Periodic Saddle Research Articles

Existence, Number and Stability of Periodic Orbits Induced by Homoclinic Loops in Three-Dimensional Piecewise Linear Systems with an Admissible Saddle-Focus

For a class of three-dimensional piecewise linear systems with an admissible saddle-focus, the existence of three kinds of homoclinic loops is shown. Moreover, the birth and number of the periodic orbits induced by homoclinic bifurcation are investigated, and various sufficient conditions are obtained to guarantee the appearance of only one periodic orbit, finitely many periodic orbits or countably infinitely many periodic orbits. Furthermore, the stability of these newborn periodic orbits is analyzed in detail and some conclusions are made about them to be periodic saddle orbits or periodic sinks. Finally, some examples are given.

The Birth of a Hidden Attractor Through Boundary Crisis

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.

Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surface

For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.

Periodic sinks and periodic saddle orbits induced by heteroclinic bifurcation in three-dimensional piecewise linear systems with two zones

For general three-dimensional piecewise linear systems, some explicit sufficient conditions are achieved for the existence of a heteroclinic loop connecting a saddle-focus and a saddle with purely real eigenvalues. Furthermore, certain sufficient conditions are obtained for the existence and number of periodic orbits induced by the heteroclinic bifurcation, through close analysis of the fixed points of the parameterized Poincaré map constructed along the hereroclinic loop. It turns out that the number can be zero, one, finite number or countable infinity, as the case may be. Some sufficient conditions are also acquired that guarantee these periodic orbits to be periodic sinks or periodic saddle orbits, respectively, and the main results are illustrated lastly by some examples.

A numerical renormalization method for quasi–conservative periodic attractors

We describe a renormalization method in maps of the plane $ (x, y) $, with constant Jacobian $ b $ and a second parameter $ a $ acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. $ |b| = 1-\varepsilon $), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the $ (x, y, a) $ space, in sequences of diverging period, that we call 'branches'. We define a renormalization approach, by 'hopping' among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter $ a $ (see [3]) and in other ranges of the period for the dynamical plane $ (x, y) $. For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane $ (x, y) $. We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map.The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [7] for highly dissipative systems.

Singular cycles connecting saddle periodic orbit and saddle equilibrium in piecewise smooth systems

For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involved in saddle periodic orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle periodic orbit, and the other is the piecewise affine system having an admissible saddle- focus and an admissible saddle periodic orbit. Precisely, several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise affine systems, respectively. In addition, some examples are presented to illustrate the results.

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

On the Existence of Chaos in a Discontinuous Area-Preserving Map Arising in Financial Markets

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

The existence and stability analyses of periodic orbits in 3-dimensional piecewise affine systems

This paper investigates the existence and stability of periodic orbits in a class of 3-dimensional piecewise affine systems with two zones. Two results on the existence of periodic orbits are obtained by applying some fixed points theorems to a Poincaré map. Precisely, we prove the existence of a periodic sink in one case and a periodic saddle in another case. Finally, some concrete examples are given to illustrate the main results.

Cellular flow in a partially filled rotating drum: regular and chaotic advection

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

A Geometric Model of Mixing Lyapunov Exponents Inside Homoclinic Classes in Dimension Three

For $C^1$ diffeomorphisms of three dimensional closed manifolds, we provide a geometric model of mixing Lyapunov exponents inside a homoclinic class of a periodic saddle $p$ with non-real eigenvalues. Suppose $p$ has stable index two and the sum of the largest two Lyapunov exponents is greater than $\log(1-\delta)$, then $\delta$-weak contracting eigenvalues are obtained by an arbitrarily small $C^1$ perturbation. Using this result, we give a sufficient condition for stabilizing a homoclinic tangency within a given $C^1$ perturbation range.

Locally Topologically Generic Diffeomorphisms with Lyapunov Unstable Milnor Attractors

We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. D\'iaz and E. R. Pujals that, for a $C^1$ topologically generic diffeomorphism of a closed manifold, either any homoclinic class admits some dominated splitting, or this diffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism has an unstable Milnor attractor. The same results hold for statistical and minimal attractors.

Lyapunov unstable milnor attractors

We show that the Lyapunov instability of Milnor attractors is a locally topologically generic dynamical phenomenon that accompanies persistent homoclinic tangencies associated with sectionally dissipative periodic saddles.

Topological analysis of the periodic structures in a harmonically driven bubble oscillator near Blake's critical threshold: Infinite sequence of two-sided Farey ordering trees

The topology of the stable periodic orbits of a harmonically driven bubble oscillator, the Rayleigh–Plesset equation, in the space of the excitation parameters (pressure amplitude and frequency) has been revealed numerically. This topology is governed by a hierarchy of two-sided Farey trees initiated from a unique primary structure defined also by a simple asymmetric Farey tree. The sub-topology of each of these building blocks is driven by a homoclinic tangency of a periodic saddle. This self-similar organisation is a suitable basis for a general description, since it is in good agreement with partial results obtained in other periodically forced oscillators and iterated maps. The applied ambient pressure in the model is near but still below Blake's critical threshold. Therefore, this paper is also a straightforward continuation of the work of Hegedűs [1], who first found numerical evidence for the existence of stable, period 1 solutions beyond Blake's threshold. The present findings are crucial for the extension of the available numerical results from period 1 to arbitrary periodicity.

Backbones in the parameter plane of the Hénon map.

Parameter plane (b, a) of the real Hénon map has been investigated for curves of bifurcation, curves of homoclinic heteroclinic onsets, and also searching for borders of areas variously characterized. Such curves are, in general, complicated and show singularities. Pieces of two monotone curves, spanning the (b, a) parameter plane of the real Hénon map, can be detected in four quite different studies appeared along the years 1982-2008. We study the extent of their similarity to read and interpret them into the same curves. To us, these two curves are the accumulation loci of bifurcation curves of two principal families of periodic sinks of type "period-adding machine." We call them "backbones," because they are monotone; moreover, they are the borders of some important regions in the (b, a)-plane. Hamouly and Mira in 1982 [C. R. Acad. Sc. Paris1 293, 525-528 (1982)] studied the structure of bifurcation of periodic orbits and their mutual position and intersection. Gonchenko et al. [SIAM J. Appl. Dyn. Syst. 4, 407-436 (2005)] display the continuation (in parameter plane) of the first heteroclinic connection and of the first homoclinic connection between the two fixed points of the map. Alligood and Sauer [Commun. Math. Phys. 120, 105-119 (1988)] studied parameter regions characterized by the same rotation number of the "accessible" periodic saddle. Finally, Lorenz [Physica D 237, 1689-1704 (2008)] in 2008 draws areas in the parameter plane statistically characterized by a finite attractor. In this paper, we show how these criteria interact. We therefore conjecture that the wealth of curves of homoclinic onsets could be in general hierarchized by the structure of accessible saddles.

Specification and partial hyperbolicity for flows

We prove that if a flow exhibits a partially hyperbolic attractor Λ with splitting and two periodic saddles with different indices such that the stable index of one of them coincides with the dimension of Es then it does not satisfy the specification property. In particular, every sectional-hyperbolic attractor with the specification property is hyperbolic. As an application, we prove that no geometric Lorenz attractor satisfies the specification property.

Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies

This paper studies the distribution of characteristic multipliers, the structure of submanifolds, the phase diagram, bifurcations and chaotic motions in the potential field of rotating highly irregular-shaped celestial bodies (hereafter called irregular bodies). The topological structure of the submanifolds for the orbits in the potential field of an irregular body is shown to be classified into 34 different cases, including 6 ordinary cases, 3 collisional cases, 3 degenerate real saddle cases, 7 periodic cases, 7 period-doubling cases, 1 periodic and collisional case, 1 periodic and degenerate real saddle case, 1 period-doubling and collisional case, 1 period-doubling and degenerate real saddle case, and 4 periodic and period-doubling cases. The different distribution of the characteristic multipliers has been shown to fix the structure of the submanifolds, the type of orbits, the dynamical behaviour and the phase diagram of the motion. Classifications and properties for each case are presented. Moreover, tangent bifurcations, period-doubling bifurcations, Neimark-Sacker bifurcations and the real saddle bifurcations of periodic orbits in the potential field of an irregular body are discovered. Submanifolds appear to be Mobius strips and Klein bottles when the period-doubling bifurcation occurs.

Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

Structural Stability of Planar Bimodal Linear Systems

Structural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case.

Noise-induced intermittency in Hénon map and estimation of critical strength of noise

In this paper noise-induced intermittency phenomenon in Hénon map is studied. Through numerical simulation and global analysis the mechanism of noise-induced intermittency in Hénon map is revealed. By using stochastic sensitivity function method to detect whether there will be collisions between confidence ellipse of periodic attractor and chaotic saddle, a method to estimate the critical strength of noise is proposed. Results show that the noise-induced intermittency in Hénon map is caused by interactions between cycle attractor and unstable manifold of chaotic saddle, whereas the stochastic sensitivity function method can estimate the critical noise strength in intermittency phenomenon accurately.