Singularity Theory Techniques Research Articles

Focal set of curves in the Minkowski space near lightlike points

We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal set is not defined. We use singularity theory techniques to carry out our study and investigate the focal set near lightlike points.

Solution Branching and Dive Plane Reversal of Submarines at Low Speeds

The problem of multiple steady-state solutions in the dive plane of submarines under depth control at low speeds is analyzed. This phenomenon occurs regardless of the particular means used for depth control, manual or automatic, and linear or nonlinear. It is shown that the primary bifurcation parameter is a Froude-like number based on the vehicle speed and metacentric height. Generic solution branching is shown to occur below a critical Froude number. Singularity theory techniques are employed to quantify the effects that various vehicle geometric properties and hydrodynamic characteristics have on steady-state motion. It is demonstrated that a comprehensive bifurcation study provides a systematic and effective way of predicting the phenomenon of dive plane reversal at low speeds. A complete characterization of the parameters in the problem, both in deep water and at periscope depth, is achieved through the organizing center of the pitchfork singularity.

On the geometry and dynamics of floating four-bar linkages

In this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed in part through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops

Degenerate Hopf Bifurcation and Isolas of Periodic Solutions in an Enzyme-Catalyzed Reaction Model

A point of degenerate Hopf bifurcation in an enzyme-catalyzed model is rigorously analyzed by using techniques of singularity theory and interval analysis. The existence of the point is proven by a numerical computation using interval analysis. Singularity theory as developed by M. Golubitsky and D. G. Schaeffer is then used to completely characterize the families of small amplitude periodic solutions that arise for parameters near the generate values. The normal form for the singularity is X5 + 2m0ΛX3 + Λ2X, with Λ the bifurcation parameter, m0 a model parameter. Among the results is the existence of an isolated branch (isola) of periodic solutions. Purely numerical techniques are used to confirm the results: excellent agreement is found between the bifurcation theoretic unfolding and (numerical) continuation results using pseudo-arclength continuation.

Unfolding a Point of Degenerate Hopf Bifurcation in an Enzyme-Catalyzed Reaction Model

A point of degenerate Hopf bifurcation in an enzyme-catalyzed model previously studied by Doedel [2] is rigorously analyzed by using techniques of singularity theory and interval analysis. A computation using interval analysis proves the existence of a point of degenerate Hopf bifurcation, which is a smooth function of additional parameters in the model system. Singularity theory as developed by Golubitsky and Langford [J. Differential Equations, 3 (1981), pp. 375–415.] and Golubitsky and Schaeffer [Singularities and Groups in Bifurcation Theory I, Springer-Verlag, New York, 1985.] is then used to construct universal unfoldings of the degeneracy, to completely characterize the families of small amplitude periodic solutions that arise for parameters near the degenerate values. Computations using interval analysis are employed in this proof. Excellent agreement is found between the bifurcation theoretic unfolding and (numerical) continuation results using pseudoarclength continuation.

Stability and Bifurcations of Towed Underwater Vehicles in the Dive Plane

The problem of static and dynamic loss of stability in the vertical plane in steady towing of underwater vehicles is considered. Bifurcations of steady-state equilibria are studied using singularity theory techniques and all qualitatively different bifurcation diagrams that occur locally are revealed. Analytical conditions for stability of straight-line motion are derived. Bifurcations to periodic solutions are analyzed and shown to provide paths to complicated dynamics. The incorporation of the techniques used in this work, with related studies in cable dynamics, can lead to a design methodology for safer and more efficient operations.

The search for a realistic superstring vacuum

Recent progress in understanding heterotic string compactification utilizes abstract algebraic methods. In particular, Gepner has given a prescription for constructing exactly soluble Calabi-Yau compactifications by using N = 2 minimal models. A large class of new N = 2 models, discovered by Kazama & Suzuki, can also be used. Recent progress in understanding N = 2 models and Calabi-Yau spaces by using mathematical techniques of singularity theory improves the prospects of a complete classification. A key question is whether a classical solution can give a reasonable first approximation to the exact quantum ground state even though string theory is strongly coupled and the perturbation expansion diverges.

Quadratic autocatalysis in a non-isothermal CSTR

Typical analyses of chemical systems have so far been carried out on either isothermal autocatalytic systems or deceleratory reactions with thermal feedback. In real chemical systems, however, heat may often be released by an autocatalytic reaction. The quadratic autocatalator with kinetic scheme ▪ is now studied as an example of a non-isothermal, non-adiabatic reaction in a CSTR. The techniques of singularity theory and Hopf theory are used to study the effects of thermal feedback on the steady-state and the dynamic behaviour of this four parameter system. An interesting variety of steady-state behaviour is found including breaking wave, isola and mushroom patterns. Hopf bifurcations and sustained oscillations also exist for large ranges of parameter values with a wide combination of stable and unstable limit cycles being found.

Degenerate Hopf Bifurcation and Nerve Impulse. Part II

The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...

A qualitative and quantitative study of steady-state response of towed floating bodies

The problem of multiple equilibria in steady towing of a floating body is considered. The two coordinates of the towing point are the main bifurcation parameters. An approach to bifurcation of steady-state equilibria using singularity theory reveals all qualitatively different bifurcation diagrams that occur locally. It is shown that these bifurcation problems may be viewed as paths in the universal unfolding space of the cusp catastrophe. The organizing centre for the towing problem is the pitchfork singularity. Numerical calculations suggest that results obtained by singularity-theory techniques are valid globally.