A construction of reversible extensions of dynamical systems which applies to arbitrary mappings (not necessarily with open range) is presented. It is based on calculating the maximal ideal space of C*-algebras that extends endomorphisms to partial automorphisms via partial isometric representations, and involves a newfound set of "parameters" (the role of parameters play chosen sets or ideals). Additionally, it is characterised as a universal object. As model examples, we give a thorough description of reversible extensions of logistic maps, and a classification of systems associated with compression of unitaries generating homeomorphisms of the circle.
We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.
We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.
For area-preserving Hénon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, the original method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic points in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well a…
In this paper we study R-reversible area-preserving maps f on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that Ro f=f^{-1}o R where R is an isometric involution on M. We obtain a C1-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits. As a consequence we obtain the proof of the stability conjecture for this class of maps. Along the paper we also derive the C1-closing lemma for reversible maps and other perturbation toolboxes.
An element of a group is \emph{reversible} if it is conjugate to its own inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line.
We show that every uniquely ergodic minimal Cantor system is topological orbit equivalent to the natural extension of a numeration scale associated to a logistic map.
In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks' theorem on a dichotomy of the number of periodic points of area preserving maps on the annulus to symmetric periodic points of area preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area preserving rever…
We consider the logistic family and apply the $q$-deformation $ϕ_q(x)=\frac{1-q^x}{1-q}$. We study the stability regions of the fixed points of the $q$-deformed logistic map and the regions where the dynamic is complex through topological entropy and Lyapunov exponents. Our results show that the dynamic of this deformed family is richer than that of the $q$-deformed family studied in [8].
We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map $f$ can be taken so that each such invariant measu…
We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, whe…
We introduce new models of very weakly coupled logistic and tent maps for which orbits of very long period are found. The length of these periods is far greater than one billion. The property of these models relatively to the distribution of the iterated points (invariant measure) is described.
In this paper, we show that the stability analysis in the paper A note on stability of fractional logistic maps, Appl. Math. Lett. 125 (2022) 107787 is incorrect and repeat a proof of a theorem on the convergence of a convolution of the product of a converging series with a converging sequence. We also mention that the paper commented on should have been edited more carefully.
An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group $\mathfrak{G}_n$ of formal maps of $(\mathbb{C}^n,0)$, i.e. formally-invertible $n$-tuples of formal power series in $n$ variables, with complex coefficients. The case $n=1$ was already understood. Each product $F$ of reversibles has linear part $L(F)$ of determinant $\pm1$. The main results are that for $n\ge2$ each map $F$ with det$(L(F))=\pm1$ is the product of …
A route to chaos is studied in 3-dimensional maps of logistic type. Mechanisms of period doubling for invariant closed curves (ICC) are found for specific 3-dimensional maps. These bifurcations cannot be observed for ICC in the 2-dimensional case. When the parameter of the system is modified, localized oscillations occur on the ICC that give rise to weakly chaotic rings, then to chaotic attractors, which finally disappear by contact bifurcations. These maps can be considered as models for the symbiotic interaction of three species.
We consider reversible non-conservative perturbations of the conservative cubic Hénon maps $H_3^{\pm}: \bar x = y, \bar y = -x + M_1 + M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i.e. bifurcations of fixed points with eigenvalues $e^{\pm i 2π/3}$. It follows from the work by Dullin and Meiss, this resonance is degenerate for $M_1=0, M_2=-1$ when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map $H_3^+$ and…
We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18 (2005) 2171-2192). We show that these statistics correspond to those of a composition of two random involutions, having an appropriate number of f…
We study the family of quadratic maps f_a(x) = 1 - ax^2 on the interval [-1,1] with a between 0 and 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a density which is bounded away from zero and is analogous to the density for the corresponding closed system. These results establish the exponential escape rate of Lebesgue measure from the system, despite the contraction in a neighborhood of the critical point of the map. We also prove convergence of the c…
Many natural systems are organized as networks, in which the nodes interact in a time-dependent fashion. The object of our study is to relate connectivity to the temporal behavior of a network in which the nodes are (real or complex) logistic maps, coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We investigate in particular the relationship between the system architecture and possible dynamics. In the current paper we focus on establishing the framework, terminology and pertinent questions for low-dimensional networks. A subsequen…
In a previous work by the authors the one dimensional (doubling) renormalization operator was extended to the case of quasi-periodically forced one dimensional maps. The theory was used to explain different self-similarity and universality observed numerically in the parameter space of the Forced Logistic Maps. The extension proposed was not complete in the sense that we assumed a total of four conjectures to be true. In this paper we present numerical support for these conjectures. We also discuss the applicability of this theory to the Forced Logistic Map.
In this paper we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points. A local bifurcation analysis of the system using Center Manifold is undertaken and then supported by numerical computations.This reveals the existence of reverse flip and Neimark-Sacker bifurcations.
We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values $s_n$ where the $2^n$-periodic orbit is superattracting. Moreover these parameter values lay between one period doubling and the next. Under quasi-periodic forcing, the superattracting periodic orbits give birth to two reducibility-loss bifurcations in the two dimensional parameter space of the Forced Logistic Map, both around the points $s_n$. In the present w…
In this paper, a rigorous analysis of the behavior of the standard logistic map, Logistic Tent system (LTS), Logistic-Sine system (LSS) and Tent-Sine system (TSS) is performed using 0-1 test and three state test (3ST). In this work, it has been proved that the strength of the chaotic behavior is not uniform. Through extensive experiment and analysis, the strong and weak chaotic regions of LTS, LSS and TSS have been identified. This would enable researchers using these maps, to have better choices of control parameters as key values, for stronger encryption. In addition, this paper serves as a …
We explain the relation between the $r=1$ logistic map $x_{i+1}=rx_i(1-x_i)$, $x_i\in\mathbb R$, $i=0,1,\ldots$, $r>0$ and $x_0\geq0$, and the RG flow in the multiscale analysis of zero fixed point, asymptotic free QFT models as e.g. the ultraviolet (1+1)-dimensional Gross-Neveu model and QCD, and the infrared $ϕ^4_4$. We obtain the asymptotics of the mapping, showing an inverse power decay to the fixed point $x^*=0$ Gaussian fixed point, with logarithmic-like corrections. This asymptotics is independent of the initial condition $x_0\in(0,1)$ (so, there is no constraint for $x_0$ to be small, …
It is well known that for $a>4$, the dynamical behaviors of the logistic map $f_a(x)=ax(1-x)$ on the maximal invariant compact set are "simple" which could be clearly explained by the theories of hyperbolic dynamics and symbolic dynamics. Is it possible that similar phenomena could be observed in general real polynomial maps? In this paper, we study this problem by investigating the real polynomial map $f_a(x)=ag(x)$, where $a$ is a parameter, and $g$ is a real-coefficient polynomial, which has at least two different real zeros or only one real zero.
For any infinite transitive sofic shift $X$ we construct a reversible cellular automaton (i.e. an automorphism of the shift $X$) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible CA which is sensitive with respect to all directions. As another application we prove a finitary Ryan's theorem: the automorphism group aut$(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of $S$-gap …
