scholarly journals Minimal periodic orbits for continuous maps of the interval

1984 ◽  
Vol 286 (2) ◽  
pp. 595-595
Author(s):  
Llu{í}s Alsed{à ◽  
Jaume Llibre ◽  
Rafel Serra
Keyword(s):  
Periodic Orbits ◽  
Continuous Maps

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Minimal Periodic Orbits for Continuous Maps of the Interval

1984 ◽  
Vol 286 (2) ◽  
pp. 595 ◽  
Author(s):  
Lluis Alseda ◽  
Jaume Llibre ◽  
Rafel Serra
Keyword(s):  
Periodic Orbits ◽  
Continuous Maps

scholarly journals The hidden unstable orbits of maps with gaps

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190473
Author(s):  
Mike R. Jeffrey ◽  
Simon Webber
Keyword(s):  
Periodic Orbits ◽  
Smooth Maps ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Piecewise Continuous

Piecewise-continuous maps consist of smooth bran- ches separated by jumps, i.e. isolated discontinui- ties. They appear not to be constrained by the same rules that come with being continuous or differentiable, able to exhibit period incrementing and period adding bifurcations in which branches of attractors seem to appear ‘out of nowhere’, and able to break the rule that ‘period three implies chaos’. We will show here that piecewise maps are not actually so free of the rules governing their continuous cousins, once they are recognized as containing numerous unstable orbits that can only be found by explicitly including the ‘gap’ in the map’s definition. The addition of these ‘hidden’ orbits—which possess an iterate that lies on the discontinuity—bring the theory of piecewise-continuous maps closer to continuous maps. They restore the connections between branches of stable periodic orbits that are missing if the gap is not fully accounted for, showing that stability changes must occur in discontinuous maps via stability changes not so different to smooth maps, and bringing piecewise maps back under the powerful umbrella of Sharkovskii’s theorem. Hidden orbits are also vital for understanding what happens if the discontinuity is smoothed out to render the map continuous and/or differentiable.

Adaptive Control of Transient Dynamics to Periodic Orbits

2014 ◽  
Vol 2 ◽  
pp. 82-85
Author(s):  
Hiroyasu Ando ◽  
Kazuyuki Aihara
Keyword(s):  
Adaptive Control ◽  
Periodic Orbits ◽  
Transient Dynamics

LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

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