Modal decomposition and normal form for hydrodynamic flows: Examples from cellular flame patterns

Spatio-temporal complexity of hydrodynamic flows may be reduced through modal decomposition, especially in systems with symmetries. The symmetries of the most significant modes can then be used to deduce normal form equations associated with the observed state. In turn, the normal form equations can be used to deduce bifurcations to and from the given state.We illustrate this process using two spatio-temporal cellular states on a circular flame front. The first example contains a pair of uniformly rotating cells. Principle component analysis shows that two coherent structures capture most of the dynamics and suggests that the state is a broken-parity traveling mode. Other experimentally observed states, such as modulated rotating states and a heteroclinic cycle between two spatially orthonormal states result from secondary bifurcations from the rotating state. The second example, referred to as the hopping mode, visually appears to have significantly more complicated dynamics. However, modal decomposition shows that it consists of two parity broken states moving at different angular velocities. The corresponding normal form contains a codimension-three steady-state bifurcation leading to a homoclinic cycle whose spatio-temporal characteristics are similar to those of hopping states.We use these examples to propose a methodology to combine coherent structures that form a single, possibly time-dependent entity which we refer to as a generalized coherent structure. The process can reduce the number of entities needed to expand complex spatio-temporal states.The paper is dedicated to the memory of Michael Gorman, whose experiments on cellular flame fronts and relentless demands for better theoretical understanding of the patterns motivated the study.     

Cellular pattern formation in circular domains

An analysis of stationary and nonstationary cellular patterns observed in premixed flames on a circular, porous plug burner is presented. A phenomenological model is introduced, that exhibits patterns similar to the experimental states. The primary modes of the model are combinations of Fourier-Bessel functions, whose radial parts have neighboring zeros. This observation explains several features of patterns, such as the existence of concentric rings of cells and the weak coupling between rings. Properties of rotating rings of cells, including the existence of modulated rotations and heteroclinic cycles can be deduced using mode coupling. For nonstationary patterns, the modal decomposition of experimental data can be carried out using the Karhunen-Loeve (KL) analysis. Experimental states are used to demonstrate the possibility of using KL analysis to differentiate between uniform and nonuniform rotations. The methodology can be extended to study more complicated nonstationary patterns. In particular, it is shown how the complexity of "hopping states" can be unraveled through the analysis. (c) 1997 American Institute of Physics.     

Hopping behavior in the Kuramoto-Sivashinsky equation

We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments but not in truly two-dimensional computer simulations. A modal decomposition analysis of the simulated patterns, via the proper orthogonal decomposition, reveals spatio-temporal behavior in which the overall temporal dynamics is similar to that of equivalent experimental states but the spatial dynamics exhibits a few more features that are not seen in the experiments. Similarities in the temporal behavior and subtle differences in the spatial dynamics between numerical hopping states and their experimental counterparts are discussed in more detail.     

Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing

In this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi-analytically for near-grazing conditions. The experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. Smooth as well as nonsmooth bifurcations are observed, and the resulting atypical bifurcations are explained, often as an interplay between them. In order to understand the observed bifurcation scenarios, a global analysis is required, due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near-grazing conditions. The good degree of correspondence between experiment and theory fully justifies the modelling approach.     

Three-dimensional numerical simulations of cellular jet diffusion flames

Recent experimental investigations have demonstrated that the appearance of particular cellular states in circular non-premixed jet flames significantly depends on a number of parameters, including the initial mixture strength, reactant Lewis numbers, and proximity to the extinction limit (Damköhler number). For CO₂-diluted H₂/O₂ jet diffusion flames, these studies have shown that a variety of different cellular patterns or states can form. For given fuel and oxidizer compositions, several preferred states were found to co-exist, and the particular state realized was determined by the initial conditions. To elucidate the dynamics of cellular instabilities, circular non-premixed jet flames are modeled with a combination of three-dimensional numerical simulation and linear stability analysis (LSA). In both formulations, chemistry is described by a single-step, finite-rate reaction, and different reactant Lewis numbers and molecular weights are specified. The three-dimensional numerical simulations show that different cellular flames can be obtained close to extinction and that different states co-exist for the same parameter values. Similar to the experiments, the behavior of the cell structures is sensitive to (numerical) noise. During the transient blow-off process, the flame undergoes transitions to structures with different number of cells, while the flame edge close to the nozzle oscillates in the streamwise direction. For conditions similar to the experiments discussed, the LSA results reveal various cellular instabilities, typically with azimuthal wavenumber m = 1–6. Consistent with previous theoretical work, the propensity for the cellular instabilities is shown to increase with decreasing reactant Lewis number and Damköhler number.     

Two-component nonlinear Schrödinger models with a double-well potential

We introduce a model motivated by studies of Bose-Einstein condensates (BECs) trapped in double-well potentials. We assume that a mixture of two hyperfine states of the same atomic species is loaded in such a trap. The analysis is focused on symmetry-breaking bifurcations in the system, starting at the linear limit and gradually increasing the nonlinearity. Depending on values of the chemical potentials of the two species, we find numerous states, as well as symmetry-breaking bifurcations, in addition to those known in the single-component setting. These branches, which include all relevant stationary solutions of the problem, are predicted analytically by means of a two-mode approximation, and confirmed numerically. For unstable branches, outcomes of the instability development are explored in direct simulations.     

Steady state response of optical feedback in orthogonally polarized microchip Nd:YAG laser based on optical feedback rate equation

We present a rate equation model to explain and simulate the steady state response of optical feedback in orthogonally polarized microchip Nd:YAG lasers, which is observed in our experiment: When the external-cavity length is tuned, the two orthogonal polarization states perform an opposite-phase power modulation. With the increase of feedback light intensity, the power modulation of each orthogonal polarization state will present an amplitude increase, and finally become a polarization-state hopping; when the frequency difference of the two orthogonal polarization states is tuned, the power modulation amplitude performs a periodic variation. The theoretical analysis and numerical simulation are in good agreement with the experimental results.     

Universal behavior in heavy-electron materials

We present our finding that an especially simple scaling expression describes the formation of a new state of quantum matter, the Kondo Fermi liquid (KL) in heavy-electron materials. Emerging at T* as a result of the collective coherent hybridization of localized f electrons and conduction electrons, the KL possesses a non-Landau density of states varying as (1-T/T*)3/2[1+ln(T*/T)]. We show that four independent experimental probes verify this scaling behavior and that for CeIrIn5 the KL state density is in excellent agreement with the recent microscopic calculations.     

Dissipative discrete breathers: periodic,quasiperiodic, chaotic,and mobile

The properties of discrete breathers in dissipative one-dimensional lattices of nonlinear oscillators subject to periodic driving forces are reviewed. We focus on oscillobreathers in the Frenkel-Kontorova chain and rotobreathers in a ladder of Josephson junctions. Both types of exponentially localized solutions are easily obtained numerically using adiabatic continuation from the anticontinuous limit. Linear stability (Floquet) analysis allows the characterization of different types of bifurcations experienced by periodic discrete breathers. Some of these bifurcations produce nonperiodic localized solutions, namely, quasiperiodic and chaotic discrete breathers, which are generally impossible as exact solutions in Hamiltonian systems. Within a certain range of parameters, propagating breathers occur as attractors of the dissipative dynamics. General features of these excitations are discussed and the Peierls-Nabarro barrier is addressed. Numerical scattering experiments with mobile breathers reveal the existence of two-breather bound states and allow a first glimpse at the intricate phenomenology of these special multibreather configurations.     

Mode instabilities in a homogeneously broadened ring laser

A nonperturbative theory of a homogeneously broadened ring laser is presented after incorporating spatial hole-burning effects arising due to mutual interference of two counterpropagating cavity modes. It is shown that under suitable conditions the continuous-wave unidirectional lasing mode becomes unstable due to Hopf bifurcations. The time-periodic state associated with these bifurcations corresponds to self-induced pulsations either in one direction or in both directions.     

Control and characterization of spatio-temporal disorder in parametrically excited surface waves

The nonlinear interactions of parametrically excited surface waves have been shown to yield a rich family of nonlinear states. When the system is driven by two commensurate frequencies, a variety of interesting superlattice type states are generated via a number of different 3-wave resonant interactions. These states occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two different unstable modes generated by the two forcing frequencies. Near the system’s bicritical point, a well-defined region of phase space exists in which a highly disordered state, both in space and time, is observed. We first show that this state results from the competition between two distinct nonlinear super-lattice states, each with different characteristic temporal and spatial symmetries. After characterizing the type of spatio-temporal disorder that is embodied in this disordered state, we will demonstrate that it can be controlled. Control to either of its neighboring nonlinear states is achieved by the application of a small-amplitude excitation at a third frequency, where the spatial symmetry of the selected pattern is determined by the temporal symmetry of the third frequency used. This technique can also excite rapid switching between different nonlinear states.     

Imperfection sensitivity of interacting hopf and steady-state bifurcations and their classification

The bifurcation problem of interacting time-periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory. A complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurcations is presented. The sensitivity of the bifurcation diagrams to imperfections is analyzed. Normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined. A variety of phenomena, such as gaps in Hopf branches, periodic motions not stably connected to steady states and the formation of islands, is discovered, which one can expect to find in perturbed evolution equations on pure geometric grounds. Implications for physical systems are discussed.     

Multistability of twisted states in non-locally coupled Kuramoto-type models

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, -2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.     

Noise effect in metabolic networks

Constraint-based models such as flux balance analysis (FBA) are a powerful tool to study biological metabolic networks. Under the hypothesis that cells operate at an optimal growth rate as the result of evolution and natural selection, this model successfully predicts most cellular behaviours in growth rate. However, the model ignores the fact that cells can change their cellular metabolic states during evolution, leaving optimal metabolic states unstable. Here, we consider all the cellular processes that change metabolic states into a single term `noise', and assume that cells change metabolic states by randomly walking in feasible solution space. By simulating a state of a cell randomly walking in the constrained solution space of metabolic networks, we found that in a noisy environment cells in optimal states tend to travel away from these points. On considering the competition between the noise effect and the growth effect in cell evolution, we found that there exists a trade-off between these two effects. As a result, the population of the cells contains different cellular metabolic states, and the population growth rate is at suboptimal states.     

基于分叉修正面跳跃轨线的绝热和透热布居计算方法

对标准散射问题的非绝热动力学模拟,分叉修正面跳跃已被证明是一种可提高传统最少面跳跃性能的可靠方法[J. Chem. Phys. 150,164101 (2019)]. 本文研究了如何分析分叉修正面跳跃轨线,以得到可靠的绝热和透热布居. 研究了一系列一维二能级散射模型,以量子精确解和最少面跳跃结果为参考,发现分叉修正面跳跃方法可以得到接近精确解的含时布居. 特别地,不同的轨线分析方法在不同的表象下会得到明显不同的结果. 具体来说,轨线占据数方法在计算绝热态布居时表现更好,而波函数方法可以得到更可靠的透热态布居.     

Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators

A unified control theorem is presented in this paper, whose aim is to suppress the transversal intersections of stable and unstable manifolds of homoclinic and heteroclinic orbits in the Poincarè map embedding in system dynamics. Based on the control theorem, a primary resonant optimal control technique (PROCT for short) is applied to a general single-dof nonlinear oscillator. The novelty of this technique is able to obtain the unified analytical expressions of the control gain and the control parameters for suppressing the homoclinic and heteroclinic bifurcations, where the control gain can guarantee that the control region where the homoclinic and heteroclinic bifurcations do not occur can be enlarged as much as possible at least cost. The technique is applied to a nonlinear oscillator with a pair of nested homoclinic and heteroclinic orbits. By the PROCT, the transversal intersections of homoclinic and heteroclinic orbits can be suppressed, respectively. The hopping phenomenon that there coexist two kinds of chaotic attractors of Duffing-type and pendulum-type can be suppressed. On the contrary, if the first amplitude coefficient is greater than the critical heteroclinic bifurcation value, then another degenerate hopping behavior of chaos will take place again. Therefore, the phenomenon of hopping is the dominant type of chaos in this oscillator, whose suppressing or inducing is admissible from the points of practical and theoretical view.