## 27 Citations

Robust Devaney chaos in the two-dimensional border-collision normal form.

- MathematicsChaos
- 2022

The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on R2 can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove…

Multistability and hidden attractors in a multilevel DC/DC converter

- EngineeringMath. Comput. Simul.
- 2015

Two-Parameter Discontinuity-Induced bifurcations of Limit Cycles: Classification and Open Problems

- MathematicsInt. J. Bifurc. Chaos
- 2006

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations, and suggests three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate.

The structure of mode-locking regions of piecewise-linear continuous maps: I. Nearby mode-locking regions and shrinking points

- Physics
- 2016

The mode-locking regions of a dynamical system are subsets of parameter space within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a…

Mandelbrot-like bifurcation Structures in Chaos Band Scenario of Switched converter with Delayed-PWM Control

- EngineeringInt. J. Bifurc. Chaos
- 2010

Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones are reported.

Resonance near border-collision bifurcations in piecewise-smooth, continuous maps

- Mathematics
- 2010

Mode-locking regions (resonance tongues) formed by border-collision bifurcations of piecewise-smooth, continuous maps commonly exhibit a distinctive sausage-like geometry with pinch points called…

Torus-Bifurcation Mechanisms in a DC/DC Converter With Pulsewidth-Modulated Control

- EngineeringIEEE Transactions on Power Electronics
- 2011

Pulse-modulated converter systems play an important role in modern power electronics. However, by virtue of the complex interplay between ordinary (smooth) and so-called border-collision bifurcations…

Calculation of Invariant Manifolds of Piecewise-Smooth Maps

- MathematicsProceedings of the Southwest State University
- 2020

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the…

Torus bifurcations in Multilevel converter Systems

- MathematicsInt. J. Bifurc. Chaos
- 2011

The paper illustrates the coexistence of three stable tori with different resonance behaviors and shows how reconstruction of these tori takes place across the borders of different dynamical regimes.

Inclusion of higher-order terms in the border-collision normal form: persistence of chaos and applications to power converters

- Mathematics
- 2021

The dynamics near a border-collision bifurcation are approximated to leading order by a continuous, piecewise-linear map. The purpose of this paper is to consider the higher-order terms that are…

## References

SHOWING 1-10 OF 26 REFERENCES

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation.

- MathematicsChaos
- 2006

The present article reports the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation in the two-dimensional piecewise-linear normal form map.

Border-collision bifurcations including “period two to period three” for piecewise smooth systems

- Mathematics
- 1992

Torus birth bifurcations in a DC/DC converter

- MathematicsIEEE Transactions on Circuits and Systems I: Regular Papers
- 2006

A border-collision bifurcation that can lead to the appearance of quasi-periodicity in piecewise-smooth dynamical systems is described and it is demonstrated how a two-dimensional torus can arise from a periodic orbit through a bIfurcation in which two complex-conjugate Poincare characteristic multipliers jump abruptly from the inside to the outside of the unit circle.

Two-Parameter Discontinuity-Induced bifurcations of Limit Cycles: Classification and Open Problems

- MathematicsInt. J. Bifurc. Chaos
- 2006

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations, and suggests three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate.

On multi-parametric bifurcations in a scalar piecewise-linear map

- Mathematics
- 2006

In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that…

Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems

- Mathematics
- 1999

There exists a variety of physically interesting situations described by continuous maps that are nondifferentiable on some surface in phase space. Such systems exhibit novel types of bifurcations in…

Border-collision bifurcations: An explanation for observed bifurcation phenomena.

- Physics, MathematicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1994

A general criterion for the occurrence of border-collision bifurcations is given and illustrative numerical results, including transitions to chaotic attractors, are presented.

Grazing and border-collision in piecewise-smooth systems: a unified analytical framework.

- MathematicsPhysical review letters
- 2001

A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links grazings with border-collisions in nonsmooth maps.…