Search

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions., Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular the discussion on partial BDEs is updated and enlarged

International audience; A simple heuristic model of coupled decadal ocean-atmosphere modes in middle latitudes is developed. Previous studies have treated atmospheric intrinsic variability as a linear stochastic process modified by a deterministic coupling to the ocean. The present paper takes an alternative view: based on observational, as well as process modeling results, it represents this variability in terms of irregular transitions between two anomalously persistent, high-latitude and low-latitude jet-stream states. Atmospheric behavior is thus governed by an equation analogous to that describing the trajectory of a particle in a double-well potential, subject to stochastic forcing. Oceanic adjustment to a positional shift in the atmospheric jet involves persistent circulation anomalies maintained by the action of baroclinic eddies; this process is parameterized in the model as a delayed oceanic response. The associated sea-surface temperature anomalies provide heat fluxes that affect atmospheric circulation by modifying the shape of the double-well potential. If the latter coupling is strong enough, the model’s spectrum exhibits a peak at a periodicity related to the ocean’s eddy-driven adjustment time. A nearly analytical approximation of the coupled model is used to study the sensitivity of this behavior to key model parameters.

We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

The double-gyre circulation induced by a symmetric wind-stress pattern in a quasi-geostrophic model of the mid-latitude ocean is studied analytically and numerically. The model is discretized vertically by projection onto normal modes of the mean stratification. Within its horizontally rectangular domain, the numerical model captures the wind-driven circulation’s three dynamic regimes: (1) a basin-scale double-gyre circulation, cyclonic in the basin’s northern part and anticyclonic in the south, which is dominated by Sverdrup balance; (2) a swift western boundary current in either gyre, with dissipation most important near the coast and inertial balance further out; and (3) a strong recirculating dipole near the intersection of the western boundary with the symmetry line of zero wind-stress curl. The flow inside this stationary dipole is highly nonlinear, and equivalent-barotropic. An analytical solution to the potential vorticity equation with variable stratification describes the dipole, and fits well the full numerical model’s steady-state solutions. Changes in the numerical model’s solutions are investigated systematically as a function of changes in the strength of the wind stress τ and the Rossby radius of deformation LR. The main changes occur in the recirculation region, while the basin-scale gyres and the western boundary currents are affected but little. A unique symmetric dipole is observed for small τ, and agrees in its properties with the analytical solution. As τ increases, multiple asymmetric equilibria arise due to pitchfork bifurcation and are stable for large enough LR. The numerically obtained asymmetric equilibria also agree in their main properties with the analytical ones, as well as with the corresponding solutions of a shallow-water model. Increasing τ further results in two successive Hopf bifurcations, that lead to limit cycles with periods near 10 and 1 years, respectively. Both oscillatory instabilities have a strong baroclinic component. Above a certain threshold in τ the solutions become chaotic. Flow pattern evolution in this chaotic regime resembles qualitatively the circulation found in the Gulf Stream and Kuroshio current systems after their separation from the continent.

Boolean delay equations (BDEs) provide a mathematical framework to formulate and analyze conceptual models of complex multi-component systems. This framework is used here to construct a simple conceptual model for the El-Nino/Southern Oscillation (ENSO) phenomenon. ENSO involves the coupling of atmospheric and oceanic processes that are far from being completely understood. Our BDE model uses Boolean variables to represent key atmospheric and oceanic quantities and equations that involve logical operators to describe their evolution. Two distinct time-delay parameters, one for the local atmosphere–ocean coupling effects and the other for oceanic wave propagation, are introduced. Over a range of physically relevant delay values, this truly minimal model captures two essential features of ENSO’s interannual variability — its regularity and its tendency to phase-lock to the annual cycle. Oscillations with average cycle length that is an integer multiple of the seasonal cycle are prevalent and range from 2 to 7 years. Transition zones — where the average period lengths are noninteger rational multiples of the forcing period — exhibit Devil’s staircases, a signature of the quasi-periodic (QP) route to chaos. Our BDE model thus validates results from previous studies of the interaction of the seasonal cycle with ENSO’s “delayed oscillator”. It gives therewith support to the view that the observed irregularity results predominantly from low-order chaotic processes rather than from stochastic weather noise. Moreover, in the transition zone between the two integer periodicities of 2 and 3 years, a heretofore unsuspected, self-similar “fractal sunburst” pattern emerges in phase-parameter space. This pattern provides a distinct and more complex scenario than the QP route to chaos found in earlier, more detailed ENSO models. Period selection in this 2–3-year transitional region seems to play a key role in ENSO’s irregularity, as well as in the appearance of the observed quasi-biennial mode of variability.

An intermediate coupled model for the El Nino/Southern Oscillation (ENSO) phenomenon is used to examine the interaction of ENSO with the seasonal cycle and how this is related to the irregularity of ENSO time series. Under annual mean conditions, the mechanisms that give rise to an inherent ENSO frequency and spatial pattern are well understood in this model. When the annual cycle is included, the ENSO cycle tends to lock to rational multiples of the annual frequency. The role of such subharmonic frequency locking in the transition to chaos in ENSO models has recently been examined by Jin et al., Tziperman et al. and Chang et al. Here we fit this transition scenario into a broader perspective. We emphasize three main points: (1) Using Floquet theory, we show step by step the relation of the ENSO mode in the linearized annual average case to the linear and nonlinear cases with the seasonal cycle. The dynamics that determine the spatial pattern and the basic oscillation mechanism are the same, thus putting a decade of theory on the linear annual average case onto firmer ground. (2) The quasi-periodicity scenario for the transition to chaos fundamentally involves two parameters, one governing the amplitude of the ENSO cycle, the other the inherent period. Examining model behavior as a function of parameters reveals that at reasonable amplitude of the ENSO cycle, superstable frequency-locked regimes are more prevalent than chaotic regimes. (3) When stochastic forcing by atmospheric uncoupled variability-“weather noise”-is included, the resulting irregularity in frequency-locked regimes still carries the signature of the frequency-locked spectral peaks. Thus the role of subharmonic frequency-locked resonances is important whether or not irregularity is due to internal chaotic behavior or to weather noise.

A general tool to prove nonlinear stability for stationary solutions of infinite-dimensional Hamiltonian systems is the energy-Casimir method, proposed by Arnol'd for the two-dimensional, incompressible Euler equations and generalized since to other flows. Arnol'd's stability theorems are based on a uniform estimate for the second variation of the energy-Casimir functional, leading to a strict convexity condition. In this paper we develop the method of supporting functionals and show that a local convexity condition, equivalent to the convexity of the quasi-energy invariant of the linearized Euler equations, is also a sufficient condition for genuine stability of the fully nonlinear Euler equations.

What: The evidence for irregular climatic change between warm (thermos) and cold (cryos) extremes is reviewed, concentrating on the last two million years, the Quaternary. Who: The suspects in generating this irregular evolution of temperature and ice volume are the climatic subsystems, atmosphere, oceans, ice sheets, bedrock and biota. Processes within each subsystem are briefly described, including major feedback mechanisms that characterize their collective behavior. How: Simple, semi-empirical models are formulated and shown to have irregular behavior with both quasi-periodic and aperiodic components. Model behavior is compared with the evidence, to determine the likeliest set of suspects.

Two dimensions of a dynamical system given by experimental time series are distinguished. Statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe the attractor up to the accuracy of the data, taking into account sampling and noise problems. The dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data. Singular Spectrum Analysis (SSA) provides estimates of the statistical dimension. SSA also describes the main physical phenomena reflected by the data. It gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data. SSA is applied to four paleoclimatic records. The principal climatic oscillations and the regime changes in their amplitude are detected. About 10 degrees of freedom are statistically significant in the data. Large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension.