report from Kampala, Uganda, compares the situation in 1991 to the state of chaos 10 years earlier when the regime of Idi Amin had been overthrown by Milton Obote's soldiers with the help of Tanzanian troops. Soldiers went on looting sprees, and 1 victim of their marauding became a 12-year old boy who got shot for refusing to part with his bike. In contrast, in 1991 things were much more peaceful; however, the AIDS epidemic was the new threat. The government radio transmits hourly warnings on HIV. Since President Museveni came to power, economy and security have improved radically. Shops and markets are open until late at night; public transport is reliable, and small scale industry flourished. There would be optimism about the future, if AIDS was not here. There is no doubt that the economy will soon be affected. According to the Kampala blood bank, 40% of the healthy population is already seropositive. In the hospitals the majority of admissions suffer from AIDS with diarrhea and an itching dermatitis; there is more cancer of the cervix and lymphoma; appendicitis is on the increase; and tuberculous lymph nodes are now quite common. Many of these patients have clinical AIDS. The government is frank about the situation and is active in preventive measures and education. Private charities and foreign aid organizations contribute. But the epidemic is so overwhelming, that some Western organizations might soon lose interest owning to meager returns on their efforts. A 6-year-old boy has grossly swollen lymph nodes around his neck, both parotids are painfully swollen, pus pours from the ears. A nonspecific cough and mild diarrhea are also present with an itching and sore herpes zoster on his left chest. the mother is frightened of losing him, and demurs at the hint of AIDS, since for her, AIDS means sexual promiscuity.

Human Life in the Balance

We are clearly in a state of chaos about the status of human life in our writes David C. Thomasma in his bookHuman Life in the Balance. This is a well-thought-out, well-researched, and incredibly well-referenced plea for each of us to begin to affirm the value of human life and then to participate in social action in support of those views. It is not a call for civil disobedience (an excellent discussion of this can be found in Francis Schaeffer'sA Christian Manifesto) but a cry for dialogue. Thomasma's nightmare is no consensus about the status of human life. He persuades that a commitment to an irreducible value inherent in human beings does not confine us to one particular moral, political, or religious position. In the process, he outlines three possible positions that we, as individuals and as a society, can take and still have a baseline

Infinitely many co-existing sinks from degenerate homoclinic tangencies

The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families { F t } \{ {F_t}\} of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, P t ; F t n ( P t ) = P t {P_t};\;F_t^n({P_t}) = {P_t} , and | det D F t n ( P t ) | > 1 |\det DF_t^n({P_t})| > 1 . We also require the stable and unstable manifolds of P t {P_t} to form homoclinic tangencies as the parameter t t varies through t 0 {t_0} . Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values t t near t 0 {t_0} . We show that there are parameter values t t near t 0 {t_0} at which F t {F_t} has infinitely many co-existing periodic sinks.

Chaotic interactions in a pendulum-energy source system

There is a'long history of research on pendulum systems, which has provided some striking discoveries, since it is simple to observe the phenomena in these widely occurring systems with their various properties, e.g., such systems were first used in observations on parametric resonance [ 11, 15] and high-frequency stabilization [2, 8]. Essentially novel features have been observed as regards states of chaos in pendulum systems [ 1, 5, 6, 13, 14]. The history of research on pendulum systems reflects advances in mathematics, mechanics, and physics. Here we consider the steady states in the resonant motion of a pendulum when the point of support is vibrated by a mechanism with restricted power, and also when chaos occurs in the interactions. Particular attention is given to how the source features affect the state. Deterministic chaos occurs [5, 6, 9] with a simple pendulum only if there is an external force, and extensive numerical data exist on the behavior of pendulums with inputs described by explicit time functions [6, 13]. However, if there is restricted excitation [3, 4], where the operation of the exciting device is dependent on the dynamic state, chaotic oscillations should give rise to analogous behavior, and consequently the system is subject to a chaotically varying input. One therefore has to consider how the individual features of the device affect the oscillations. Before deterministic chaos was discovered, a reduction principle was used in researching complex systems: the system was split up into parts, each examined separately. The recognition of chaotic states led to the view that the combined system may have fairly complicated behavior because of the interaction between several components [7], so one anticipates that a pendulum acted on by a restricted force will show behavior that gives much more information on the interaction rather than the properties of the pendulum. Here one cannot neglect the coupling of the pendulum to the mechanism on the argument that it is weak (which is mathematically expressed as small terms describing it) because even a weak interaction produces an essentially different motion pattern. We consider the Fig. 1 system. The crank mechanism is coupled via the rod b to the support of a physical pendulum. When the drive a rotates through an angle O, the slider together with the suspension shows a displacement u(t) = altos O + (al/4)(1 + cos 20)], in which a i ffi a/b. An Oxz cartesian coordinate system is used to describe the pendulum oscillations, and the kinetic energy of the entire system, with the mass of the slider taken as negligible, is [3, 12, 13]

Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3 laser.

Detailed studies of digitized recordings of periodic and chaotic intensity pulsing of an unidirectional far-infrared ${\mathrm{NH}}_{3}$ ring laser at 81.5 \ensuremath{\mu}m reveal common features of a broad range of different pulsing patterns as well as systematic relationships among entropies, dimensions, and decay rates of the autocorrelation function. Spiral-type ``Lorenz-like'' chaos of many different spiraling rates and of differing modulation depths has an underlying attractor dimension of 2.0--2.3 and an entropy rate that lies mainly in the range (0.2--0.7)${T}^{\mathrm{\ensuremath{-}}1}$, where T is the average intensity pulsing period. Over a wide range of types of pulsing (spiral chaos, period-doubling chaos, and periodic states) the decay rate of the envelope of the autocorrelation function over short times provides an estimator of the entropy. The results are in excellent agreement with the characteristics of the Lorenz-Haken model for similar operating parameters.

Reduction of flexural vibrations by damping of axial motions

To realize a simple passive method for the reduction of the amplitude of flexural vibrations in structures, a method of damping the axial displacements of a structural component is introduced. The non-linear deformation of slender beam is considered, and the axial force acting at one end of it is assumed to be linearly proportional to the beams' axial velocity there. This, in turn, is expressed in terms of an integral over the length of the beam of the time derivative of the quantity ν s 2, where ν s = ∂ν ∂s, s denoting the arc length along the axis of the beam. This is a second order non-linear effect. The derivation of the equation of motion of the beam includes non-linear effects to the third order. The beam is then assumed to be driven by a transverse force that varies sinusoidally in time. For a simply supported beam, a one-term Galerkin approximation is applied to remove the spatial dependence of the partial differential equation. The novel feature of this differential equation is the presence of a non-linear damping term that consists of the product of the transverse velocity and the square of the transverse displacement. Non-linear stiffness and inertia terms also appear in this reduced equation of motion. A system of non-linear first order differential equations is then obtained. For the case of the steady state response, the dependence of the steady state amplitude upon the frequency ratio is determined and a plot of its variation is reported for several values of the non-linear damping parameter. The Runge-Kutta method is used to determine the variation of the amplitude of the motion as a function of time when the driving frequency is equal to the natural frequency of the system. Numerical calculations reveal that significant reductions in the transverse amplitudes of the motion can be realized through an increase in the degree of damping that is exerted on the axial motion of the beam. Poincaré maps are plotted for undamped and non-linearly damped beam motions. A transition into and then out of a state of chaos is observed as the value of the non-linear damping parameter is increased.

Experimental evidence of period doubling of tori during an electrochemical reaction

Chaos and period doubling of tori during copper electrodissolution in acidic chloride solutions are studied. The thickness of a surface film acts as a slowly varying parameter. We present experimental evidence for the sequence one-band chaos, two-band chaos, double torus, torus, limit cycle, and steady state.