Feigenbaum Route Research Articles

Adjoint-based sensitivity analysis of periodic orbits by the Fourier–Galerkin method

Sensitivity of periodic solutions of time-dependent partial differential equations is commonly computed using time-consuming direct and adjoint time integrations. Particular attention must be provided to the periodicity condition in order to obtain accurate results. Furthermore, stabilization techniques are required if the orbit is unstable. The present article aims to propose an alternative methodology to evaluate the sensitivity of periodic flows via the Fourier–Galerkin method. Unstable periodic orbits are directly computed and continued without any stabilizing technique. The stability of the periodic state is determined via Hill's method: the frequency-domain counterpart of Floquet analysis. Sensitivity maps, used for open-loop control and physical instability identification, are directly evaluated using the adjoint of the projected operator. Furthermore, we propose an efficient and robust iterative algorithm for the resolution of underlying linear systems. First of all, the new approach is applied on the Feigenbaum route to chaos in the Lorenz system. Second, the transition to a three-dimensional state in the periodic vortex-shedding past a circular cylinder is investigated. Such a flow case allows the validation of the sensitivity approach by a systematic comparison with previous results presented in the literature. Finally, the transition to a quasi-periodic state past two side-by-side cylinders is considered. These last two cases also served to test the performance of the proposed iterative algorithm.

Transition to Chaos of Buoyant-Thermocapillary Convection in Large-Scale Liquid Bridges

To cooperate with the Chinese TG-2 space experiment project, the transition routes to chaos of buoyant-thermocapillary convection were experimentally researched in large-scale liquid bridge of 2cSt silicone oil with 20 mm in diameter. A Nanovoltmeter with high resolution was adopted to measure the dynamical temperature oscillations, as well as to detect the convective transitional behaviors which were nonlinear and non-stationary. The existence of the quasi-periodic route and the Feigenbaum route has been confirmed in large-scale LBs, and a novel periodic oscillation state was discussed in detail for the first time. The chaotic characteristics were verified by analyzing the maximum Lyapunov exponent and correlation dimension on the basis of a phase-space reconstruction. Additionally, it was found that bifurcation could potentially lead to the reconstruction of flow fields.

Transition to chaos in thermocapillary convection

We present experiments on transition to chaos in thermocapillary convection in a rectangular pool of silicone oil. The applied temperature difference between the two sidewalls is adjusted in the range of 0–43°C to observe various dynamic states. The applied temperature gradient along the fluid–gas interface drives shear flow along the free surface from hot to cold and a back flow in the underlying layer. With the increase of the temperature gradient, the thermocapillary convection will transit from steady flow to regularly oscillatory flow, and finally to chaos. A temperature measurement system, which consists of thermocouple, voltmeter and data-acquiring computer, is used to record the temperature of the liquid dynamically. In order to identify the different dynamic regimes from steady flow to chaos, fast Fourier transform and fractal theory are used to analyze the experimental data. The critical conditions for transition have been obtained and discussed by non-dimensional analysis. The quasi-periodic route and Feigenbaum route were observed for different experimental conditions, and the relationship between oscillatory frequency and Marangoni number Ma has been discussed.

Nonlinear Dynamics of Self-Sustained Supersonic Reaction Waves: Fickett’s Detonation Analogue

The present study investigates the spatiotemporal variability in the dynamics of self-sustained supersonic reaction waves propagating through an excitable medium. The model is an extension of Fickett's detonation model with a state-dependent energy addition term. Stable and pulsating supersonic waves are predicted. With increasing sensitivity of the reaction rate, the reaction wave transits from steady propagation to stable limit cycles and eventually to chaos through the classical Feigenbaum route. The physical pulsation mechanism is explained by the coherence between internal wave motion and energy release. The results obtained clarify the physical origin of detonation wave instability in chemical detonations previously observed experimentally.

Effects of periodic perturbations on the oscillatory behavior in theNO+H2reaction on Pt(100)

The mean field model proposed by Makeev and Nieuwenhuys [J. Chem. Phys. 108, 3740 (1998)] simulates the oscillatory behavior experimentally observed in the NO+H2 reaction on the surface Pt(100). This model reproduces quite well the kinetic oscillations and the transition to chaos via the Feigenbaum route, that is to say, through bifurcations involving period doubling. From this model, we analyze the response of the natural oscillations of period-1 (P1, one maximum) to periodic perturbations superposed to the partial pressure of one of the reactants. The perturbed model reproduces the periodic states found in the autonomous model, the route to chaos through bifurcations with period doubling, and the appearance of chaos via the route of intermittency, which shows alternation of periodic oscillations with intervals of disordered oscillations in the same time evolution. Experimentally it has been observed that the reaction shows a great sensitivity to reactant partial pressures and temperature. In experimental conditions slightly different to those considered in Makeev and Nieuwenhuys (MN) model, oscillations with period-3 (P3, three maxima) have been observed. At T=457 K and certain pressures, these P3 oscillations do not appear in MN model, although they appear at T=456 K . The same effect (P3 oscillations) is obtained at T=457 K in our perturbed model, due to the modulation of pH2. In a second step we show how the modulation of the perturbing frequency influences on the oscillations P1 of the perturbed system. The results show that the periodic behavior loses its regularity at low values of the normalized amplitude and of the modulated frequency of the perturbation. Other aspect observed in the perturbed model is that the amount of products varies in relation to nonperturbed model. When the oscillations are periodic or they follow the Feigenbaum route to chaos, the production average decreases or slightly increases, whereas it always increases if there are intermittencies, the most significant percentage increase being for NH3 (nearly 10%).

Non-linear analysis and modelling of the cellular mechanisms that regulate arterial vasomotion

Non-linearity intrinsic to the ion transport systems that regulate intracellular [Ca2+] and smooth muscle tone allows the emergence of fluctuations in vascular diameter and resistance in isolated arteries (vasomotion) that can be classified as chaotic. Correlation analysis suggests an underlying low dimensional system and a four-dimensional model of vasomotion has been formulated to simulate the effects of pharmacological manipulation of smooth muscle tone or nitric oxide (NO) synthesis by the vascular endothelium. The oscillatory patterns observed experimentally and in modelling studies may be considered ‘universal’ in the sense that they also occur in many physico-chemical systems and include: (a) period-doubling, a feature of the Feigenbaum route to chaos; (b) mode-locking and quasiperiodicity, which reflect the interaction of two nonlinear oscillatory subsystems; and (c) intermittency, in which segments of nearly periodic oscillations of variable duration are interrupted by short chaotic bursts. Low dimensionality allows the construction of iterative maps that confirm the existence of types I and III Pomeau-Manneville intermittency in vascular dynamics, with attractor reconstructions indicating that the reinjection mechanism underlying the type III scenario involves a Shil'nikov-type homoclinic trajectory. Dimensional analysis of experimental data and corresponding surrogate time series generated by randomization of Fourier phase also provide evidence for underlying nonlinear structure in fluctuations in red cell velocity and arteriolar calibre in vivo.

Numerical investigation of the instability for one-dimensional Chapman–Jouguet detonations with chain-branching kinetics

The dynamics of one-dimensional Chapman–Jouguet detonations driven by chain-branching kinetics is studied using numerical simulations. The chemical kinetic model is based on a two-step reaction mechanism, consisting of a thermally neutral induction step followed by a main reaction layer, both governed by Arrhenius kinetics. Results are in agreement with previous studies that detonations become unstable when the induction zone dominates over the main reaction layer. To study the nonlinear dynamics, a bifurcation diagram is constructed from the computational results. Similar to previous results obtained with a single-step Arrhenius rate law, it is shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade. The corresponding Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be close to the universal value of 4.669. The present parametric analysis determines quantitatively the relevant non-dimensional parameter χ, defined as the activation energy for the induction process ϵ I multiplied by the ratio of the induction length Δ I to the reaction length Δ R . The reaction length Δ R is estimated by the inverse of the maximum thermicity (1/ max) multiplied by the Chapman–Jouguet particle velocity u CJ . An attempt is made to provide a physical explanation of this stability parameter from the coherence concept. A series of computations is carried out to obtain the neutral stability curve for one-dimensional detonation waves over a wide range of chemical parameters for the model. These results are compared with those obtained from numerical simulations using detailed chemistry for some common gaseous combustible mixtures.

Analysis of a Ferroresonant Circuit Using Bifurcation Theory and Continuation Techniques

In this contribution, the analysis of the dynamics of a ferroresonant circuit are presented using continuation techniques and bifurcation theory. Despite its great simplicity, this circuit can assume a diverse range of steady-state regimes including fundamental and subharmonic ferroresonance, quasiperiodic oscillations, and chaos. The system dynamics are explored through the continuation of periodic solutions of the associated circuit equations. A detailed picture is drawn of various transitions between the individual periodic steady-state regimes of the circuit. Bifurcation points are computed, revealing a clearly defined succession of periodic steady-state regimes, including the Feigenbaum route to chaos through a cascade of period doublings. The analysis presented is performed using the freely available software package XPPAUT. The contribution of this paper is to provide a detailed description of how to define the circuit equations in XPPAUT and how to conduct the interactive bifurcation analysis. The proposed approach is shown to be both computationally efficient and robust, as it eliminates the need for numerically critical and long lasting transient simulations.

Chaos in 1D Radiative Edge Plasmas

Bifurcation and chaos in radiative edge plasmas are investigated on the basis of a periodically driven reaction-diffusion equation which results from the time dependent 1d heat conduction equation including a given periodically time-modulated impurity density. The temporal problem shows the transition to chaos through the Feigenbaum route. In 1d and time dependent plasmas Hopf bifurcation and intermittency phenomena exist. The application to a carbon seeded plasma shows the existence of different oscillation regimes.

Potts model on a Cayley tree and logistic equation

The q -state Potts model on a Cayley tree can be solved by a recursion formula depending on the properties at the surface. The model on a Bethe lattice is obtained by extrapolating the interior of a Cayley tree sufficiently far from the surface in order to have a stable fixed point of the recursion. For q >2 we find a second-order transition of percolation type at the Bethe-Peierls temperature and a first-order transition at a higher temperature. For coordination number z = 3 the recursion extrapolated to q = 1 is identical to the logistic equation. The Feigenbaum route to chaos appears for antiferromagnetic coupling of the Potts model. The first period doubling corresponds to a multicritical point in the phase diagram of the Potts model.

Phase multistability of synchronous chaotic oscillations

The paper describes the sequence of bifurcations leading to multistability of periodic and chaotic synchronous attractors for the coupled Rössler systems which individually demonstrate the Feigenbaum route to chaos. We investigate how a frequency mismatch affects this phenomenon. The role of a set of coexisting synchronous regimes in the transitions to and between different forms of synchronization is studied.

Role of multistability in the transition to chaotic phase synchronization.

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.

Oscillation Phenomena Leading to Chaos in a Stochastic Surface Reaction Model

A microscopic lattice gas model for the $\mathrm{CO}+\mathrm{NO}$ reaction on Pt(100) is studied by means of Monte Carlo simulations. It shows different kinetical phenomena such as steady state reaction, damped, regular, and irregular oscillations, as well as a transition into chaotical behavior via the Feigenbaum route. Because of its small number of parameters, each with a specific physical meaning, it enables the investigation of the whole parameter regime leading to a deeper insight to the mechanisms which create the oscillations and chaotical behavior.

Coexistence of multiple periodic and chaotic regimes in biochemical oscillations with phase shifts.

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a multiplicity of stable coexisting states: birhythmicity, trirhythmicity, hard excitation and quasiperiodic with chaotic regimes. For different initial functions in the phase space one may observe the coexistence of two different quasiperiodic motions, the existence of a stable steady state with a stable torus, and the existence of a strange attractor with different stable regimes (chaos with torus, chaos with bursting motion, and chaos with different periodic regimes). For a single range of the control parameter values our system may exhibit different bifurcation diagrams: in one case a Feigenbaum route to chaos coexists with a finite number of successive periodic bifurcations, in other conditions it is possible to observe the coexistence of two quasiperiodicity routes to chaos. These studies were obtained both at constant input flux and under forcing conditions.

CHAOS IN A HYDRAULIC CONTROL VALVE

In this paper we have studied the instability and chaos occurring in a pilot-type poppet valve circuit. The system consists of a poppet valve, an upstream plenum chamber, a supply pipeline and an orifice inserted between the pelnum and the pipeline. Although the poppet valve rests on the seat stably for a supply pressure lower than the cracking pressure, the circuit becomes unstable for an initial disturbance beyond a critical value and develops a self-excited vibration. In this unstable region, chaotic vibration appears at the period-doubling bifurcation. We have investigated the stability of the circuit and the chaotic phenomenon numerically, and elucidated it by power spectra, a bifurcation diagram and Lyapunov exponent calculations, showing that the phenomenon follows the Feigenbaum route to chaos.

Intermittency route to chaos in a biochemical system

The numerical analysis of a glycolytic model performed through the construction of a system of three differential-delay equations reveals a phenomenon of intermittency route to chaos. In our biochemical system, the consideration of delay time variations under constant input flux as well as frequency variations of the periodic substrate input flux allows us, in both cases, to observe a type of transition to chaos different from the ‘Feigenbaum route’.

Bifurcations of two coupled classical spin oscillators

Two classical, damped and driven spin oscillators with an isotropic exchange interaction are considered. They represent a nontrivial physical system whose equations of motion are shown to allow for an analytic treatment of local codimension-one and -two bifurcations. In addition, numerical results are presented which exhibit a Feigenbaum route to chaos.

Rayleigh-Be´nard Convection in a Small Aspect Ratio Enclosure: Part II—Bifurcation to Chaos

The present numerical study documents bifurcation sequences for Rayleigh-Be´nard convection in a rectangular enclosure with insulated sidewalls. The aspect ratios are 3.5 and 2.1 and the Boussinesq fluid is water (average temperature of 70°C) with a Prandtl number of 2.5. The transition to chaos observed in the simulations and experiments is similar to the period-doubling (Feigenbaum) route to chaos. However, special symmetry conditions must be imposed numerically, otherwise the route to chaos is different (Ruelle-Takens-Newhouse). In particular, the Feigenbaum route to chaos can be realized only if the oscillating velocity and temperature field preserves the fourfold symmetry that is observed in the mean flow in the horizontal plane.

A hierarchy of transitions to mixed mode oscillations in an electrochemical system

The dynamic behavior of the electrocatalytic oxidation of hydrogen on a platinum anode in the presence of Cu 2+ and Cl − ions under galvanostatic conditions has been studied experimentally in two control parameters: current density and copper ion concentration. This parameter plane can roughly be divided into three broad regions: A steady state is found at low values of the current density, at medium values the system shows small amplitude oscillations, whereas at high current densities typical mixed mode oscillations (MMOs) are found. Depending on the Cu 2+ concentration, the transition from the small amplitude to the mixed mode regime takes place from a simple periodic, period doubled, or chaotic attractor arising from a Feigenbaum route. In this paper we focus on the characteristics of these different transitions to MMOs and try to construct a bifurcation diagram. The bifurcation from the small chaotic attractor to MMOs very likely constitutes an interior crisis (a bifurcation at which the small chaotic attractor is destroyed by colliding with a stable manifold of a saddle type limit set). At lower copper ion concentrations, the transition to MMOs moved through the period-double cascade. We compile different properties of the bifurcation diagram and discuss a scenario which fits the observations in a consistent way.

One-dimensional dynamics for a discontinuous map

Abstract A discontinuous, two-parameter family of one-dimensional maps is shown to posses many of the features of the quadratic (“logistic”) map; for instance, a unique, twice differentiable maximum. However, paths may be chosen in the two-dimensional parameter space which give rise to a wealth of possible routes to chaos. In particular, paths may be found for the Feigenbaum route to chaos — period-doubling cascade of bifurcations — with a resulting single parameter in the map converging geometrically with a constant ω n directly to chaotic motion.