Dynamical systems with Lipschitz inverse shadowing properties

In this paper, the notion of the Lipschitz inverse shadowing property with respect to two classes of d-methods that generate pseudotrajectories of dynamical systems is introduced. It is shown that if a diffeomorphism of a Euclidean space has the Lipschitz inverse shadowing property on the trajectory of an individual point, then the Mañé analytic strong transversality condition must be satisfied at this point. This result is used in the proof of the main theorem: a diffeomorphism of a smooth closed manifold that has the Lipschitz inverse shadowing property is structurally stable.     

Lipschitz distributions and Anosov flows

We show that if a distribution is locally spanned by Lipschitz vector fields and is involutive a.e., then it is uniquely integrable giving rise to a Lipschitz foliation with leaves of class . As a consequence, we show that every codimension-one Anosov flow on a compact manifold of dimension such that the sum of its strong distributions is Lipschitz, admits a global cross section.

     

A shadowing lemma for quasi-hyperbolic strings of flows

In this paper we prove a shadowing lemma for pseudo orbits made by quasi-hyperbolic strings. We allow singularities in question and hence, in particular, the quasi-hyperbolic strings are formulated by the rescaled linear Poincaré flow instead of the usual linear Poincaré flow. We also introduce the sectional Poincaré map and rescaled sectional Poincaré map for Lipschitz vector fields on Banach spaces in the article.     

On integral stability and Lipschitz stability of motion

We establish new conditions of integral stability and uniform Lipschitz stability based on the use of the comparison principle and a matrix-valued Lyapunov function. Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal Vol. 49, No. 1, pp. 77–83, January, 1997.     

Structural stability of vector fields with shadowing

Let X be a C¹ vector field without singularities. In this paper, we show that X is in the C¹ interior of the set of vector fields with the shadowing property if and only if X satisfies both Axiom A and the strong transversality condition; that is, X is structurally stable.     

Lipschitz stability of controlled invariant subspaces

Let (A,B)∈C^n×n×C^n×m and M be an (A, B)-invariant subspace. In this paper the following results are presented: (i) If M∩ImB={0}, necessary and sufficient conditions for the Lipschitz stability of M are given. (ii) If M contains the controllability subspace of the pair (A, B), sufficient conditions for the Lipschitz stability of the subspace M are given.     

Lipschitz selections and stability for quasiconvex programs

Stability of ε-optimal solutions for quasiconvex programs is studied.     

Lipschitz stability of operators in Banach spaces

We consider approximations of an arbitrarymap F: X → Y between Banach spaces X and Y by an affine operator A: X → Y in the Lipschitz metric: the difference F — A has to be Lipschitz continuous with a small constant ? > 0. In the case Y = ? we show that if F can be affinely ?-approximated on any straight line in X, then it can be globally 2?-approximated by an affine operator on X. The constant 2? is sharp. Generalizations of this result to arbitrary dual Banach spaces Y are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.     

Generalizations in the problem of Lipschitz stability

The problem of Lipschitz stability of nonlinear systems of differential equations is considered. Theorems extending previous results are stated. Translated fromDinamicheskie Sistemy. Vol. 12, pp 101–105, 1993.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

A remark on Lipschitz stability for inverse problems

An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse medium problem for the Helmholtz equation.     

On genericity of shadowing and periodic shadowing property

Let M be a smooth compact manifold (possibly with boundary) or compact manifold with dimM?3 or dimM?6. We prove that generic homeomorphism on M has the periodic shadowing property.     

Energy stability of channel flows

Zusammenfassung Die Energiemethode wird zur Untersuchung der Stabilität einer beliebigen Kanalströmung angewandt. Es wird eine auf Eigenwertabschätzungen basierte qualitative Beschreibung der Fläche (, ) gegeben, welche die Schranke für den Instabilitätsbereich in Abhängigkeit von Längs- bzw. Querwellenzahl der Störungen darstellt. Eine allgemeine untere Grenze für die minimalisierende Querwellenzahl und den zugehörigen Eigenwert (0, ) wird angegeben. Das Problem der Energiestabilität wird dann für eine Klasse von Geschwindigkeitsprofilen mit Wendepunkt gelöst. Die durch numerische Integration erhaltenen Ergebnisse werden mit den Ergebnissen der linearen Stabilitätstheorie verglichen.     

Lipschitz stability in determining density and two Lamé coefficients

We consider an inverse problem of determining spatially varying density and two Lamé coefficients in a non-stationary isotropic elastic equation by a single measurement of data on the whole lateral boundary. We prove the Lipschitz stability provided that initial data are suitably chosen. The proof is based on a Carleman estimate which can be obtained by the decomposition of the Lamé system into the rotation and the divergence components.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

On the Lipschitz stability of complex systems of differential equations

Mathematical models of real physical processes and phenomena can be described by the so-called complex systems of differential equations. Among them are systems that decompose into interacting subsystems and intrasystemic connections. They have significant nonlinearities. In this connection the application of the existing methods of study of the dynamics of the behavior of such systems is quite difficult. We propose some approaches to the study of Lipschitz stability connected with the technique of applying Lyapunov functions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 12–16.     

Lipschitz stability criteria for a generalized delayed Kolmogorov model

Lipschitz stability and Lipschitz? _° - equistability of the functional differential equation $x' = B(x)f(t,x,x_t ), x_{t_ \circ = \theta _ \circ } $ are discussed. Sufficient conditions are given using the comparison with the corresponding scalar equation.