We prove that the boundary of a component $U$ of the basin of an attracting periodic cycle (of period greater than 1) for an exponential map on the complex plane has Hausdorff dimension greater than 1 and less than 2. Moreover, the set of points in the boundary of $U$ which do not escape to infinity has Hausdorff dimension (in fact: hyperbolic dimension) greater than 1, while the set of points in the boundary of $U$ which escape to infinity has Hausdorff dimension 1.
We give sufficient conditions under which the set of eventually periodic points in the intersection of immediate attracting basins boundaries is non-empty. We give other conditions under which this set is dense in the intersection.
The present work deals with the recently introduced restricted six body-problem with square configuration. It is determined that the total number of libration points are twelve and twenty for the mass parameter $0< μ< 0.25$. The multivariate form of Newton-Raphson scheme is used to discuss the basin of attraction. Different aspects of the basin of attraction are investigated and explained in detail. The complex combination of the different basins is found along the boundaries. The concept of basin entropy is used to unveil the nature of the boundaries. For $μ= 0.22$ and $0.23$, the basin of at…
Two-dimensional maps can model interactions between populations. Despite their simplicity, these dynamical systems can show some complex situations, as multistability or fractal boundaries between basins that lead to remarkable pictures. Some of them are shown and explained here for three different 2D discrete models.
Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$. Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embeddings of ${\widehat{I}}_t$ into the sphere. We also construct a family $\{χ_t\colon S^2\to S^2\}$ of sphere homeomorphisms with the property that each $χ_t$ is a factor of ${\widehat{f}}_t$, by…
We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent $λ_x$ {\bfac on the nonattracting set} is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a parti…
In this paper we study the dynamics of a family of diffeomorphisms in $\bR^2$ defined by $ F(x,y)=(g(x)+h(y),h(x)), $ where $ g(x) $ is a unimodal $C^2$-map which has the same dynamical properties as the logistic map $P(x)=μx(1-x)$, and $h(x) $ is a $C^2$ map which is a small perturbation of a linear map. For certain maps of this form we show that there are exactly two periodic points, namely an attracting fixed point and a saddle fixed point and the boundary of the basin of attraction is the stable manifold of the saddle. The basin boundary also has the same regularity as $F$, in contrast to …
The paper presents an analytical approach for predicting the safe basins (SB) in a plane of initial conditions (IC) for escape of classical particle from the potential well under harmonic forcing. The solution is based on the approximation of isolated resonance, which reduces the dynamics to conservative flow on a two-dimensional resonance manifold (RM). Such a reduction allows easy distinction between escaping and non-escaping ICs. As a benchmark potential, we choose a common parabolic-quartic well with truncation at varying energy levels. The method allows accurate predictions of the SB boun…
In this paper we investigate some properties of equilibrium points in n-dimensional linear control systems with saturated state feedback. We provide an index formula for equilibrium points and discuss its relation to boundaries of attraction basins in feedback systems with single input. In addition, we also touch upon convexity of attraction basin.
We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $α\geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $α< 2$ the boundaries between the different …
We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number $1/q$ is polynomial sense in the smooth case, and exponentially dense in the analytic case.
In this paper, we study the attracting basins of the origin in C^(k+1) for the polynomial lifts of Lattes examples. We show that their boundaries are obtained as quotient of a spherical hypersurface and we explicit the singularities that appear. These domains are surprising, because they are very close to the ball, and admit a non injective self proper holomorphic map. For the proof, we desingularize these boundaries into a spherical hypersurface in a line bundle over a complex torus by using theta fonctions. Finally, we discuss some examples in dimension 2. We show that some critically finite…
The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by C.G. Gal and the author. Additionally, it was established by the author that the problem admits a finite dimensional global attractor and a robust family of exponential attractors in the case where singularly perturbed memory kernels defined on the interior of the domain and on the boundary of the domain coincide. In the present article we report advances concerning the asymptotic behavior of this heat transfer model when the memory kernels do {\em not}…
