Manifold Projection Research Articles

Projections of transnormal manifolds

Transnormal manifolds are generalizations of convex hypersurfaces of constant width (see [7]). For such hypersurfaces it is known that every orthogonal projection onto a hyperplane has an outline which is of constant width, too. Orthogonal projections of transnormal manifolds have been studied by F.J. Craveiro de Carvalho [1] in the case when the projection is an immersion onto a convex hypersurface of constant width in a suitable affine subspace of the ambient Euclidean space. Here it will be shown that transnormality is not preserved under orthogonal projection onto hyperplanes without assuming that the manifold has the topological type of a sphere. This implies another general version of the transnormal graph theorem (see [2]). Furthermore, in the case of closed transnormal curves in 3-space also non-orthogonal parallel projection onto planes cannot preserve transnormality as is shown in the last section.

Non-Boussinesq effects on transitions in Hele–Shaw convection

Mode jumping between buoyancy-induced convection patterns in a slender Hele–Shaw cell (height>width) is studied with a center manifold projection technique at a critical aspect ratio, where both the single-roll and double-roll convection patterns destabilize simultaneously. In contradiction to some experimental observations, the Boussinesq model with insulated or isothermal side wall conditions indicates that the single-roll mode never surrenders its stability and yields to the double-roll mode as the Rayleigh number is increased. By adding the temperature dependence of fluid density and conductivity to the heat equation, it is shown that the single-roll mode can now undergo secondary bifurcations. Hysteretic jumps to the double-roll mode and smooth transitions to a mixed mode, and even to a time-periodic convection pattern, are then possible.

Dynamics of delayed systems under feedback control

Abstract We study the dynamics of a general system under linear feedback control subject to system, measurement and actuator delays. There are two general classes of dynamic behavior, resonant and nonresonant, which are pertinent to the tuning of the controllers and to the nonlinear closed-loop dynamics. For non-resonant systems, the corss-over frequency and ultimate gain are monotonic and continuous functions of the delay and conventional Ziegler—Nichols type tuning methods are appropriate. For resonant systems, however, the cross-over frequency is a discontinuous function of the delay and at certain critical values of the delay, two cross-over frequencies coexist at the same ultimate gain. Such systems cannot be tuned with the Ziegler—Nichols methods. These critical delays of a resonant system correspond to double-Hopf bifurcation points with two distinct pairs of purely imaginary eigenvalues (closed-loop poles). Using center manifold projection and normal form analyses, we are able to predict all nonlinear dynamics in the vicinity of these singularities. In particular, a two-frequency torus is always found. This is verified experimentally in a cascaded liquid level control system. There is also evidence that the torus degenerates into a chaotic attractor as the feedback gain is increased.

Process dynamic models for heterogeneous chemical reactors - an application of dynamic singularity theory

Abstract By analyzing bifurcations from the marginal gain settings of a nonlinear reactor under PI-control, several characteristics of the closed-loop reactor dynamics are revealed via the center manifold projection and normal form techniques of dynamic singularity theory. Of particular practical interests are the effects of set-point error which can often destabilize a nonlinear reactor. It is shown that judicious choice of the set-point developed here can reduce such undesirable effects. The analysis also offers extremely low dimensional nonlinear models which faithfully reproduce all the closed-loop dynamics and are especially accurate near the marginal gains. These reduced models are hence ideal for the design and tuning of reactor controllers.

Domain of feasible motions in certain mechanical systems

Domains of feasible motions — projections of integral manifolds onto a configuration space — are studied. It is shown that in systems with symmetry and in Liouville systems the reorganization of such domains is possible only simultaneously with the reorganization of the integral manifolds. An example is presented of a system with gyroscopic forces, in which the domains of feasible motions change topological type outside a bifurcation set.

Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds

Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds