Nonsmooth Perturbations Research Articles

Closed-form periodic solutions for piecewise-linear vibrating systems

In this paper, closed-form asymptotic solutions are derived for piecewise-linear one- and two-degrees-of-freedom systems. Deviations from perfectly linear restoring force characteristics play the role of small but nonsmooth perturbations. It is shown that the nonsmooth transformation of temporal argument enables one of justifying at least first several steps of the classic perturbation procedure which eventually gives the unit-form solutions. The form of the solutions is suitable for further manipulations including possible generalization on non-periodic cases.

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.

Chaos prediction in the Duffing-type system with non-smooth periodic perturbation and bounded parametric excitation

In this paper, the dynamics of Duffing-type system with non-smooth periodic perturbation and bounded noise was studied. The theory of Fourier series was used in this system to deal with the non-smooth character for the first time. The analytical condition for the appearance of chaos was given using stochastic Melnikov method. The numerical simulations confirm the validity of this method.

A Duality Approach for the Boundary Variation of Neumann Problems

In two dimensions, we study the stability of the solution of an elliptic equation with Neumann boundary conditions for nonsmooth perturbations of the geometric domain. Using harmonic conjugates, we relate this problem to the shape stability of the solution of an elliptic equation with Dirichlet boundary conditions. As a particular case, we prove the stability of the solution under a topological constraint (uniform number of holes), which is analogous to Sverak's result for Dirichlet boundary conditions.

Semi-Hyperbolic Mappings, Condensing Operators, and Neutral Delay Equations

Semi-hyperbolic mappings in Banach spaces are Lipschitz continuous and not necessarily invertible. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. It is shown that semi-hyperbolic mappings are locallyψ-contracting, whereψis the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it isψ-contracting and has no spectral values on the unit circle. A bishadowing result, which combines both direct and indirect forms of shadowing, is extended to semi-hyperbolic mappings in Banach spaces with locally condensing continuous comparison mappings. The result is applied to linear neutral delay equations with nonsmooth perturbations.

Robustness of dynamical systems to a class of nonsmooth perturbations

Robustness of dynamical systems to a class of nonsmooth perturbations

The geometry and motion of sharp fronts within geochemical transport problems

We consider some reactive geochemical transport problems in groundwater sys­tems. When incoming fluid is in disequilibrium with the mineralogy, sharp tran­sition fronts may develop. We show that this is a generic property for a class of systems where the time scales associated with reaction and diffusion phenomena are much shorter than those associated with advective transport. Such multi­ple timescale problems are relevant to a variety of processes in natural systems: mathematically, methods of singular perturbation theory reduce the dimension of the problems to be solved locally. Furthermore, we consider how spatial heteroge­neous mineralogy can make an impact upon the propagation of sharp geochemical fronts. We develop an asymptotic approach in which we solve equations for the evolv­ing geometry of the front and indicate how the non-smooth perturbations, due to natural heterogeneity of the mineralogy on underlying groundwater flow field, are balanced against the smoothing effect of diffusion-dispersive processes. Fronts are curvature damped, and the results here indicate the generic nature of sepa­rate front propagation within both model (idealized) and natural (heterogeneous) geochemical systems.

Sharp Fronts within Geochemical Transport Problems

AbstractWe consider some reactive geochemical transport problems in groundwater systems. When incoming fluid is in disequilibrium with the mineralogy sharp transition fronts may develop. We show that this is a generic property for a class of systems where the timescales associated with reaction and diffusion phenomena are much shorter than those associated with advective transport. Such multiple timescale problems are relevant to a variety of processes in natural systems: mathematically methods of singular perturbation theory reduce the dimension of the problems to be solved locally. Furthermore we consider how spatial heterogeneous mineralogy can impact upon the propagation of sharp geochemical fronts.We develop an asymptotic approach in which we solve equations for the evolving geometry of the front and indicate how the non-smooth perturbations due to natural heterogeneity of the mineralogy on underlying ground water flow field are balanced against the smoothing effect of diffusion/dispersive processes. Fronts are curvature damped, and the results here indicate the generic nature of separate front propagation within both model (idealized) and natural (heterogeneous) geochemical systems.