Novel routes to chaos through torus breakdown in non-invertible maps

Zhusubaliyev, Zhanybai T.; Mosekilde, Erik

doi:10.1016/j.physd.2008.12.012

Cited by 21 publications

(10 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Moreover, symmetrical trajectories are generated by symmetrical initial conditions leading to symmetrical basins of attraction [22]. Novel routes to chaos through torus-breakdown mechanism of this system were reported in [9] and different dynamics were characterized in the parameter region given by ( , ). Figure 1 shows the dynamic behavior of (1) when is varied and two symmetrical initial conditions are considered.…”

Section: Coupled Logistic Maps: Linearmentioning

confidence: 99%

“…Moreover, by using the information obtained from bifurcation diagrams and basins of attraction, it is possible to compute the minimum control effort required to stabilize the target orbit in the defined region. In particular, the coexistence of periodic solutions in coupled logistic maps [8,9,22] is controlled by widening the basin of attraction of a period-orbit that coexists with another one, and the minimum control effort is computed aided by the unstable period-2 orbit for the linear coupling and by unstable period-1 orbits for the nonlinear coupling case.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

“…Interesting nonlinear phenomena have been reported in them. For example, in [8,9], coupled logistic maps were analyzed and new scenarios for transition to chaos were found via the creation and destruction of multilayered tori. In those papers, novel routes to chaos were described, and authors found that, depending on the coupling constant value, the system approaches different periodic attractors.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing and Controlling Basins of Attraction in Multistability Scenarios

Taborda

Angulo

2015

Mathematical Problems in Engineering

View full text Add to dashboard Cite

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

show abstract

Section: Coupled Logistic Maps: Linearmentioning

confidence: 99%

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing and Controlling Basins of Attraction in Multistability Scenarios

Taborda

Angulo

2015

Mathematical Problems in Engineering

View full text Add to dashboard Cite

show abstract

“…By Smale and Shilnikov [18,19], in the neighborhood of the transversal intersection 𝑊 𝑢 (𝑂) ∩ 𝑊 𝑠+ (𝑂) there exist countably many periodic orbits of the same type as the fixed point 𝑂 which is a saddle-focus of (1,2)-type. If the stable invariant curve 𝐿 breaks down (by Afraimovich-Shilnikov [32] or due to some other scenario [33,34,35,36]) giving chaotic attractor (torus-chaos), then, on some subinterval inside 𝜀 ∈ (𝜀 3 , 𝜀 4 ), it can contain the fixed point 𝑂 together with the nontrivial hyperbolic set of (1,2)-type and become hyperchaotic.…”

Section: Phenomenological Part Of the Scenario Of Discrete Shilnikov ...mentioning

confidence: 99%

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Shykhmamedov¹,

Karatetskaia²,

Казаков³

et al. 2020

Preprint

View full text Add to dashboard Cite

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

show abstract

“…Такая кривая является объединением неустойчивых многообразий двух седловых периодических орбит с устойчивым циклом. В последующих исследованиях [12][13][14] было установлено, что это явление -типичное свойство двумерных эндоморфизмов, причем число «слоев» замкнутой инвариантной кривой может быть и больше двух. Такую кривую мы назвали «многослойной».…”

Section: Introductionunclassified

Рождение И Разрушение Многослойной Замкнутой Кривой В Необратимых Отображениях

2010

AND

View full text Add to dashboard Cite

В статье обсуждается новый феномен, открытый авторами совсем недавно в двумерных эндоморфизмах, демонстрирующих переход к хаосу через разрушение замкнутой инвариантной кривой. Как известно, при рациональном числе вращения на этой кривой имеется четное число периодических орбит, половина из которых устойчивые, а половина -седловые, а сама кривая образована замыканием неустойчивых могогообразий седловых циклов. Как оказалось, если отображение необратимо, то седловая периодическая орбита, лежащая на замнкутой инвариантной кривой, обычно претерпевает бифуркацию удвоения периода или бифуркацию вилки. В результате этого на инвариантной кривой мягко рождаются два новых «слоя», образованные неустойчивыми многообразиями седлового цикла удвоенного периода или неустойчивыми многообразиями двух седловых циклов того же периода в случае бифуркации вилки.

show abstract

Novel routes to chaos through torus breakdown in non-invertible maps

Cited by 21 publications

References 37 publications

Computing and Controlling Basins of Attraction in Multistability Scenarios

Computing and Controlling Basins of Attraction in Multistability Scenarios

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Рождение И Разрушение Многослойной Замкнутой Кривой В Необратимых Отображениях

Contact Info

Product

Resources

About