On structurally stable diffeomorphisms with codimension one expanding attractors

We show that if a closed -manifold admits a structurally stable diffeomorphism with an orientable expanding attractor of codimension one, then is homotopy equivalent to the -torus and is homeomorphic to for . Moreover, there are no nontrivial basic sets of different from . This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on , .

     

On the creation of stable periodic behavior of parametrically excited dynamical systems

The problem of the realization of stable periodic behavior of dynamical systems is considered. It is shown analytically that in certain cases it is possible to achieve by parametric perturbation stable periodic behavior of systems that in the autonomous case possess only unstable oscillatory or stationary regimes or are in a stable equilibrium position.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 507–512, September, 1995.     

Compact leaves of structurally stable foliations

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspension foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary compact manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in the sense of Ehresmann and Reeb.     

On the class of infinite horizon optimal control problems affine,which are structurally stable

We prove in this article that there is in the set of all problems we consider a subset, which is residual. Every problem in this subset is shown to be structurally stable and defines a dynamical system, which looks like the graph of the figure given in Section  1 , contrary to what happens for ordinary dynamical systems, that is, the ones associated with ODEs. There, the initial value problem (in the smooth case) is uniquely solvable; the structurally stable systems look like the figure given in Section  1 , but sources are equilibria. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponentially asymptotically stable dynamical systems

In this paper, we give some sufficient conditions which guarantee that the zero solution of the differential equation with two delayed terms x(t) = a(t)x(t-p(t)) - b(t)x(t-r(t))is uniformly stable     

On a Bernoulli property of some piecewiseC 2-diffeomorphisms in ℝ d

Some piecewiseC ²-diffeomorphisms which expand distances uniformly (in all directions) are considered. It is shown that they have the so-called Bernoulli property, i.e. their natural extensions are isomorphic to Bernoulli shifts.     

Stable and structurally stable regulators for singular systems

This paper deals with the structure, the existence conditions and the design methods of the stable and structurally stable regulators for the singular system . Two classes of normal regulators are proposed: the first class of them, using the measurementy as its input, is applicable to the situation where the structural parameter pecturbations (especiallyE) is constrained. On the other hand, the second class of them, usingy and as its input, is applicable to arbitrsry perturbations in the sense for singular systems of the structural parameters. But this is done at the cost of additional equipment to measure the (the speed ofy).     

A dynamical characterization of evolutionarily stable states

Evolutionary stability, the central solution concept in evolutionary game theory, is closely related to local asymptotic stability in a certain nonlinear dynamical system operating on the state space, the so-called "replicator dynamics". However, a purely dynamical characterization of evolutionary stability is not available in an elementary manner. This characterization can be achieved by investigating so-called "derived games" which consist of mixed strategies corresponding to successful states in the original game. Using well-known facts, several characterization results are obtained within this context. These also may shed light on the extremality properties of evolutionary stability.     

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability(in particular, stationarity, periodicity, quasiperiodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations(ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.     

Construction and asymptotic stability of structurally stable internal layer solutions

We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined.

Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues.

An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection.

We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.

     

Flight-test-based construction of structurally stable models for the dynamics of large space structures

A method is proposed for constructing dynamic models of large space structures (LSS) when their parameter values are uncertain and LSS state measurements in actual operation conditions are incomplete and subject to errors. The method can be used to construct structurally stable dynamic models of LSS when ground-based LSS tests are impossible.

     

On the duals of the stable rank 3 reflexive sheaves on P3 withc 3 maximal

Summary One proves some properties of the spectrum of the duals of the stable rank 3 reflexive sheaves onP ³ withc ₁=0 andc ₃=c ₂ ² -c ₂ and one determines the Chern classes of these duals.

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Riassunto Si prova alcune proprietà dello spettro degli duali degli fasci riflessivi stabili di rango 3 suP ³ conc ₁=0 ec ₃=c ₂ ² -c ₂ e si determina le classi di Chern di questi duali.

     

On the transformations of the dynamical equations

In this issue we bring to the reader’s attention a translation of Levi-Civita’s work “Sulle trasformazioni delle equazioni dinamiche”. This paper, written by Levi-Civita at the onset of his career, is remarkable in many respects. Both the main result and the method developed in the paper brought the author in line with the greatest mathematicians of his day and seriously influenced the further progress of geometry and the theory of integrable systems. Speaking modern language the main result of his paper is the deduction of the general geodesic equivalence equation in invariant form and local classification of geodesically equivalent Riemannian metrics in the case of arbitrary dimension, i.e., metrics having the same geodesics considered as unparameterized curves (this classification problem was formulated by Beltrami in 1865). Levi-Civita’s work produced a great impact on further development of the theory of geodesically equivalent metrics and geodesic mappings, and still remains one of the most important tools in this area of differential geometry. In this paper the author uses a new method based on the concept of Riemannian connection, which later has been also referred to as the Levi-Civita connection. This paper is truly a pioneering work in the sense that the real power of covariant differentiation techniques in solving a concrete and highly nontrivial problem from the theory of dynamical systems was demonstrated. The author skillfully operates and weaves together many of the most advanced (for that times) algebraic, geometric and analytic methods. Moreover, an attentive reader can also notice several forerunning ideas of the method of moving frames, which was developed a few decades later by E. Cartan. We hope that the reader will appreciate the style of exposition as well. This work, focused on the essence of the problem and free of manipulation with abstract mathematical terms, is a good example of a classical text of the late 19th century. Owing to this, the paper is easy to read and understand in spite of some different notation and terminology. The Editorial Board is very grateful to Professor Sergio Benenti for the translation of the original Italian text and valuable comments (see marginal notes at the end of the text, p. 612).     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Existence of uniformly stable solutions of nonautonomous discontinuous dynamical systems

We are concerned here with the existence of uniformly Lyapunov stable integrable solution of linear and nonlinear nonautonomous discontinuous dynamical systems.     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).     

Discrete dynamical systems with invariant asymptotically stable toroidal manifold

We obtain conditions for asymptotic stability of quasiperiodic trajectories of discrete dynamical systems in the case of infinite-dimensional Banach space. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 466–471, April, 1999.