Calculation of Invariant Manifolds of Piecewise-Smooth Maps

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

Time-Invariant Working Memory Representations in the Presence of Code-Morphing in the Lateral Prefrontal Cortex

Endogenous processes allow the maintenance of working memories. These processes presumably involve prefrontal networks with strong recurrent connections. Distractors evoke a morphing of the population code, even when memories are stable. But it is unclear whether these dynamic population responses contain stable memory information. Here we show that dynamic prefrontal activity contains stable memory information, and the stability depends on parallel movement of trajectories associated with different memories in state space. We used an optimization algorithm to find a subspace with stable memory information. In correct trials the stability extended to periods that were not used to find the subspace, but in error trials the information and the stability were reduced. A bump attractor model was able to replicate these behaviors. The model provided predictions that could be confirmed with the neural data. We conclude that downstream regions could read memory information from a stable subspace.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

Stable Subspaces of Positive Maps of Matrix Algebras

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.