Paper analyzes coexisting hidden and self-excited attractors in an economic system.
problem Existence of coexisting hidden and self-excited attractors in economic systems.
method Integer and fractional order analysis of an economic system.
result Integer-order system exhibits multiple combinations of coexisting hidden and self-excited attractors.
Characterizes knotted toroidal sets as attractors in 3D.
problem Characterizing knotted toroidal sets as attractors in R3. method Global and local dynamical systems analysis, homeomorphisms and flows.
result Sufficient conditions for incompressible surfaces as attractors.
The study examines attractors in Cartan foliations and their properties.
problem Existence of attractors in Cartan foliations.
method Reduction of the problem to the action of Lie groups and holonomy groups.
result Conditions for the existence of attractors in reductive Cartan foliations.
This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
problem Classifying expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
method Using the derived Anosov (DA) expanding attractor and the Franks-Williams manifold, the paper proves the uniqueness of these structures.
result The DA expanding attractor and the Franks-Williams non-transitive Anosov flow are the unique structures supported by N0 and M0 respectively. Convolutional attractor nets improve image completion and super-resolution.
problem Construct coherent neural states from noisy data.
method Revisit and extend attractor networks with convolutional bipartite architecture.
result Demonstrates potential for image completion and super-resolution.
We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a cha…
The paper characterizes toroidal sets as attractors for flows and homeomorphisms of R^3.
problem Characterizing toroidal sets as attractors for flows and homeomorphisms in R^3.
method Defined self-geometric index and genus to analyze toroidal sets and their attractor properties.
result Characterized toroidal sets that can be realized as attractors for flows but not for homeomorphisms.
Generative memory model avoids vanishing gradients to robustly retrieve patterns.
problem Robust retrieval of stored patterns in the presence of interference and noise.
method Training a generative distributed memory without explicitly simulating attractor dynamics, using a likelihood-based Lyapunov function.
result The model converges to correct patterns upon iterative retrieval and achieves competitive performance as a memory model and a generative model.
The paper studies dimensions of attractors for modified Leray-alpha equation on various surfaces.
problem Investigate attractor dimensions of the modified Leray-alpha equation.
method Existence and uniqueness of weak solutions, global attractor existence, estimates for vorticity scalar equations, Kolmogorov flows.
result Established upper and lower bounds for Hausdorff and fractal dimensions of global attractors on S2 and T2. Study bounds topological entropy of toroidal attractors.
problem Bounding entropy of toroidal attractors.
method Analyzing topological properties of toroidal sets to bound entropy.
result Entropy of toroidal attractors is bounded from below.
We prove a theorem on structural stability of smooth attractor-repellor endomorphisms of compact manifolds, with singularities. By attractor-repellor, we mean that the non-wandering set of the dynamics f is the disjoint union of a repulsive compact subset with a hyperbolic attractor on which f acts bijectively. The…
The paper shows knots and solenoids that cannot be attractors of self-homeomorphisms of R^3.
problem Deciding when a continuum can be an attractor for a homeomorphism of R^3.
method Introducing toroidal sets and defining their genus, then using these tools to find counterexamples.
result Knots and solenoids exist that cannot be attractors of self-homeomorphisms of R^3.
If there exists a diffeomorphism f on a closed, orientable n-manifold M such that the non-wandering set Ω(f) consists of finitely many orientable (±) attractors derived from expanding maps, then M must be a rational homology sphere; moreover all those attractors are of topological dimension n−2. Expandi…
The study finds a local attractor for tetrahedron transformations.
problem Analyzing the dynamical properties of tetrahedron transformations.
method Analysis of a geometric tetrahedron transformation on a specific space.
result Existence of a local attractor coinciding with the set of regular tetrahedra.
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
New method interprets recurrent neural networks via excitable network attractors.
problem Understanding the inner workings of recurrent neural networks.
method Excitable network attractors for mechanistic interpretation.
result Recurrent neural networks' behavior can be interpreted using excitable network attractors.
New method reconstructs hidden dynamics from low-dimensional time series.
problem Reconstructing hidden dynamics from limited experimental data.
method Autoencoder trained with a novel latent-space loss function.
result Reconstructs strange attractors better than existing techniques.
Study analyzes Echo State Network parameters for Rossler attractor dynamics.
problem Understanding the influence of network type on Echo State Network performance.
method Experimental analysis of Echo State Network parameters using Rossler attractor.
result Exploration of how network type affects Echo State Network performance.
Paper refines RNN training by analyzing smoothness and attractors.
problem Exploding and vanishing gradients in RNNs.
method Refined concept of exploding gradients using cost function smoothness.
result RNNs need to learn attractors to fully use their power.
Study the evolution of the Lorenz strange set using Conley index theory.
problem Understanding the evolution of the Lorenz strange set through parameter changes.
method Application of Conley index theory to analyze the global attractor and its Morse decompositions.
result Identification and analysis of bifurcations and the role of the strange set in these transformations.
New risk models use chaotic attractors to predict extreme events.
problem Predicting Black Swan events in financial markets.
method Combining heavy-tailed priors with chaotic dynamics (Lorenz and Rossler systems).
result Models generate volatility clustering, fat tails, and extreme events.
This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R3. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…
In this paper we consider the realization of DE attractors by self-diffeomorphisms of manifolds. For any expanding self-map φ:M→M of a connected, closed p-dimensional manifold M, one can always realize a (p,q)-type attractor derived from φ by a compactly-supported self-diffeomorphsm of $\RR^{p+q}$, as long…
The paper generalizes two-field α-attractor models using geometrically finite hyperbolic surfaces.
problem Modeling inflationary dynamics in curved spacetime.
method Coupling four-dimensional gravity to a non-linear sigma model with a hyperbolic scalar manifold.
result Generalized two-field α-attractor models can be parameterized by a surface group and scalar potential.
New method for cohomological Conley index simplifies complex dynamics.
problem Computing cohomological Conley index for complex dynamics.
method Attractor-repeller decomposition and summation of power series in cohomology.
result Simple dynamical interpretation of first cohomological Conley index.
Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? T…
Learning three data points can generate all types of periodic orbits in a neural network.
problem Can learning three data points generate all types of periodic orbits in a neural network?
method Investigated a continuous one-dimensional map with period three in a random neural network in its thermodynamic limit.
result Almost all learned periods are unstable, and each network has its own characteristic attractors.
Unified Bayesian framework predicts cryptocurrency market dynamics and volatility.
problem Predicting cryptocurrency market trends and volatility.
method Bayesian framework based on potential field theory and Gaussian Process.
result Attractors and repellers from the potential field are reliable market indicators.
Deep neural networks can store and recall data efficiently.
problem Identifying computational mechanisms for memorization and retrieval of data.
method Training overparameterized autoencoders and sequence encoders using standard optimization methods.
result Overparameterized autoencoders and sequence encoders store and recall data efficiently as attractors.
New study shows min-max algorithms can converge to non-stationary points.
problem Challenges in min-max optimization due to periodic cycles and spurious attractors.
method Analyzed state-of-the-art algorithms and heuristics in non-convex/non-concave problems.
result Spurious attractors can prevent min-max algorithms from reaching true optima.
An iterated function system Φ consisting of contractive similarity mappings has a unique attractor F⊆Rd which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling T of the con…
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
Let M be a manifold or (more generally) a locally compact, metrizable ANR. If K is an attractor for a flow in M, with basin of attraction A(K), it is well known that the inclusion i:K⊆A(K) is always a shape equivalence. In this paper we investigate to what extent this generaliz…
ERDMD discovers sparse, nonuniformly timed DMD models from chaotic attractors.
problem Discovering high-fidelity, nonuniformly timed DMD models from chaotic data.
method Entropic regression for nonlinear information flow detection, combined with multi-step DMD.
result ERDMD produces highly efficient and robust models with minimal complexity.
Reverse engineered RNNs reveal line attractor dynamics for sentiment classification.
problem Understanding how recurrent neural networks solve sequential tasks like sentiment classification.
method Dynamical systems analysis to reverse engineer trained RNNs, identifying fixed points and linearized dynamics.
result Trained RNNs converge to low-dimensional line attractor dynamics, providing interpretable solutions.
This paper will be splited into two papers and submited later.
Static spacetimes are stable attractors in a flow equation.
problem Stability of static spacetimes with negative cosmological constant.
method New expander entropy for Ricci-harmonic flow.
result Static metrics are stable if and only if a positive mass theorem holds.
We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…
This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the understanding of the existence and structure of attractors as well as dynamical and topolo…
GRUs exhibit diverse dynamical behaviors but cannot mimic continuous attractors.
problem Understanding and predicting the dynamics of GRUs for neural data.
method Continuous time dynamical systems analysis of GRU networks.
result GRUs can represent stable limit cycles, multi-stable dynamics, and homoclinic bifurcations but not continuous attractors.
We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk D that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeom…
Deep learning methods improve overlapping speaker separation across languages and noise.
problem Overlapping speaker separation in realistic scenarios.
method Deep clustering and deep attractor networks.
result Deep learning methods are effective for a broad range of languages and can handle untrained languages with common features.
Paper tackles incremental few-shot learning with novel classes.
problem Learning new classes with limited data and without re-training.
method Attention Attractor Network (AAN) for incremental few-shot learning.
result AAN helps recognize new classes without forgetting old ones.
New framework improves restless bandit policies for large numbers of arms.
problem Efficiently compute policies for large numbers of arms in restless bandit problems.
method Follow-the-Virtual-Advice framework, converting single-armed policies to N-armed policies.
result Achieves an O(1/\sqrt{N}) optimality gap in both discrete and continuous settings.
We propose a construction which transforms a self-similar zipper in Rn to a self-affine zipper Rn+1 whose attractor is a smooth curve.
Meta-learning approach for fast and compressive energy-based memory models.
problem Learning associative memory with fast and compressive models for complex data.
method Meta-learning approach to energy-based memory models (EBMM) using arbitrary neural architectures.
result Demonstrated associative retrieval outperforming existing systems in reconstruction error and compression rate.
Neural networks can model chaos efficiently by becoming geometrically chaotic.
problem Lack of theoretical understanding of how neural networks learn chaos.
method Employed a geometric perspective to show neural networks can model chaotic dynamics.
result Neural networks can reconstruct strange attractors and accurately predict local divergence rates.
Low-connectivity reservoirs outperform standard designs in chaotic system forecasting.
problem Forecasting chaotic systems with high accuracy and low computational resources.
method Used Bayesian optimization to find optimal reservoir configurations, focusing on global system climate rather than short-term prediction.
result Optimized reservoirs with very low connectivity perform well in forecasting chaotic systems, challenging existing design heuristics.