Dynamical Systems
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New submissions for Thu, 25 Nov 21
 [1] arXiv:2111.12171 [pdf, other]

Title: Domains which are integrable close to the boundary and close to the circular ones are ellipsesAuthors: Illya KovalComments: 46 pages, 3 figuresSubjects: Dynamical Systems (math.DS)
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. We generalize the result of another paper, where integrability was proven only for specific values, proving its main conjecture. In particular, we show that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding actionangle coordinates with respect to the eccentricity parameter, as well as using irrationality to prove the main nondegeneracy condition.
 [2] arXiv:2111.12222 [pdf, ps, other]

Title: Inclusion of higherorder terms in the bordercollision normal form: persistence of chaos and applications to power convertersSubjects: Dynamical Systems (math.DS)
The dynamics near a bordercollision bifurcation are approximated to leading order by a continuous, piecewiselinear map. The purpose of this paper is to consider the higherorder terms that are neglected when forming this approximation. For twodimensional maps we establish conditions under which a chaotic attractor created in a bordercollision bifurcation persists for an open interval of parameters beyond the bifurcation. We apply the results to a prototypical power converter model to prove the model exhibits robust chaos.
 [3] arXiv:2111.12254 [pdf]

Title: Emergence of dynamic properties in network hypermotifsSubjects: Dynamical Systems (math.DS)
Networks are fundamental for our understanding of complex systems. Interactions between individual nodes in networks generate network motifs  small recurrent patterns that can be considered the network's buildingblock components, providing certain dynamical properties. However, it remains unclear how network motifs are arranged within networks and what properties emerge from interactions between network motifs. Here we develop a framework to explore the mesoscalelevel behavior of complex networks. Considering network motifs as hypernodes, we define the rules for their interaction at the network's next level of organization. We infer the favorable arrangements of interactions between network motifs into hypermotifs from real evolved and designed networks data including biological, neuronal, social, linguistic and electronic networks. We mathematically explore the emergent properties of these higherorder circuits and their relations to the properties of the individual minimal circuit components they combine. This framework provides a basis for exploring the mesoscale structure and behavior of complex systems where it can be used to reveal intermediate patterns in complex networks and to identify specific nodes and links in the network that are the key drivers of the network's emergent properties.
 [4] arXiv:2111.12316 [pdf, ps, other]

Title: A comment on stabilizing reinforcement learningSubjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Systems and Control (eess.SY)
This is a short comment on the paper "Asymptotically Stable AdaptiveOptimal Control Algorithm With Saturating Actuators and Relaxed Persistence of Excitation" by Vamvoudakis et al. The question of stability of reinforcement learning (RL) agents remains hard and the said work suggested an onpolicy approach with a suitable stability property using a technique from adaptive control  a robustifying term to be added to the action. However, there is an issue with this approach to stabilizing RL, which we will explain in this note. Furthermore, Vamvoudakis et al. seems to have made a fallacious assumption on the Hamiltonian under a generic policy. To provide a positive result, we will not only indicate this mistake, but show critic neural network weight convergence under a stochastic, continuoustime environment, provided certain conditions on the behavior policy hold.
 [5] arXiv:2111.12377 [pdf, other]

Title: Sliding motion on tangential sets of Filippov systemsSubjects: Dynamical Systems (math.DS)
We consider piecewise smooth vector fields $Z=(Z_+, Z_)$ defined in $\mathbb{R}^n$ where both vector fields are tangent to the switching manifold $\Sigma$ along a manifold $M$. Our main purpose is to study the existence of an invariant vector field defined on $M$, that we call tangential sliding vector field. We provide necessary and sufficient conditions under $Z$ to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. The results are applied to study a Filippov model for intermittent treatment of HIV.
 [6] arXiv:2111.12416 [pdf, other]

Title: Slow passage through a Hopflike bifurcation in piecewise linear systems: application to elliptic burstingSubjects: Dynamical Systems (math.DS)
The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multipletimescale dynamical systems, where a slowlyvarying quantity replacing a static parameter induces the solutions of the resulting slowfast system to feel the effect of a Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slowfast dynamical systems. In the present work, we study for the first time this phenomenon in piecewise linear (PWL) slowfast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slowfast system to exhibit a slow passage through a Hopflike bifurcation, in link with the number of linearity zones considered in the system and possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the socalled wayin/wayout function. Finally, we investigate this slow passage effect in the DoiKumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.
 [7] arXiv:2111.12461 [pdf, other]

Title: Stability and Dynamics of Complex Order Fractional Difference EquationsComments: 21 pages, 17 figuresSubjects: Dynamical Systems (math.DS)
We extend the definition of $n$dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$dimensional matrix $A$ and derive conditions for the stability of equilibrium points for linear systems. For the onedimensional case where $A =\lambda \in \mathbb {C}$, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is selfintersecting. Even for $ \lambda \in \mathbb{R} $, the solutions can be complex and dynamics in onedimension is richer than the case for $ \alpha\in \mathbb{R} $. These results can be extended to $n$dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.
 [8] arXiv:2111.12633 [pdf, other]

Title: Untangling the role of temporal and spatial variations in persistance of populationsSubjects: Dynamical Systems (math.DS); Probability (math.PR); Populations and Evolution (qbio.PE)
We consider a population distributed between two habitats, in each of which it experiences a growth rate that switches periodically between two values, $1 \varepsilon > 0$ or $  (1 + \varepsilon) < 0$. We study the specific case where the growth rate is positive in one habitat and negative in the other one for the first half of the period, and conversely for the second half of the period, that we refer as the $(\pm 1)$ model. In the absence of migration, the population goes to $0$ exponentially fast in each environment. In this paper, we show that, when the period is sufficiently large, a small dispersal between the two patches is able to produce a very high positive exponential growth rate for the whole population, a phenomena called inflation. We prove in particular that the threshold of the dispersal rate at which the inflation appears is exponentially small with the period. We show that inflation is robust to random perturbation, by considering a model where the values of the growth rate in each patch are switched at random times: we prove, using theory of Piecewise Deterministic Markov Processes (PDMP) that inflation occurs for low switching rate and small dispersal. Finally, we provide some extensions to more complicated models, especially epidemiological and density dependent models.
Crosslists for Thu, 25 Nov 21
 [9] arXiv:2111.12164 (crosslist from mathph) [pdf, ps, other]

Title: Anisothermal Chemical Reactions: OnsagerMachlup and Macroscopic Fluctuation TheoryAuthors: D. R. Michiel RengerSubjects: Mathematical Physics (mathph); Statistical Mechanics (condmat.statmech); Dynamical Systems (math.DS)
We study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the largedeviation rate of the microscopic invariant measure. The second result is an application of modern OnsagerMachlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.
Replacements for Thu, 25 Nov 21
 [10] arXiv:1906.07680 (replaced) [pdf, other]

Title: Recognizing topological polynomials by lifting treesComments: 57 pages, 32 figures; accepted to Duke Mathematical JournalSubjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
 [11] arXiv:2105.00548 (replaced) [pdf, ps, other]

Title: Quenched limit theorems for expanding on average cocyclesComments: 48 pages, comments are welcome !Subjects: Dynamical Systems (math.DS); Probability (math.PR)
 [12] arXiv:2105.03339 (replaced) [pdf, other]

Title: Existence of physical measures in some ExcitationInhibition NetworksSubjects: Dynamical Systems (math.DS)
 [13] arXiv:2108.04670 (replaced) [pdf, ps, other]

Title: Polynomial growth, comparison, and the small boundary propertyAuthors: Petr NaryshkinComments: changes from the first version: fixed mistakes in the proofs; added Lemma 3.1Subjects: Dynamical Systems (math.DS); Operator Algebras (math.OA)
 [14] arXiv:1809.08512 (replaced) [pdf, ps, other]

Title: Rational points on certain homogeneous varietiesAuthors: Pengyu YangComments: 11 pages. Added an exampleSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Dynamical Systems (math.DS)
 [15] arXiv:2005.01537 (replaced) [pdf, ps, other]

Title: Simultaneous supersingular reductions of CM elliptic curvesComments: 46 pages. Revised according to the referee's commentsSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
 [16] arXiv:2011.09468 (replaced) [pdf, other]

Title: Gradient Starvation: A Learning Proclivity in Neural NetworksAuthors: Mohammad Pezeshki, SékouOumar Kaba, Yoshua Bengio, Aaron Courville, Doina Precup, Guillaume LajoieComments: Proceeding of NeurIPS 2021Subjects: Machine Learning (cs.LG); Dynamical Systems (math.DS); Machine Learning (stat.ML)
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