Cusp Bifurcation Research Articles

Generalised Biological Function

A physiological function can be described as a cycle based on a cusp bifurcation set in catastrophe theory. This cycle involves four phases that are successively developed along a functional potential, which is used to perform a given physiological act. The work we present is firstly based on a detailed study of the global function of vision, which covers a vast field extending from the molecular to cerebral scale. We then present other examples of generalised functions by expanding the frame of reference, from the scale of an organ to the scale of a whole organism with predation function, and to the scale of a social group with crisis management. We observed that the criteria remain the same, whether the function is considered on the scale of a cell membrane or a group of individuals organised into a cohort. Moreover, a so-called generalised function can be broken down into simpler sub-functions, each phase of the initial function containing the physiological act of each sub-function. It seems thus that a fractal extension may characterise the four-cusp spiral. This spiral is increasingly large as the types of sub-functions are multiplied. In functions relative to sociological science, such as crisis management, which are not natural but built and designed by humans, close attention must be paid to following the rules of these generalised biological functions.

Dayside magnetosphere‐ionosphere coupling during IMF clock angle ∼90°: Longitudinal cusp bifurcation, quasi‐periodic cusp‐like auroras, and traveling convection vortices

In spite of the great progress achieved in the understanding the Earth's magnetosphere‐ionosphere system, its configuration and dynamics during periods with large horizontal interplanetary magnetic field (IMF ∣By∣ ≫ 0) is still poorly investigated. In such time intervals, the cusp/low‐latitude boundary layer (LLBL) entry regions are characterized by a more complex magnetic reconnection topology, than during IMF Bz‐dominated periods, which in turn leads to peculiar magnetosheath plasma injections into the magnetosphere and associated ionospheric signatures. In this context, we discuss the 19 December 2002 dayside aurora activity, observed by the ITACA2 twin all‐sky camera system, along the Greenland‐Svalbard sector, during a nearly horizontal IMF (By ∼ 15 nT and Bz ∼ 0). The event originates from a longitudinally bifurcated cusp configuration and then develops into quasi‐periodic red aurora activations, between about 1000 and 1140 UT. By analyzing the Greenlandic magnetometer data, we identify a series of westward traveling convection vortices (TCVs), moving away from magnetic noon at 2–3 km/s and synchronized with the transit of red‐dominated auroral forms. We find that the auroral emission takes place in coincidence with the ionospheric footprint of the downward field‐aligned currents (counterclockwise TCVs), instead of the upward ones (clockwise TCVs), as it would be expected for precipitating electrons. This fact and several analogies with previously reported quasi‐periodic TCV activity, observed during IMF By‐dominated periods, suggest that the geomagnetic activity could be associated with FACs that are triggered by the magnetic merging on the dayside, either by inducing azimuthal pressure gradients at the magnetopause or by driving magnetopause/LLBL wave activity.

Multiple attractors via CUSP bifurcation in periodically varying environments

Periodically forced (non-autonomous) single species population models support multiple attractors via tangent bifurcations, where the corresponding autonomous models support single attractors. Elaydi and Sacker obtained conditions for the existence of single attractors in periodically forced discrete-time models. In this paper, the Cusp Bifurcation Theorem is used to provide a general framework for the occurrence of multiple attractors in such periodic dynamical systems.

Taylor–Couette flow with independently rotating end plates

Results are presented from a combined numerical and experimental study of steady bifurcation phenomena in a modified Taylor–Couette geometry where the end plates of the flow domain are allowed to rotate independently of the inner cylinder. The ends rotate synchronously and the ratio between the rate of rotation of the ends ωe and the inner cylinder ωi defines a control parameter Ω:=ωe/ωi. Stationary ends favour inward motion along the end walls whereas rotating walls promote outward flow. We study the exchange between such states and focus on two-cell flows, which are found in the parameter range between Ω=0 and Ω=1 for Γ=2. Hence Ω is used as an unfolding parameter. A cusp bifurcation is uncovered as the organizing centre for the stability exchange between the two states. Symmetry breaking bifurcations, which lead to flows that break the mid-plane symmetry are also revealed. Overall, excellent agreement is found between numerical and experimental results.

Standing waves in flow between finite counterrotating cylinders.

Experimental evidence for standing waves resulting from a supercritical Hopf bifurcation that appears as the first pattern-forming instability in counterrotating Taylor-Couette flow is presented. Depending on the aspect ratio two different types of standing waves, denoted as SW0 and SW(pi), could be observed. Both modes have an azimuthal wave number m=1 but differ in symmetry. While for SW(pi), a spatiotemporal glide-reflection symmetry could be found, SW0 is purely spatial reflection symmetric. The transition between the two modes is found to be organized in a cusp bifurcation unfolded by variations of the aspect ratio. The "classical" spiral vortex flow appears in this control parameter regime only as a result of a secondary steady bifurcation from SW0. This transition is found to be either subcritical or supercritical. The experimentally observed bifurcation structure has been predicted by theory of Hopf bifurcation to spiral vortex flow in finite counterrotating Taylor-Couette systems.

Feedback Regulation in the Lactose Operon: A Mathematical Modeling Study and Comparison with Experimental Data

A mathematical model for the regulation of induction in the lac operon in Escherichia coli is presented. This model takes into account the dynamics of the permease facilitating the internalization of external lactose; internal lactose; beta-galactosidase, which is involved in the conversion of lactose to allolactose, glucose and galactose; the allolactose interactions with the lac repressor; and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We have paid particular attention to the estimation of the parameters in the model. We have tested our model against two sets of beta-galactosidase activity versus time data, as well as a set of data on beta-galactosidase activity during periodic phosphate feeding. In all three cases we find excellent agreement between the data and the model predictions. Analytical and numerical studies also indicate that for physiologically realistic values of the external lactose and the bacterial growth rate, a regime exists where there may be bistable steady-state behavior, and that this corresponds to a cusp bifurcation in the model dynamics.

Multi-site observations of the association between aurora and plasma convection in the cusp/polar cap during a southeastward(<i>B<sub>y</sub></i> <u>~</u> |<i>B<sub>z</sub></i>|) IMF orientation

Abstract. In a case study we demonstrate the spatiotemporal structure of aurora and plasma convection in the cusp/polar cap when the interplanetary magnetic field (IMF) Bz < 0 and By ~ | Bz | (clock angle in GSM Y - Z plane: ~ 135°). This IMF orientation elicited a response different from that corresponding to strongly northward and southward IMF. Our study of this "intermediate state" is based on a combination of ground observations of optical auroral emissions and ionospheric plasma convection. Utilizing all-sky cameras at NyAlesund, Svalbard and Heiss Island (Russian arctic), we are able to monitor the high-latitude auroral activity within the ~10:00–15:00 MLT sector. Information on plasma convection is obtained from the SuperDARN radars, with emphasis placed on line of sight observations from the radar situated in Hankasalmi, Finland (Cutlass). A central feature of the auroral observations in the cusp/polar cap region is a ~ 30-min long sequence of four brightening events, some of which consists of latitudinally and longitudinally separated forms, which are found to be associated with pulsed ionospheric flows in merging and lobe convection cells. The auroral/convection events may be separated into different forms/cells and phases, reflecting a spatiotem-poral evolution of the reconnection process on the dayside magnetopause. The initial phase consists of a brightening in the postnoon sector (~ 12:00–14:00 MLT) at ~ 73° MLAT, accompanied by a pulse of enhanced westward convection in the postnoon merging cell. Thereafter, the event evolution comprises two phenomena which occur almost simultaneously: (1) westward expansion of the auroral brightening (equatorward boundary intensification) across noon, into the ~ 10:00–12:00 MLT sector, where the plasma convection subsequently turns almost due north, in the convection throat, and where classical poleward moving auroral forms (PMAFs) are observed; and (2) auroral brightening at slightly higher latitudes (~ 75° MLAT) in the postnoon lobe cell, with expansion towards noon, giving rise to a clear cusp bifurcation. The fading phase of PMAFs is accompanied by a "patch" of enhanced (~ 1 km/s) poleward-directed merging cell convection at high latitudes (75–82° MLAT), e.g. more than 500 km poleward of the cusp equatorward boundary. The major aurora/convection events are recurring at ~ 5–10 min intervals. Key words. Magnetospheric physics (auroral phenomena; magnetopause, cusp, and boundary layers; plasma convection)

Effect of the Reynolds number on streamline bifurcations in a double-lid-driven cavity with free surfaces

The two-dimensional Navier–Stokes equations for a Newtonian fluid in the absence of body forces are considered in a rectangular double-lid-driven cavity with free surface side walls, the cavity aspect ratio A and three cases of the ratio ( S=0,−1,1) of the upper to the lower lid speed. Using a finite element formulation with a mesh which is adaptively refined to facilitate the location of stagnation points, the effect of Reynolds numbers ( Re) in the range [0,100] on the streamline patterns and their bifurcations is investigated as A is varied for each S. For Re→0 and each S as A is decreased, a sequence of pitchfork bifurcations at a stagnation point on x=0 is identified as seen in the work of Gaskell et al. [Proc Instn Mech Engrs Sci Part C 212 (1998) 387]. As Re increases for S=0 and decreasing A the stagnation point on x=0 disappears and away from x=0 cusp (saddle–node) bifurcations arise rather than the pitchfork bifurcation whereas for S=−1 and Re∈[0,100] the origin is always a stagnation point at which the same type of bifurcations arises.

Fluctuations and bistability in a “hybrid” atomistic model for CO oxidation on nanofacets: An effective potential analysis

We analyze fluctuations in a “hybrid” atomistic model mimicking CO oxidation on nanoscale facets of metal(100) catalyst surfaces. The model incorporates a mean-field-like treatment of infinitely mobile CO, and a lattice-gas treatment of the superlattice ordering of immobile O. For an infinite system, it exhibits an Ising-type order–disorder transition for O, together with mean-field-like bistability disappearing at a cusp bifurcation. For finite systems, we use kinetic Monte Carlo simulation to study the probability distribution for the population of adsorbed species, from which bistability can be observed, together with fluctuation-induced transitions between the two stable states. An effective potential picture emerges from our analyses that can be used to quantify both the system size dependence of fluctuations and the transition rates. Thus, our hybrid atomistic model displays fluctuation behavior analogous to traditional mean-field models. This qualitative behavior can be understood by approximate treatments of population dynamics using master equations and Fokker–Planck equations. A generalized model with finite mobility of CO is also analyzed for comparison with the hybrid model. In contrast, it exhibits fluctuation behavior akin to equilibrium systems with Ising-type first-order transitions.

Nonlinear observer design for bifurcating systems

This paper is a geometric study of the nonlinear observer design for bifurcating systems. First, we obtain necessary and sufficient conditions for local exponential observers for nonlinear bifurcating systems under some stability assumptions. Next, using our characterization of local exponential observers, we derive a construction procedure for exponential observers for the bifurcating systems. As applications of our observer design, we consider codimension one bifurcations such as the pitchfork and Hopf bifurcations, and codimension two bifurcations such as the cusp bifurcation.

Nonlinear observer design for discrete-time bifurcating systems

This paper is a geometric study of the nonlinear observer design for discrete-time bifurcating systems. First, we obtain necessary and sufficient conditions for local exponential observers for discrete-time nonlinear bifurcating systems under some stability assumptions. Next, using our characterization of local exponential observers, we derive a construction procedure for exponential observers for the discrete-time bifurcating systems. As applications of our observer design, we consider codimension one bifurcations such as the pitchfork bifurcation, and codimension two bifurcations such as the cusp bifurcation.

Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

{Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ Rn , α∈ Rm can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c1 n+ c2 m+ c3 , ci∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.}

Catalytic CO oxidation on nanoscale Pt facets: effect of interfacet CO diffusion on bifurcation and fluctuation behavior.

We present lattice-gas modeling of the steady-state behavior in CO oxidation on the facets of nanoscale metal clusters, with coupling via interfacet CO diffusion. The model incorporates the key aspects of the reaction process, such as rapid CO mobility within each facet and strong nearest-neighbor repulsion between adsorbed O. The former justifies our use of a "hybrid" simulation approach treating the CO coverage as a mean-field parameter. For an isolated facet, there is one bistable region where the system can exist in either a reactive state (with high oxygen coverage) or a (nearly CO-poisoned) inactive state. Diffusion between two facets is shown to induce complex multistability in the steady states of the system. The bifurcation diagram exhibits two regions with bistabilities due to the difference between adsorption properties of the facets. We explore the role of enhanced fluctuations in the proximity of a cusp bifurcation point associated with one facet in producing transitions between stable states on that facet, as well as their influence on fluctuations on the other facet. The results are expected to shed more light on the reaction kinetics for supported catalysts.

Fluctuations and critical phenomena in catalytic CO oxidation on nanoscale Pt facets

Local fluctuations and fluctuation-induced transitions in catalytic CO oxidation are studied with field electron microscopy on the (112) facets of a [100]-oriented Pt field emitter tip. The reaction is investigated in the bistable range close to the cusp point (critical point) that terminates the bistability range in ${p}_{\mathrm{CO}},$ T-parameter space. The amplitude and the spatial coherence of the fluctuations increase on approaching the critical point. The fluctuations are spatially well correlated on each flat (112) facet, but their correlation decays rapidly across stepped regions that terminate the flat facets. On smaller (112) facets, an onset of fluctuation-induced transitions is observed earlier (i.e., further away from the critical point) than for larger (112) facets. The behavior of the reaction system near the cusp point appears to be similar to that of an equilibrium system near the critical point. The observed fluctuations are mimicked in a simple reaction model for CO oxidation on surfaces that incorporates both rapid diffusion of adsorbed CO, and superlattice ordering of adsorbed immobile oxygen. The steady states of the model exhibit a cusp bifurcation, from a regime of bistability to one of monostability. The fluctuations increase near this cusp point, as in experiment. This behavior is analyzed via kinetic Monte Carlo simulations and analytic procedures, focusing on the consequences for fluctuation-induced transitions.

CO-oxidation model with superlattice ordering of adsorbed oxygen. I. Steady-state bifurcations

We analyze a model for CO oxidation on surfaces which incorporates both rapid diffusion of adsorbed CO, and superlattice ordering of adsorbed immobile oxygen on a square lattice of adsorption sites. The superlattice ordering derives from an “eight-site adsorption rule,” wherein diatomic oxygen adsorbs dissociatively on diagonally adjacent empty sites, provided that none of the six additional neighboring sites are occupied by oxygen. A “hybrid” formalism is applied to implement the model. Highly mobile adsorbed CO is assumed randomly distributed on sites not occupied by oxygen (which is justified if one neglects CO–CO and CO–O adspecies interactions), and is thus treated within a mean-field framework. In contrast, the distribution of immobile adsorbed oxygen is treated within a lattice–gas framework. Exact master equations are presented for the model, together with some exact relationships for the coverages and reaction rate. A precise description of steady-state bifurcation behavior is provided utilizing both conventional and “constant-coverage ensemble” Monte Carlo simulations. This behavior is compared with predictions of a suitable analytic pair approximation derived from the master equations. The model exhibits the expected bistability, i.e., coexistence of highly reactive and relatively inactive states, which disappears at a cusp bifurcation. In addition, we show that the oxygen superlattice ordering produces a symmetry-breaking transition, and associated coarsening phenomena, not present in conventional Ziff–Gulari–Barshad-type reaction models.

Ideal glass-glass transitions and logarithmic decay of correlations in a simple system

We calculate the ideal-glass-transition line for adhesive hard spheres in the temperature-volume-fraction plane within the framework of the mode-coupling theory. We find two intersecting lines, controlled by the hard-core and the adhesive part of the potential respectively, giving rise to two different mechanisms for structural arrest. In the glass region we identify the presence of a glass-glass-transition line ending in a cusp bifurcation which causes, even in the close by liquid region, a logarithmic decay of correlations.

Numerical Normalization Techniques for All Codim 2 Bifurcations of Equilibria in ODE's

Explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous ODEs are derived. They include second-order coefficients for the Bogdanov--Takens bifurcation, third-order coefficients for the cusp and fold-Hopf bifurcations, and coefficients of the fifth-order terms for the generalized Hopf (Bautin) and double Hopf bifurcations. The formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the Jacobian matrix of the right-hand sides and its transpose, as well as multilinear functions from the Taylor expansion of the right-hand sides at the critical equilibrium. The normal form coefficients for the fold-Hopf bifurcation in the "new" Lorenz model are computed using the derived formulas, proving the existence of a nontrivial invariant set in the system.

Dynamics of the aurora and associated convection currents during a cusp bifurcation event

We report the detailed two‐dimensional evolution of a transient auroral bifurcation in the cusp region which occurred during an interval of eastward (By>0) oriented interplanetary magnetic field (IMF). The actual auroral bifurcation is characterized by the near‐simultaneous activation and intensification of two latitudinally separated auroral forms which both expanded westward across the 1200 MLT meridian, after their initial appearance in the postnoon sector. The northernmost form (called type 2) is typical of periods of northward IMF orientation while the southernmost form (called type 1) represents a continuation and intensification of a type of auroral activity which is characteristic of intervals of southward IMF orientation. The auroral bifurcation is commonly observed when the IMF is in an intermediate regime of clock angles (∼45°–90°), i.e., when By is the dominating IMF component. We explain the transient cusp bifurcation in terms of a short‐lived activation of high‐latitude reconnection and an associated lobe cell convection/current system and type 2 aurora embedded in a long interval of active low‐latitude reconnection/type 1 aurora.

The Hénon and Heiles Problem in Three Dimensions.

The second part of a research on the Hénon and Heiles system in three dimensions is presented. We focus on motions around the origin, where the system may be treated as a case of three perturbed isotropic harmonic oscillators. It is the only Hamiltonian of the family of axially symmetric cubic potentials in 1–1–1 resonance, which needs at least order four in the normalization to reach the features of the system. The fourth order normalized system is analyzed, considering orbits at any inclination. We study and classify the relative equilibria of the system finding pitchfork, centre-saddle and cusp bifurcations in the reduced orbit space.

Theory of the periodic orbits of a chaotic quantum well

A theory is developed for the periodic orbits of an electron trapped in a rectangular potential well under the influence of an electric field normal to the barriers and a magnetic field. When the magnetic field is parallel to the electric field the dynamics of an electron in the well is integrable; however, when it is tilted by an angle $\ensuremath{\theta}$ the system undergoes a transition to chaos. Motivated by recent experimental and theoretical studies of magnetotunneling in quantum wells that emphasize the role of periodic orbits, we present here a unified theory of all the periodic orbits within the well that are of relevance to experiments. We define the appropriate scaled variables for the problem, which we divide into two qualitatively different cases, the single-barrier model (depending on two parameters) and the double-barrier model (depending on three parameters). We show that in both cases all relevant orbits are related to bifurcations of period-one traversing orbits. A full analytic theory is derived for the period and stability of these traversing orbits; and analytic and numerical results are obtained for the important period-two and period-three orbits. An unusual feature of the classical mechanics of the double-barrier is a discontinuity in the classical Poincar\'e map, which leads to a new type of bifurcation that we term a cusp bifurcation. We show that all the periodic orbits that traverse the well exist only in finite intervals of voltage and magnetic field, appearing and disappearing in bifurcations. These intervals are shown to correspond to the appearance of new resonance peaks in the experimental data, laying the foundation for a quantitative semiclassical treatment of the system.