Dimension estimates for iterated function systems and repellers. Part II

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

Computing Birational Polynomial Surface Parametrizations without Base Points

In this paper, we present an algorithm for reparametrizing birational surface parametrizations into birational polynomial surface parametrizations without base points, if they exist. For this purpose, we impose a transversality condition to the base points of the input parametrization.

Elements of analytical solutions constructor in a class of time-optimal control problems with the break of curvature of a target set

A planar velocity control problem with a disc indicatrix and a target set with a smooth boundary having finite discontinuities of second-order derivatives of coordinate functions is considered. We have studied pseudo-vertices-special points of the goal boundary that generate a singularity for the optimal control function. For non-stationary pseudo-vertices with discontinuous curvature, one-way markers are found, the values of which are necessary for analytical and numerical construction of branches of a singular set. It is proved that the markers lie on the border of the spectrum-the region of possible values. One of them is equal to zero, the other takes an invalid value -∞. In their calculation, asymptotic expansions of a nonlinear equation expressing the transversality condition are applied. Exact formulas for the extreme points of branches of a singular set are also obtained based on markers. An example of a control problem is presented, in which the constructive elements are obtained using the developed methods (pseudo-vertex, its markers, and the extreme point of a singular set), are sufficient to construct a singular set and an optimal result function in an explicit analytical form over the entire area of consideration.

The conditions of minimum for a smooth functionon the boundary of a quasidifferntiable set

In this paper, we consider problems of mathematical programming with nonsmoothconstraints of equality type given by quasidifferentiable functions. By using thetechnique of upper convex approximations, developed by B. N. Pshenichy, necessary conditionsof extremum for such problems are established. The Lagrange multipliers signsare specified by virtue of the fact that one can construct whole familers of upper convexapproximations for quasidifferentiable function and thus the minimum points in such extremalproblems are characterized more precisely. Also the simplest problem of calculusof variations with free right hand side is considered, where the left end of the trajectorystarts on the boundary of the convex set. The transversality condition at the left end of thetrajectory is improved provided sertain sufficient conditons hold

Solving Nonholonomic Systems with the Tau Method

A numerical procedure based on the spectral Tau method to solve nonholonomic systems is provided. Nonholonomic systems are characterized as systems with constraints imposed on the motion. The dynamics is described by a system of differential equations involving control functions and several problems that arise from nonholonomic systems can be formulated as optimal control problems. Applying the Pontryagins maximum principle, the necessary optimality conditions along with the transversality condition, a boundary value problem is obtained. Finally, a numerical approach to tackle the boundary value problem is required. Here we propose the Lanczos spectral Tau method to obtain an approximate solution of these problems exploiting the Tau toolbox software library, which allows for ease of use as well as accurate results.

Ways for detection of the exact number of limit cycles of autonomous systems on the cylinder

For real autonomous systems of differential equations with continuously differentiable right-hand sides, the problem of detecting the exact number and localization of the second-kind limit cycles on the cylinder is considered. To solve this problem in the absence of equilibria of the system on the cylinder, we have developed our previously proposed ways consisting in a sequential two-step application of the Dulac – Cherkas test or the Dulac test. Additionally, a new way has been worked out using the generalization of the Dulac – Cherkas or Dulac test at the second step, where the requirement of constant sign for divergence is replaced by the transversality condition of the curves on which the divergence vanishes. With the help of the developed ways, closed transversal curves are found that divide the cylinder into subdomains surrounding it, in each of which the system has exactly one second-kind limit cycle.The practical efficiency of the mentioned ways is demonstrated by the example of a pendulum-type system, for which, in the absence of equilibria, the existence of exactly three second-kind limit cycles on the entire phase cylinder is proved.

Dynamics of post-critically finite maps in higher dimension

We study the dynamics of post-critically finite endomorphisms of $\mathbb{P}^{k}(\mathbb{C})$. We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins; thus, partially extending to any dimension is a result of Fornaess–Sibony and Rong holding in the case $k=2$.

Peeling of an Extensible Soft Microbeam to Solid: Large Deformation Analysis

Adhesion and peeling of slender structures at micro and nanoscale have attracted great attention of scientists and engineers, which hold great implications in a number of domains. In this study, we present the model formulation on the peeling of a soft beam under a concentrated force, by considering its large deformation and axis extensibility. The governing equation group and the transversality condition are then derived, according to the variation on the energy functional with movable boundary conditions. We find that the key parameters in peeling, such as the adhered segment length, applied force and length increment of the beam, are correlated with two dimensionless variables, i.e., the non-dimensional maximum displacement of the beam and non-dimensional work of adhesion. The calculated results are in excellent agreement with the experimental data, which are also compared with the infinitesimal deformation model and large deformation model with inextensible axis. These findings shed light on the design of elementary structures in micro and nanodevices, fabrication of nanofiber materials, and better application of micro/nanoprinting technique, etc.