Nonautonomous Maps Research Articles

Parametric topological entropy on orbits of arbitrary multivalued maps in compact Hausdorff spaces

The Adler-Konheim-McAndrew type definitions and the Bowen-Dinaburg-Hood type definitions of parametric topological entropy will be considered on orbits and coincidence orbits of nonautonomous multivalued maps in compact Hausdorff spaces. Their mutual relationship and their link to various further types of definitions like those of (parametric) preimage entropy, will be investigated. In this way, several recent results of the other authors will be generalized and extended into a new setting.

Full-deautonomisation of a class of second-order mappings in ancillary form

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

Non-visible transformations of chaotic attractors due to their ultra-low density in AC–DC power factor correction converters

Recently, it has been shown that DC–AC and AC–DC power converters whose dynamics is governed by two vastly different frequencies lead to a special class of piecewise smooth models characterized by a practically unpredictable number of switching manifolds between partitions in the state space associated with different dynamics. In the previous publications, we have shown that transformations of chaotic attractors in such models can be caused by their interactions with unstable cycles occurring in the regions of the attractors associated with an ultra-low invariant density, which is beyond a practical observability in physical or numerical experiments. The appearance of such low-density regions has been explained by the so-called spiking phenomenon. In the present paper, we make the next step and ask the natural question what are the reasons behind the appearance of spikes. Applying a novel technique (cobweb diagrams for non-autonomous maps), we explain a connection between spiking and the presence of two vastly different frequencies.

Chaotic driven maps: Non-stationary hyperbolic attractor and hyperchaos

In this paper we study simple examples of non-autonomous maps having different changing in time chaotic attractors. We present the definition of non-stationary hyperbolic attractor of the driven maps. We rigorously prove the existence of non-stationary hyperbolic attractor in 2D driven map and introduce a hyperchaotic attractor for autonomous 3D map of master-slave structure. Our analysis is based on the auxiliary systems approach and the construction of invariant cones.

Historic behaviour for nonautonomous contraction mappings

We consider a parametrised perturbation of a diffeomorphism on a closed smooth Riemannian manifold with , modeled by nonautonomous dynamical systems. A point without time averages for a (nonautonomous) dynamical system is said to have historic behaviour. It is known that for any diffeomorphism, the observability of historic behaviour, in the sense of the existence of a positive Lebesgue measure set consisting of points with historic behaviour, disappears under absolutely continuous, independent and identically distributed (i.i.d.) noise. By contrast, we show that the observability of historic behaviour can appear by a non-i.i.d. noise: we consider a contraction mapping for which the set of points with historic behaviour is of zero Lebesgue measure and provide an absolutely continuous, non-i.i.d. noise under which the set of points with historic behaviour is of positive Lebesgue measure.

Studies on spaces of initial conditions for non-autonomous mappings of the plane

Studies on spaces of initial conditions for non-autonomous mappings of the plane

Deautonomizations of integrable equations and their reductions

We present a deautonomization procedure for partial difference and differential-difference equations (with the latter defining symmetries of the former) which uses the integrability conditions as integrability detector. This procedure is applied to Hirota’s Korteweg–de Vries and all the ABS equations and leads to non-autonomous equations and their non-autonomous generalized symmetries of order two, all of which depend on arbitrary periodic functions and are related to the same two-quad equation and its symmetries. We show how reductions of the derived differential-difference equations lead to alternating QRT maps, and periodic reductions of the difference equations result to non-autonomous maps and discrete Painleve type equations.

On the Periodic Logistic Map

In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map $${x_{n + 1}} = {r_n}{x_n}\left( {1 - {x_n}} \right),n = 0,1, \ldots ,$$ where {r n } is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel–Roberts–Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai’s classification.

Global transport in a nonautonomous periodic standard map

A non-autonomous version of the standard map with a periodic variation of the perturbation parameter is introduced and studied via an autonomous map obtained from the iteration of the nonautonomous map over a period. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers of invariant sets. The role of the nonautonomous dynamics on period-one orbits, stability and bifurcation is studied. The critical boundaries for the global transport and for the destruction of invariant circles with fixed rotation number are studied in detail using direct computation and a continuation method. In the case of global transport, the critical boundary has a particular symmetrical horn shape. The results are contrasted with similar calculations found in the literature.

On the global dynamics of periodic triangular maps

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular maps. We show that, under certain conditions, the orbit of every point in a periodic non-autonomous triangular map converges to a fixed point (respectively, periodic orbit of period p) if and only if there is no periodic orbit of prime period two (respectively, periodic orbits of prime period greater than p).

Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

Bifurcations of the class $A_{\mu}$, in Arnold's classification, in non-autonomous $p$-periodic difference equations generated by parameter depending families with $p$ maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relatively to cyclic order of compositions for any natural number $\mu$. The main tool for the proofs is the local topological conjugacy. Invariance results are essential for proper definition of bifurcations of the class $A_{\mu}$ and associated lower codimension bifurcations, using all possible cyclic compositions of fiber families of maps of the $p$-periodic difference equation. Finally, we present two examples of the class $A_{3}$ or swallowtail bifurcation occurring in period two difference equations for which bifurcation conditions are invariant.

Inverse period-doubling bifurcations determine complex structure of bursting in a one-dimensional non-autonomous map.

We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.

Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautonomous Hénon Map

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set.

Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model

Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees-of-freedom. The model is formulated as a large set of $N$ coupled standard-like area-preserving twist maps in which the amplitude and phase of the perturbation, rather than being constant like in the standard map, are dynamical variables. Of particular interest is the study of the impact of periodic orbits on the chaotic transport and coherent structures. Numerical simulations show that self-consistency leads to the formation of a coherent macro-particle trapped around the elliptic fixed point of the system that appears together with an asymptotic periodic behavior of the mean field. To model this asymptotic state, we introduced a non-autonomous map that allows a detailed study of the onset of global transport. A turnstile-type transport mechanism that allows transport across instantaneous KAM invariant circles in non-autonomous systems is discussed. As a first step to understand transport, we study a special type of orbits referred to as sequential periodic orbits. Using symmetry properties we show that, through replication, high-dimensional sequential periodic orbits can be generated starting from low-dimensional periodic orbits. We show that sequential periodic orbits in the self-consistent map can be continued from trivial (uncoupled) periodic orbits of standard-like maps using numerical and asymptotic methods. Normal forms are used to describe these orbits and to find the values of the map parameters that guarantee their existence. Numerical simulations are used to verify the prediction from the asymptotic methods.

Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems – a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the “singular sets” of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the “chaotic saddle” for autonomous and nonautonomous versions of the Hénon map.

Dynamics from Multivariable Longitudinal Data

We introduce a method of analysing longitudinal data in n≥1 variables and a population of K≥1 observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space Sn. At each time, information of each observation is coded to a point (x,y)∈Sn, where x is the physical condition of the observation and y is an ordering of variables. Orbit of each observation in Sn is described by a map that dynamically rearranges order of variables at each time step, eventually placing the most stable, least frequently changing variable to the left and the most frequently changing variable to the right. By this operation, we are able to extract dynamics from data and visualise the orbit of each observation. In addition, clustering of data in the stable variables is revealed. All possible paths that any observation can take in Sn are given by a subshift of finite type (SFT). We discuss mathematical properties of the transition matrix associated to this SFT. Dynamics of the population is a nonautonomous multivalued map equivalent to a nonstationary SFT. We illustrate the method using a longitudinal data of a population of households from Agincourt, South Africa.

LOCAL BIFURCATION IN ONE-DIMENSIONAL NONAUTONOMOUS PERIODIC DIFFERENCE EQUATIONS

In this article we extend the theory of local bifurcation in one-dimensional autonomous maps to one-dimensional nonautonomous periodic maps. We give the necessary conditions for the main types of local bifurcation in one-dimensional periodic maps.

Application of the companion factorization to linear non-autonomous area-preserving maps

For a large class of discrete matrix difference equations many qualitative problems remain unsolved. The companion matrix factorization is applied here to the shift matrices associated to linear non-autonomous area-preserving maps. It permits us to introduce second order linear difference equations, which provide a faster computation of the transition matrices with respect to numerical algorithms based on the standard product of matrices. In addition, compact representations for the main elements of these discrete planar systems can be provided when using the well-known solutions of linear difference equations. Some properties and applications of current interest are presented.

The Szego matrix recurrence and its associated linear non-autonomous area-preserving map

A  change  to  the  Szegö matrix  recurrence  relation,  satisfied  by  orthonormal  poly- nomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU (1; 1). The companion factorization of such matrices, via 2nd-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU (1; 1) ≃ SL(2; R) enables the introduction of a linear non-autonomous area-preserving  map.   This  dynamical  system  has  counterparts  in  those  from  the  complex Szegö recurrence relation, and some basic results are outlined.