On the existence of low-period orbits in n -dimensional piecewise linear discontinuous maps (original) (raw)

Border Collision Bifurcations in n-Dimensional Piecewise Linear Discontinuous Maps

Eprint Arxiv Nlin 0601038, 2006

In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. Proceeding along similar lines, we obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map.

Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

International Journal of Bifurcation and Chaos, 2014

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, we prove that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

Multi-parametric bifurcations in a piecewise–linear discontinuous map

Nonlinearity, 2006

In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

Chaos, Solitons & Fractals, 2011

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.

Analysis of atypical orbits in one dimensional linear piecewise-smooth discontinuous map

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous map are examined analytically. It is shown that, under certain parameter conditions, the map exhibits atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomena occurs. The paper also illustrate that there exists a specific parameter region in which as parameter is varied, there is a smooth transition from stable to unstable periodic orbits. Moreover, we have derived the expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics that exist at this value of parameter is also found out.Mathematics Subject Classification (2020) 39A23 · 39A28 · 39A33

BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

International Journal of Bifurcation and Chaos, 2010

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.