One-dimensional Unstable Manifold Research Articles

On Morse–Smale diffeomorphisms on simply connected manifolds

We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold Mn. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point σ of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from σ, then Mn is simply connected. This fact does not follow from Morse inequalities that give only restrictions on homology groups of Mn.

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Similar to period-doubling bifurcation of fixed points, periodic orbits, it has been found since 1980's that a corresponding doubling bifurcation can also be found in the case of quasiperiodic orbits. Doubling bifurcations of quasiperiodic orbits has an important consequence on the dynamics of the system under consideration. Recently, it has been shown that subsequent doublings of quasiperiodic closed invariant curves lead to the formation of Shilnikov attractors. In this contribution, we illustrate for the first time in a discrete neuron system, the phenomenon of doubling of closed invariant curves. We also show the presence of mode-locked orbits and the geometry of one-dimensional unstable manifolds associated with them resulting in the formation of a resonant closed invariant curve. Moreover, we illustrate the phenomenon of crisis and multistability in the system.

A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds

A quadratic Lyapunov function is demonstrated for the non-invertible planar Ricker map which shows that for , and all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of ). Our approach bypasses and improves on methods that rely on monotonicity, which require . We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle.

Boxing-in of a blender in a Hénon-like family

IntroductionThe extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists.MethodsIn this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold.ResultsThis constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively.DiscussionIn particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.

Mapping the shape and dimension of three-dimensional Lagrangian coherent structures and invariant manifolds

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

Sketching 1D Stable Manifolds of 2D Maps Without the Inverse

Saddle fixed points are the centerpieces of complicated dynamics in a system. The one-dimensional stable and unstable manifolds of these saddle-points are crucial to understanding the dynamics of such systems. While the problem of sketching the unstable manifold is simple, plotting the stable manifold is not as easy. Several algorithms exist to compute the stable manifold of saddle-points, but they have their limitations, especially when the system is not invertible. In this paper, we present a new algorithm to compute the stable manifold of two-dimensional systems which can also be used for noninvertible systems. After outlining the logic of the algorithm, we demonstrate the output of the algorithm on several examples.

Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds

This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similar to in a non-chaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficiently computable ergodic average. In this process, we develop new algorithms, which may be useful beyond linear response, to compute i) a fundamental statistical quantity we introduce called the density gradient, and ii) the unstable derivatives of the regularized tangent vector field and the unstable direction. We prove that the S3 algorithm, which combines these computational ingredients that enter the stable and unstable contribution, converges like a Monte Carlo approximation of Ruelle's formula. The algorithm presented here is hence a first step toward full-fledged applications of sensitivity analysis in chaotic systems, wherever such applications have been limited due to lack of availability of long-term sensitivities.

Entropy charts and bifurcations for Lorenz maps with infinite derivatives.

This paper deals with one-dimensional factor maps for the geometric model of Lorenz-type attractors in the form of two-parameter family of Lorenz maps on the interval I=[-1,1] given by Tc,ν(x)=(-1+c⋅|x|ν)⋅sign(x). This is the normal form for splitting the homoclinic loop with additional degeneracy in flows with symmetry that have a saddle equilibrium with a one-dimensional unstable manifold. Due to L. P. Shilnikov' results, such a bifurcation (under certain conditions) corresponds to the birth of the Lorenz attractor. We indicate those regions in the parameter plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps

We study scenarios of the appearance of strange homoclinic attractors (which contain only one fixed point of saddle type) for one-parameter families of three-dimensional non-orientable maps. We describe several types of such scenarios that lead to the appearance of discrete homoclinic attractors including Lorenz-like and figure-8 attractors (which contain a saddle fixed point) as well as two types of attractors of spiral chaos (which contain saddle-focus fixed points with the one-dimensional and two-dimensional unstable manifolds, respectively). We also emphasize peculiarities of the scenarios and compare them with the known scenarios in the orientable case. Examples of the implementation of the non-orientable scenarios are given in the case of three-dimensional non-orientable generalized Hénon maps.

Scenarios of hyperchaos occurrence in 4D Rössler system.

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a one-dimensional unstable invariant manifold and Neimark-Sacker bifurcations of stable cycles. The onset of the discrete hyperchaotic Shilnikov attractors containing a saddle-focus cycle with a two-dimensional unstable invariant manifold is confirmed numerically in a Poincaré map of the model. A new phenomenon, "jump of hyperchaoticity," when the attractor under consideration becomes hyperchaotic due to the boundary crisis of some other attractor, is discovered.

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

Abstract We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T c,ɛ (x) = (−1 + c|x| 1−ɛ ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of Pacman Renormalization Theory that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.

Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps

For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such “subsumed” homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy [Formula: see text], in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterizations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in M at C ont M, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. M at C ont M provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features.

Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

ABSTRACTConsider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.

A Note on Hidden Transient Chaos in the Lorenz System

Abstract In this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number ρ $\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry

We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.

Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann–Lemaître–Robertson–Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ‘attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation.

Hölder shadowing in the case of cubic tangency

The paper is concerned with a model example of a diffeomorphism f of a two-dimensional surface with Axiom A in the case of cubic tangency of one-dimensional stable and unstable manifolds. It is proved that if a mapping f is C 2-smooth, then it has the Holder shadowing property with Holder exponent 1/3. An example is constructed of a C 1-smooth diffeomorphism f satisfying the Holder shadowing property with Holder exponent 1/4, but failing to have the Holder shadowing property with Holder exponent γ > 1/4.

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors’ previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.