Stable Manifolds Research Articles

On Shilnikov attractors of three-dimensional flows and maps

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results are illustrated with various specific examples.

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in [formula omitted] with [formula omitted]-symmetry and integral of motion

We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Chaotic and regular dynamics of a morphing shell with a vanishing-stiffness mode

Thin elastic shells are almost inextensible but easy to bend. In the presence of prestresses, geometric frustrations can produce complex elastic energetic landscapes, which have been tailored for the design of morphing structures with multiple stable equilibria or neutrally stable manifolds. We show that the co-existence of stiff and floppy modes leads to unexploited dynamical features. We build a neutrally stable cylindrical shell that under dynamical excitation alternates a chaotic behavior with a surprisingly regular regimes with a continuous precession of the curvature axis at a constant speed. We explain the experimental findings with a minimal model, showing how the intriguing dynamics is due to the subtle coupling between the prestress, geometrical nonlinearity, material anisotropy and inertial effects. Our results shed a new light on morphing structures dynamics and can be exploited in engineering applications such as energy harvesting.

Open-flow mixing and transfer operators.

We study finite-time mixing in time-periodic open flow systems. We describe the transport of densities in terms of a transfer operator, which is represented by the transition matrix of a finite-state Markov chain. The transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix. We use different measures to quantify the degree of mixing and show that they give consistent results in parameter studies of two model systems. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Dynamics of nonlinear hyperbolic equations of Kirchhoff type

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$\begin{aligned} u_{tt}-\left( a \int _\text{\O}mega |\nabla u|^2 \mathrm {d}x +b\right) \Delta u = \lambda u+ |u|^{p-1}u , \end{aligned}$$ where a, $$b>0$$ , $$p>1$$ , $$\lambda \in {\mathbb {R}}$$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of a, b, $$\lambda $$ and the initial data for the following cases: (i) $$1<p<3$$ , (ii) $$p=3$$ and $$a>1/\Lambda $$ , (iii) $$p=3$$ , $$a \le 1/\Lambda $$ and $$\lambda <b\lambda _1$$ , (iv) $$p=3$$ , $$a < 1/\Lambda $$ and $$\lambda >b\lambda _1$$ , (v) $$p>3$$ and $$\lambda \le b\lambda _1$$ , (vi) $$p>3$$ and $$\lambda > b\lambda _1$$ , where $$\lambda _1 = \inf \left\{ \Vert \nabla u\Vert ^2_2 :~ u\in H^1_0(\text{\O}mega )\ \mathrm{and}\ \Vert u\Vert _2 =1\right\} $$ , and $$\Lambda = \inf \left\{ \Vert \nabla u\Vert ^4_2 :~ u\in H^1_0(\text{\O}mega )\ \mathrm{and}\ \Vert u\Vert _4 =1\right\} $$ . Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable a, b and $$\lambda $$ , and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for $$a>0$$ and $$a=0$$ .

A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.

Immersion and invariance‐based adaptive pose tracking control under disturbances and input saturation

This paper addresses the simultaneous attitude and position tracking of a rigid body subject to general bounded disturbances and input saturation in the framework of dual quaternions, which provide a compact and integrated description of the coupled rotation and translation. Treating disturbances as unknown parameters, a modular adaptive saturated pose tracking control scheme is then derived which consists of two separately designed parts. One part is the noncertainty equivalent disturbance estimator designed via the immersion and invariance method. Most existing adaptive pose controllers are based on the certainty equivalence principle. Since its parameter adaption process is unpredictable and acts like a forcing disturbance imposed on the deterministic system, it can cause arbitrary degradation of the closed-loop performance. Instead of invoking the certainty equivalence principle, the adaption law designed based on the novel immersion and invariance method counters the effects of the uncertain parameters from a robustness perspective and eliminates the performance degradation by introducing a stable attracting manifold in the adaption process. For both constant and time-varying disturbances, the proposed estimator can effectively attenuate their effects on the closed-loop performance. The other part is the saturated pose tracking controller where the estimated disturbances are directly used. Based on two useful lemmas, it is proved that the saturation constraints can be removed in finite time, which enables the Lyapunov stability analysis of the closed-loop system. Simulation results demonstrate improvements of the proposed control scheme in closed-loop performance and control precision when compared with a certainty equivalent controller and a sliding mode controller.

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.

Complexified stable and unstable manifolds and chaotic tunneling

Quantum tunneling in nonintegrable systems is studied based on the complex semiclassical analysis in the time domain. The theory of complex dynamical systems predicts that complexified stable and unstable manifolds form the skeleton of the dynamics in phase space, and mixing and ergodicity hold even for the mixed system. After surveying some rigorous results derived for the Hénon map, we numerically verify that some fundamental properties thus predicted can actually be observed, and then investigate natures of complexified stable and unstable manifolds since they guide the complex orbits controlling the tunneling transition in the mixed phase space. In particular, for a properly designed scattering map, we examine the imaginary action associated with the stable manifolds for the periodic orbits in the real plane and find that the orbits tending to the sticky zone of the KAM region provide the most dominant contribution in the semiclassical sum. It is worth emphasizing that the behavior of such dominant orbits is affected by the presence of chaos in the complex plane, which is an aspect far beyond the scope of the dynamics in the real plane.

Collisions of micron-sized charged water droplets in still air

We show that fixed points and their stable manifolds determine collision outcomes of oppositely charged water droplets settling in still air. For droplets with large enough opposite charges, the stable manifold of a saddle forms a separatrix that divides colliding trajectories from those that do not collide. A consequence of our findings is that for large charges, collision outcomes are not sensitive to the breakdown of hydrodynamics at small distances.

Symmetric replicator dynamics with depletable resources.

The replicator equation is a standard model of evolutionary population game dynamics. In this article, we consider a modification of replicator dynamics, in which playing a particular strategy depletes an associated resource, and the payoff for that strategy is a function of the availability of the resource. Resources are assumed to replenish themselves, given time. Overuse of a resource causes it to crash. If the depletion rate is low enough, most trajectories converge to a stable equilibrium at which all initially present strategies are equally popular. As the depletion rate increases, these fixed points vanish in bifurcations. The phase space is periodic in each of the resource variables, and it is possible for trajectories to whirl around different numbers of times in these variables before converging to the stable equilibrium, resulting in a wide variety of topological types of orbits. Numerical solutions in a low-dimensional case show that in a cross section of the phase space, the topological types are separated by intricately folded separatrices. Once the depletion rate is high enough that the stable equilibrium in the interior of the phase space vanishes, the dynamics immediately become chaotic, without going through a period-doubling cascade; a numerical method reveals horseshoes in a Poincaré map. It appears that the multitude of topological types of orbits present before this final bifurcation generate this chaotic behavior. A periodic orbit of saddle type can be found using the symmetries of the dynamics, and its stable and unstable manifolds may generate a homoclinic tangle.

Morse Index of Saddle Equilibria of Gradient-Like Flows on Connected Sums of $$\mathbb S^{n-1}\times \mathbb S^1$$

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

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

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 that throughout an open region of parameter space, this family has an attractor satisfying Devaney's definition of chaos. This strengthens the existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally, we identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed.

Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

A generic framework of adiabatic approximation for nonlinear evolutions

In this paper, we continue the development of a generic adiabatic scheme for nonlinear evolutions. We consider an abstract gradient flow of some energy functional, together with a given manifold of static solutions arising from broken symmetries. First, we list a number of explicit and generic conditions on the energy functional that ensures the validity of our adiabatic scheme. Then, we construct some explicit low-energy but no necessarily static configurations, which form a stable manifold with finite codimensions for the given gradient flow. Thirdly, we show that the gradient flow is globally well-posed with initial configuration from the stable manifold. Finally, we show that any solution to the full gradient flow starting from the stable manifold is essentially governed by an effective equation on the manifold of static solutions, up to a uniformly small and dissipating error term.

SubtStream: Online subtractive stream clustering algorithm

AbstractReal‐time stream data processing has gained high importance with the rapid rise of big data trends in different areas such as social media, finance, business, science, and bioinformatics. Stream data can be characterized as fast, unstable, and big data sets. Due to these properties of stream data, it cannot be processed effectively with traditional algorithms. Just like stream data processing, clustering is also a difficult task. However, researchers have attempted to classify stream data by modifying traditional algorithms or designing new ones. So, in previous studies, incremental methods were used for clustering the stream data. This paper highlights the need to develop an efficient real‐time clustering algorithm for data streams in the presence of concept high drift and an adaptive algorithm for different dimensions. The proposed clustering algorithm, SubtStream, combines decremental (subtractive property) and incremental (additivity property) strategies to overcome the high drift. It also introduces a new dimension‐based approach to adopt the dimension change. We use three radius parameters, Predefined User Parameter, Proactive Adaptive Parameter, and Reactive Adaptive Parameter, to achieve adaptability. The proposed method, SubtStream, showed better performance on synthetic and real data sets.

Critical parameters of the synchronisation's stability for coupled maps in regular graphs

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters – coupling strength, maximum Lyapunov exponent, and link density – that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: k-cycles (i.e., regular lattices of even degree k, which can be embedded in Tk tori) and k-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree k = 3. Our results highlight differences in the synchronisation manifold's stability of these graphs – even for identical node degrees – in the finite size and infinite size limit.

Almost blenders and parablenders

AbstractA blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds robustly contains an open set. Introduced by Bonatti and Díaz in the 1990s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper we introduce analogous notions in a measurable setting. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of $\mathbb {R}^2$ leaving invariant the continuation of a hyperbolic basic set. When an inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger, and complements previous works on the construction of blenders by Avila, Crovisier, and Wilkinson or by Moreira and Silva. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak, and Urbański, we study families of skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

Decoding Digital Visual Stimulation From Neural Manifold With Fuzzy Leaning on Cortical Oscillatory Dynamics.

A crucial point in neuroscience is how to correctly decode cognitive information from brain dynamics for motion control and neural rehabilitation. However, due to the instability and high dimensions of electroencephalogram (EEG) recordings, it is difficult to directly obtain information from original data. Thus, in this work, we design visual experiments and propose a novel decoding method based on the neural manifold of cortical activity to find critical visual information. First, we studied four major frequency bands divided from EEG and found that the responses of the EEG alpha band (8–15 Hz) in the frontal and occipital lobes to visual stimuli occupy a prominent place. Besides, the essential features of EEG data in the alpha band are further mined via two manifold learning methods. We connect temporally consecutive brain states in the t distribution random adjacency embedded (t-SNE) map on the trial-by-trial level and find the brain state dynamics to form a cyclic manifold, with the different tasks forming distinct loops. Meanwhile, it is proved that the latent factors of brain activities estimated by t-SNE can be used for more accurate decoding and the stable neural manifold is found. Taking the latent factors of the manifold as independent inputs, a fuzzy system-based Takagi-Sugeno-Kang model is established and further trained to identify visual EEG signals. The combination of t-SNE and fuzzy learning can highly improve the accuracy of visual cognitive decoding to 81.98%. Moreover, by optimizing the features, it is found that the combination of the frontal lobe, the parietal lobe, and the occipital lobe is the most effective factor for visual decoding with 83.05% accuracy. This work provides a potential tool for decoding visual EEG signals with the help of low-dimensional manifold dynamics, especially contributing to the brain–computer interface (BCI) control, brain function research, and neural rehabilitation.