Narrow Results

Reservoir Computing (RC) is a type of machine learning inspired by neural processes, which excels at handling complex and time-dependent data while maintaining low training costs. RC systems generate diverse reservoir states by extracting features from raw input and projecting them into a high-dimensional space. One key advantage of RC networks is that only the readout layer needs training, reducing overall training expenses. Memristors have gained popularity due to their similarities to biological synapses and compatibility with hardware implementation using various devices and systems. Chaotic events, which are highly sensitive to initial conditions, undergo drastic changes with minor adjustments. Cascade chaotic maps, in particular, possess greater chaotic properties, making them difficult to predict with memoryless devices. This study aims to predict 1D and 2D cascade chaotic time series using a memristor-based hierarchical RC system.

There are two natural definitions of the Julia set for complex Hénon maps: the sets $J$ and $J^{\star }$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set $J^{\star }$, under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann.334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann.334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used.

We show that $J=J^{\star }$ also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of ${ℂ}^2$. VIII. Quasi-expansion. Amer. J. Math.124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on $J^{\star }$. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set $J$ were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The P_k=k×2ⁿ, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.

We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numerical techniques. The symbol space is naturally given by components of the complement of the stable and unstable manifolds. The shift map defining the dynamics is a factor of a subshift of finite type, and is obtained from a graph related to the tangle. The entropy of this shift map is a lower bound for the topological entropy of the planar diffeomorphism. We give examples arising from the Hénon family.

For invertible, area-contracting planar maps, much work has been done on the study of the crucial role played by certain unstable periodic orbits, distinguished by being "accessible", in understanding some global bifurcations such as metamorphoses of basin boundaries and crisis of attractors. In this paper we concentrate on a one-parameter family of Hénon maps whose attractors present a type of discontinuous change characterized by a sudden replacement of the accessible orbits as the parameter is varied. The change is quantified by a jump in the corresponding accessible rotation number. The rotation rate describes a devil's staircase as a function of the parameter. We estimate the Hausdorff dimensions of the Hénon attractors — via the Mendès France fractal dimension — for the same interval of parameter values. Our purpose is to show the strong connection between these two functions: the variation of the dimension stays minimal for the parameters under the same plateau in the staircase, while it is markedly greater when the parameter moves from one plateau to another.

We study hyperbolic dynamics and bifurcations for generalized Hénon maps in the form (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.

This paper presents an efficient method for finding horseshoes in dynamical systems by using several simple results on topological horseshoes. In this method, a series of points from an attractor of a map (or a Poincaré map) are firstly computed. By dealing with the series, we can not only find the approximate location of each short unstable periodic orbit (UPO), but also learn the dynamics of almost every small neighborhood of the attractor under the map or the reverse map, which is very helpful for finding a horseshoe. The method is illustrated with the Hénon map and two other examples. Since it can be implemented with a computer software, it becomes easy to study the existence of chaos and topological entropy by virtue of topological horseshoe.

An optimal box-counting algorithm for estimating the fractal dimension of a nonempty set which changes over time is given. This nonstationary environment is characterized by the insertion of new points into the set and in many cases the deletion of some existing points from the set. In this setting, the issue at hand is to update the box-counting result at appropriate time intervals with low computational cost. The proposed algorithm tackles the dynamic box-counting problem by using computational geometry methods. In particular, we use a sequence of compressed Box Quadtrees to store the data points. This storage permits the fast and efficient application of our box-counting approach to compute what we call the "dynamic fractal dimension". For a nonempty set of points in the d-dimensional space ℝ^d (for constant d ≥ 1), the time complexity of the proposed algorithm is shown to be O(n log n) while the space complexity is O(n), where n is the number of considered points. In addition, we show that the time complexity of an insertion, or a deletion is O(log n), and that the above time and space complexity is optimal. Experimental results of the proposed approach illustrated on the well-known and widely studied Hénon map are presented.

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically.

The quadratic map of the interval displays one attractor for each parameter value. Conservative maps of the plane display infinite coexistence of stability islands around periodic orbits. Between these two extremes, dissipative systems of the plane are known to have infinite coexistence of sinks as a generic property, yet very hard to detect. We investigate how more and more coexistence is gained as the area-contraction rate b → 1. In this paper, we show a sequence of simple sinks gaining coexistence, and investigate the convergence properties of its bifurcation values. The sinks are simple, or primary, due to their geometrical structure.

The onset and bifurcation points of n-cycles of several polynomial maps are located through a characteristic equation connecting cyclic polynomials of the cycle points. The polynomials satisfied by the parameters of the logistic, Hénon, and cubic maps at the onset and bifurcation points are obtained for n up to 14, 9, and 9, respectively.

The topological entropy of finite representations of the Hénon map is studied. Efficient methods to compute the topological entropy of finite representations of maps are presented. Accurate finite representations of the Hénon map and its iterates are constructed and the topological entropy of these representations is calculated. The relation between the topological entropy of the Hénon map and the topological entropy of its finite representations is discussed.

Thanks to the complex characteristics of ergodicity, pseudo-randomness and sensitivity in initial conditions, chaotic systems have been widely applied in the field of cryptography. By cascading the Hénon map and the Chebyshev map, a new two-dimensional Hénon–Chebyshev modulation map (2D-HCMM) is proposed in this paper. Several methods of objective assessment, including phase diagrams, bifurcation diagrams, Lyapunov exponents and information entropy, are utilized to analyze the dynamics of the 2D-HCMM. The results show that the 2D-HCMM possesses better ergodicity and unpredictability, with larger chaotic ranges, compared with the original chaotic maps. By using the proposed map and the essential principles of genetic recombination and genetic mutation, a new image encryption scheme is proposed. In this scheme, the bit planes of image are substituted by genetic recombination operation, and the pixel values are scrambled randomly by genetic mutation operation. The simulation results and security analysis demonstrate that the proposed scheme not only can resist various conventional attacks, but also possesses a fast speed, achieving a good trade-off between security and efficiency.

We propose a fast nonlinear method for assessing quantitatively both the existence and directionality of linear and nonlinear couplings between a pair of time series. We test this method, called Boolean Slope Coherence (BSC), on bivariate time series generated by various models, and compare our results with those obtained from different well-known methods. A similar approach is employed to test the BSC’s capability to determine the prevalent coupling directionality. Our results show that the BSC method is successful for both quantifying the coupling level between a pair of signals and determining their directionality. Moreover, the BSC method also works for noisy as well as chaotic signals and, as an example of its application to real data, we tested it by analyzing neurophysiological recordings from visual cortices.

The renowned 2D invertible Hénon map turns into 1D noninvertible quadratic map when its leading parameter $b$ becomes zero. This well-known link was studied by Mira who demonstrated that the bifurcation set of Hénon diffeomorphism is similar to his “box-within-a-box” bifurcation structure of 1D endomorphism. In general, such similarity has not been strictly established, especially in multidimensional cases. In this paper, we proved that the Mira bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional generalized Hénon map. The changes of periodic and homoclinic orbits and chaotic attractors at this transition are described. We proved the existence of Newhouse regions is different from those Mira boxes that accumulate to the homoclinic bifurcations.

The method described here relies on interval arithmetic and graph theory to compute guaranteed coverings of strange attractors like Hénon attractor. It copes with infinite intervals, using either a geometric method or a new directed projective interval arithmetic.

In this paper, we propose a hybrid method called sliding-window amplitude-based dispersion entropy, which combines dispersion entropy with sliding-window amplitude, to characterize nonlinear time series. This hybrid method not only inherits the fast calculation speed and the ability to characterize nonlinear time series of dispersion entropy, but also has higher noise resistance than dispersion entropy. We firstly utilize three artificial data (logistic map, Hénon map, ARFIMA model) to qualify the effectiveness of the proposed method, results show that our method can correctly characterize the nonlinear time series, and has stronger robustness to noise. Next, the method is applied to analyze stock market system, the data of stock market are composed of six main indices from different countries, the result shows that the proposed method can easily distinguish the emerging markets and developed markets, and can reveal some features under the financial time series.

Reservoir Computing (RC) is a type of machine learning inspired by neural processes, which excels at handling complex and time-dependent data while maintaining low training costs. RC systems generate diverse reservoir states by extracting features from raw input and projecting them into a high-dimensional space. One key advantage of RC networks is that only the readout layer needs training, reducing overall training expenses. Memristors have gained popularity due to their similarities to biological synapses and compatibility with hardware implementation using various devices and systems. Chaotic events, which are highly sensitive to initial conditions, undergo drastic changes with minor adjustments. Cascade chaotic maps, in particular, possess greater chaotic properties, making them difficult to predict with memoryless devices. This study aims to predict 1D and 2D cascade chaotic time series using a memristor-based hierarchical RC system.

We give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models. We argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos. Setting the stochastic models within the framework of non-linear autoregression, we introduce the notion of a generalized partial autocorrelation and an order. We approach the estimation of the embedding dimension via order determination of an unknown non-linear autoregression by cross-validation, and give justification by proving its consistency under global boundedness. As a by-product, we provide a theoretical justification of the final prediction error approach of Auestad and Tjøstheim. Some illustrations based on the Hénon map and several real data sets are given. The bias of the residual sum of squares as essentially a noise variance estimator is quantified.