The domination monoid in o-minimal theories

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of [Formula: see text]-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic 60(3) (2019) 333–351], this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.

Stable invariant manifolds with application to control problems

<p style='text-indent:20px;'>In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.</p>

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

Dynamic Analysis and Circuit Implementation of a New 4D Lorenz-Type Hyperchaotic System

This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic system while exploring new analytical results concerns its local and global dynamics. In particular, the equilibrium points of the system along with solution’s continuous dependence on initial conditions are examined. Then, a detailed Z2 symmetrical Bogdanov-Takens bifurcation analysis of the hyperchaotic system is performed. Also, the possible first integrals and global invariant surfaces which exist in system’s phase space are analytically found. Theoretical results reveal the rich dynamics and the complexity of system behavior. Finally, numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validate the present analytical study of the system.

Bell's conspiracy, Schrödinger's black cat and global invariant sets

A locally causal hidden-variable theory of quantum physics need not be constrained by the Bell inequalities if this theory also partially violates the measurement independence condition. However, such violation can appear unphysical, implying implausible conspiratorial correlations between the hidden variables of particles being measured and earlier determinants of instrumental settings. A novel physically plausible explanation for such correlations is proposed, based on the hypothesis that states of physical reality lie precisely on a non-computational measure-zero dynamically invariant set in the state space of the universe: the Cosmological Invariant Set Postulate. To illustrate the relevance of the concept of a global invariant set, a simple analogy is considered where a massive object is propelled into a black hole depending on the decay of a radioactive atom. It is claimed that a locally causal hidden-variable theory constrained by the Cosmological Invariant Set Postulate can violate the Clauser–Horne–Shimony–Holt inequality without being conspiratorial, superdeterministic, fine-tuned or retrocausal, and the theory readily accommodates the classical compatibilist notion of (experimenter) free will.