Feigenbaum Scenario Research Articles

Route to and from the NMR chaos in diamagnets

The route to and from the chaos via period doubling bifurcations in nuclear spin system with dipole-dipole interactions is investigated. The transition points are found. It is shown that route from the chaos proceeds according the Feigenbaum scenario.

Some universal properties of multifractal chaos at nuclear giant resonance

It is shown that multifractal chaos observed at nuclear giant resonance belongs to the class of critical multifractal processes with infinite critical temperature. Furthermore, numerical value of multifractal critical index of the isovector transitions at the nuclear giant resonance turns out to be remarkably close to that corresponding to the universal critical chaos generated by the map $\ensuremath{\varphi}(x)=1\ensuremath{-}{\mathrm{ax}}^{2}$ via the Feigenbaum scenario.

Surface restructuring, kinetic oscillations, and chaos in heterogeneous catalytic reactions

We present comprehensive Monte Carlo simulations of isothermal kinetic oscillations and chaos in catalytic reactions accompanied by adsorbate-induced surface restructuring. Our analysis is based on the lattice-gas model describing surface restructuring in terms of the statistical theory of first-order phase transitions. As an example, we treat the kinetics of the NO-H2 reaction on the Pt(100) surface. A proposed reduced mechanism of this reaction includes NO adsorption, desorption, and decomposition occurring on the restructured patches of the surface (the decomposition products are rapidly removed from the surface via N2 desorption and H2O formation and desorption). Calculations are performed with a qualitatively realistic ratio between the rates of different elementary steps. In particular, NO diffusion is several orders of magnitude faster compared to the other steps. On the nm scale, the model predicts formation of restructured islands with atomically sharp boundaries. The shape of the islands is found to change dramatically with varying reaction conditions. Despite phase separation on the surface, the transition from almost harmonic oscillations (with relatively small separate islands) to chaos (with merging islands) is demonstrated to occur via the standard Feigenbaum scenario. Near the critical point, the dependence of the amplitude of oscillations on the governing parameter is shown to be close to that predicted for the Hopf supercritical bifurcation.

Multifractal critical phenomena in traffic and economic processes

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose.

Computer-simulation investigation of nonlinear transport dynamics in a compensated semiconductor during low-temperature electric breakdown

The possibility of chaotic behavior of the current in a partially compensated semiconductor during low-temperature electrical breakdown is explored by computer simulation. The influence of random fluctuations in the parameters or the variables of the mathematical model, as well as the effect of some weak periodic disturbances of the current density in the semiconductor, are discussed. As a result, various pictures of chaotic oscillations of the current are obtained, including a transition to a chaotic regime through period doubling, which is characteristic of the Feigenbaum scenario.

Dynamics of Heat Integrated Homogeneous Tubular Reactors

A detailed bifurcation analysis of heat integrated homogeneous tubular reactors is presented. The analysis is based on a one-dimensional discrete map obtained from the PDE model by means of the method of characteristics. It is shown that, under operating conditions in which the heat integration provides a positive feedback mechanism, the reactor may destabilize through a saddle-node bifurcation, resulting in multiple steady states. Under conditions in which the heat integration provides a negative feedback mechanism, the reactor may destabilize through a flip bifurcation, resulting in periodic solutions. It is shown that these periodic solutions subsequently may undergo a cascade of period doubling bifurcations into chaos. The period doubling cascade is found to follow the proposed universal Feigenbaum scenario. Higher periodic solutions and chaos is found to exist in a relatively large area of parameter space.

The Feigenbaum scenario as a model of the limits of conscious information processing

The Feigenbaum scenario of the mathematical period doubling sequence from order to deterministic chaos has led to new insights about the nonlinear dynamics of a wide variety of physical and biological systems. Multiple realms from purely mechanical systems, fluid dynamics and weather to the patterns of biological growth and the dynamic of the heart, hormone and brain rhythms have been found to exhibit aspects of the Feigenbaum period doubling sequence. We explore the possibility that the Feigenbaum scenario can be extended to experiences of sensation, perception and human cognition as well. We also review the empirical data that supports the view that the Feigenbaum scenario of the period doubling sequence may portray an important limit in conscious information processing. We conclude that a major function of consciousness may be to transform the nonlinear, irrational and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psychodynamics that make human experience and social life possible.

Undamped self-oscillations and dynamical chaos under conditions of impurity breakdown of a semiconductor in the fixed-current regime

It is shown theoretically that chaotic oscillations of the Hall field, the Hall constant, and the magnetoresistance appear in a compensated semiconductor under impurity breakdown conditions. It is shown that the transition to chaos occurs via the Feigenbaum scenario.

The feigenbaum scenario in a unified science of life and mind

In exploring the possibilities for creating a model of the creative cosmos, it is important to seek mathematical models that can express the common relationships unifying the sciences of matter, life and mind. Current candidates for such all embracing mathematical models come from the burgeoning fields of non‐linear dynamics that are described under varying labels as Chaos Theory, Self‐Organization Theory and Adaptive Complexity Theory. In this paper we will limit ourselves to only one mathematical model called, “The Feigenbaum Scenario.” A major virtue of The Feigenbaum Scenario is that it is relatively easy to understand by non‐mathematicians and it has led to new depths of profound understanding in a wide variety of the physical and biological sciences. Everything from purely mechanical systems, fluid dynamics and the weather to the patterns of biological growth in nature and the dynamics of heart, hormone and brain rhythms have been found to exhibit aspects of the Feigenbaum scenario. We explore the possibility that the Feigenbaum scenario can be extended to experiences of mind, sensation, perception and human behavior as well. We conclude that a major function of consciousness may be to transform the non‐linear, irrational, unconscious and difficult to predict dynamics of unconscious nature into the more linear, rational and predictable psycho‐dynamics that make human experience and social life possible.

Chaotic mode competition in the shape oscillations of pulsating bubbles

A possible mechanism for the occurrence of the phenomenon of erratic drift of bubbles in liquids subjected to acoustic waves was proposed by Benjamin & Ellis (1990) who showed that nonlinear interactions between adjacent perturbation modes expressed in terms of spherical harmonics of any order may lead to the excitation of mode 1 which is equivalent to a displacement of the bubble centroid. We show that indeed such a mechanism can give rise to a chaotic process at least under the conditions experimentally investigated by Benjamin & Ellis (1990). In fact we examine the case in which the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the natural frequency of mode n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n + 1. The value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation. A classical multiple scale analysis allows us to follow the development of these perturbations in the weakly nonlinear regime to find an autonomous system of quadratically coupled nonlinear differential equations governing the evolution of the amplitudes of the perturbations on a slow time scale. As obtained by Gu & Sethna (1987) for the Faraday resonance problem, we find both regular and chaotic solutions of the above system. Chaos is found to develop for large enough values of the amplitude of the acoustic excitation within some region in the parameter space and is reached through a period-doubling sequence displaying the typical characteristics of Feigenbaum scenario.

Bifurcation diagram and power spectrum of chaotic domain-wall motion

The chaotic oscillation of a domain wall is investigated by computer simulation. The bifurcation diagrams of the wall velocity and the power spectrum are calculated on the frequency of the external magnetic field. The Feigenbaum scenario by period doubling bifurcation is found to be one of the routes to chaos in this system.

CHAOTIZATION OF IONIZATION WAVES IN A DISCHARGE PLASMA

Self-generated ionization waves (moving striations) in a rare gases discharge positive column were studied experimentally on the boundaries between regular and chaotic oscillations. When the current was slightly changed, sequences of period-doubling bifurcations in the local intensity of plasma radiation were observed, which condensed to the critical point of chaotization. The “window” of periodic motion inside the chaotic region, with new sequences of bifurcations which gave rise to chaos as well as sequences of inverted bifurcations (which take place at decreasing current), were detected and examined. Fundamental constants of the Feigenbaum scenario were measured at medium pressures, δ=4.6±0.1 and α=2.6±0.6, showing the quadratic nonlinearity of a logistic map. A brief theoretical survey of ionization waves showed that 1D and 2D maps may be derived from nonlinear hydrodynamic plasma equations.

PERIODIC WINDOWS INSIDE CHAOS — EXPERIMENT VERSUS THEORY

A series resonance circuit that consists of a linear inductance, a nonlinear capacitance and a sinusoidal driving is investigated. The nonlinearity arises from a ferroelectric crystal. We observed the Feigenbaum scenario, crises, periodic windows inside chaos that show a period adding behaviour and coexisting attractors of different symmetry. We conclude a Duffing equation to cover all significant properties of our dynamical system.

Synchronization in oscillatory surface reactions on single crystal surfaces

In oscillatory surface reactions on single crystal surfaces the partial pressure variations that accompany the oscillations in the reaction rate represent a global interaction between the local oscillators. The NO+CO reaction on Pt(100) exhibits both synchronized and unsynchronized oscillatory behavior depending on the substrate phase. Unsynchronized oscillations occur on the 1×1 phase but on the hex phase one finds synchronized oscillatory behavior. The mechanism leading to synchronization in the oscillations on Pt(100)-hex can be traced back to a critical dependence of the 1×1⇄hex phase transition on the partial pressures of NO and CO. At both ends of the temperature window for rate oscillations one finds well-defined transitions to a stationary reaction rate. These bifurcations are discussed in terms of a transition from synchronized to unsynchronized behavior. In particular, the occurrence of deterministic chaos in connection with a Feigenbaum scenario is interpreted as being due to such a transition.

Synchronization and spatiotemporal self-organization in the NO+CO reaction on Pt(100). II. Synchronized oscillations on the hex-substrate

The NO+CO reaction exhibits sustained rate oscillations on Pt(100) under conditions where the surface is mostly hex-reconstructed. These rate oscillations have been investigated in the 10−6 mbar range using photoemission electron microscopy as a spatially resolving method. During the rate oscillations which appear in a temperature-window ranging from 490 to 478 K, the surface reacts in a spatially homogeneous way. At the upper T-boundary of the oscillatory range, the oscillations develop via a Feigenbaum scenario leading from chaotic small amplitude oscillations at high T to regular period-1 oscillations at lower T. At the lower T-boundary of the oscillatory range, at T=478 K, target patterns appear, causing the collapse of the amplitude of the rate oscillations. As the temperature is lowered further, the parallel wave trains become increasingly irregular. Spiral waves form, and finally one observes only local reaction outbursts. A model for the synchronization mechanism in the rate oscillations is proposed based on the 1×1⇔hex-phase transition, while the origin of the chaotic oscillations in this reaction system is discussed in terms of a transition from unsynchronized to synchronized oscillations.

Scenario of a transition of oscillatory striations to chaos in a steady-state discharge in rare gases and their mixtures used in lasers

An experimental study was made of the behaviour of ionisation instabilities resulting in stratification of the positive column of discharges in argon, neon, helium, and their mixtures. The study was carried out in the vicinity of the boundaries between regions representing regular and chaotic oscillations in the discharge current—pressure plane. Variation of the current revealed sequences of period-doubling bifurcations which became more frequent as the critical point of a transition to totally chaotic behaviour of self-excited striations was reached. A 'stability window' in a chaotic region (at a current higher than the critical value), a sequence of bifurcations in this window leading to chaos again, and a series of reverse bifurcations as the current was reduced (corresponding to the left-hand limit of the transition from regular to chaotic striations) were all observed for the first time. At average pressures of 0.067–6.7 kPa the transition to chaos corresponded to the Feigenbaum scenario. The fundamental constants of this scenario, δ = 4.6 ± 0.1 and α = 2.6 ± 0.6, were determined. The presence of a stability window and the reappearance of chaos indicated that the transition to chaos involved an intermittency of the first kind (Pomeau—Manneville scenario). A bifurcation sequence was observed most clearly in discharges that took place in pure argon and neon or in the presence of small (4%—5%) amounts of helium. At higher helium concentrations the boundary regions of the transition to chaos became much narrower and they were difficult to observe in pure helium.

Bifurcation diagram and large energy dissipation caused by chaotic domain-wall motion

The chaotic oscillation of a domain wall is investigated by computer simulation. The bifurcation diagrams of the wall velocity and the power spectrum are calculated as a function of the amplitude of the external magnetic field. The Feigenbaum scenario of the period doubling bifurcations is found to be one of the routes to chaos in this system. The power spectrum has a continuous distribution, which is evidence of chaotic motion. The energy loss caused by the domain-wall motion is calculated by integrating the damping term. The value of the energy loss is discussed in connection with the bifurcation diagram. The energy loss jumps to a high value at the first transition to chaos. The energy loss in the periodic window is larger than the value in the neighboring chaotic region in spite of their having the same damping coefficient.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On Chaotic Dynamics in Nuclear Engineering Systems

The mapping ρn+1 = φ[ρn + kQ0(exp ρn — 1)] is shown to belong to the universal class of quadratic mappings with a negative Schwarzian derivative, thus rigorously providing the reasons underlying this mapping’s ability to follow the well-known Feigenbaum scenario to deterministic chaos. This scenario proceeds through an infinite cascade of period-doubling bifurcations, as noted in numerical experiments by Shabalin in a recent paper on power instabilities in periodically pulsed reactors. An analysis of this paper is also presented together with an overall perspective of the current state of research on chaotic dynamics in nuclear engineering systems.

Hopf bifurcations in the logistic map with oscillating memory

The logistic equation, with its linear part replaced by a memory term, is considered. We will investigate the modification of the Feigenbaum scenario by analytical methods. Additionally, Hopf bifurcation occurs in the case of an oscillating memory term and is discussed in detail.

Deterministic onset of chaos in a gas discharge

Sequences of bifurcations in the radiation intensity of a discharge plasma on the boundary between regular and chaotic ionization waves regions were observed by varying the discharge current at a constant gas flow velocity and pressure. Fundamental constants of the Feigenbaum scenario were measured in the experiment as δ = 4.6±0.1 and α = 2.6±0.6 by means of recording succeeding period doublings.