Special Session 50: Dynamical systems: Oseledets decomposition, ordered spaces, Lyapunov exponents, and applications

Loss of hyperbolicity as a source of critical transitions in d-concave equations: New nonautonomous bifurcation patterns

Jes\`{u}s J. Due\~{n}as Universidad de Valladolid Spain Co-Author(s):    Carmen N\`{u}\~{n}ez and Rafael Obaya

Abstract: This talk introduces bifurcation scenarios in nonautonomous scalar ordinary differential equations whose derivative with respect to the state variable is concave in measure (so-called d-concave equations). Leveraging the strong constraints that d-concavity imposes on the global dynamics, we describe bifurcation patterns that, to the best of our knowledge, have not been reported previously. In particular, using a minimal but multiply ergodic skewproduct base, we construct an example of a jump bifurcation in which two nonhyperbolic compact invariant sets coexist, and each one disappears abruptly under parameter variation--one for each of the two possible directions of change. The study of such phenomena is motivated by critical transition theory, where small perturbations in the inputs of a complex system can trigger sudden and often irreversible shifts in its response. The nonautonomous bifurcations presented here provide a potential mechanism for critical transitions distinct from the better-studied saddle-node bifurcations.

Quenched extreme value theory and hitting-time distributions for random dynamical systems via spectral perturbation of transfer operator cocycles

Gary Froyland UNSW Sydney Australia Co-Author(s):    Jason Atnip, Cecilia Gonzalez-Tokman, Sandro Vaienti

Abstract: I will describe work in a program to extract statistical properties of random dynamical systems using spectral approaches based on transfer operator cocycles and Lyapunov exponents. A random dynamical system is governed by a cocycle of nonlinear transformations on a state space and this naturally induces a cocycle of linear transfer operators acting on suitable Banach spaces. By using random perturbation theory to study derivatives of Lyapunov exponents of ``twisted`` transfer operator cocycles, we obtain elegant proofs of statistical limit laws such as a central limit theorem. Inserting random holes into the phase space, we produce a nonstationary extreme value theory for a random observation function, whereby observing an extreme value corresponds to landing in a random hole and terminating the trajectory. Considering the holes as ``targets``, we further extend this formalism to count the number of visits to random targets in a random orbit, thereby capturing the counting distribution of extreme-value occurrences in random trajectories of increasing length. All results are in the ``quenched`` sense, meaning that they hold almost surely across all random realisations of the dynamics and observations.

Lyapunov--Oseledets spectrum for transfer operator cocycles under perturbations

Cecilia Gonzalez-Tokman University of Queensland Australia Co-Author(s):    Anthony Quas

Abstract: Random dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. In recent years, the study of transfer operators has been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of such systems. The so-called Lyapunov-- Oseledets spectrum associated to the transfer operator cocycle contains fundamental information about invariant measures, exponential decay rates and coherent structures which characterize dominant global transport features of the system. While the scope of this framework is broad, it is often challenging to identify and approximate this spectrum. In this talk, we present examples of random maps where the Lyapunov-- Oseledets spectrum can be understood and analyzed under perturbations.

On the Invariance of Dynamics under Continuous Embeddings

Marek Kryspin Wroclaw University of Science and Technology Poland Co-Author(s):

Abstract: The multiplicative ergodic theorem (the Oseledets theorem), the Folquet theorem and exponential separation are related results concerning the decomposition of the phase space (more generally, of fiber bundles) on which a dynamical system acts. This decomposition is based on the system`s dynamics and exponential growth rates (Lyapunov exponents). Sometimes, when the dynamical system arises from differential equations, it may happen that after some time the associated mappings transfer elements from a Banach space $X$ into a (better) space $Y$, continuously embedded in the former. This occurs, for instance, in parabolic problems, in which an irregular initial condition becomes a smooth function, or for delay differential equations, in which an initial function of class $L_p$ becomes, after some time, continuous (and even absolutely continuous, hence differentiable almost everywhere). We will show that, on the embedded space, the dynamics of the system remains the same

Nonautonomous saddle-node bifurcations and early warning signals in scalar concave and d-concave differential equations

Iacopo P. Longo University of Exeter England Co-Author(s):    Jesus Duenas, Rafael Obaya

Abstract: We characterise the range of possible dynamical behaviours for scalar concave and d-concave nonautonomous differential equations, and demonstrate that nonautonomous saddle-node bifurcations provide a universal mechanism underlying certain critical transitions known as rate-induced tipping. Furthermore, we establish that finite-time Lyapunov exponents serve as reliable and rigorous early warning indicators of such transitions in these systems, as a change in sign occurs prior to the onset of a nonautonomous saddle-node bifurcation.

Oseledets decomposition and monotone dynamical systems

Janusz Mierczy\`nski Wroc{\l}aw University of Science and Technology Poland Co-Author(s):

Abstract: The Oseledets theory gives an invariant decomposition of a skew-product linear dynamical system into measurable subbundles according to Lyapunov exponents, that is, exponential growth rates of orbits. On the other hand, when the fibers are ordered by a cone and the dynamical system satisfies some monotonicity condition, there is an invariant decomposition into a subbundle consisting of orbits that are always positive and negative and its complementary subbundle. It is an interesting topic to investigate relations between those two approaches.

On linearized stability

Christian Poetzsche University of Klagenfurt Austria Co-Author(s):    Nestor Jara (University of Chile, Santiago, Chile)

Abstract: The role of dynamical spectra (based on Lyapunov exponents or exponential dichotomies) is to substitute eigenvalues in a time-variant framework. In doing so, such spectra indicate stability by linearizing along a reference solution and applying a suitable principle of linearized stability. We discuss related principles in the light of recent spectral notions that describe non-exponential growth/decay.

Invariant manifolds induced by stochastic delay differential equations

Sebastian Riedel FernUniversitaet Hagen Germany Co-Author(s):    Mazyar Ghani Varzaneh, Michael Scheutzow

Abstract: Stochastic delay differential equations (SDDEs) can be understood as a class of stochastic evolution equations on infinite-dimensional (path) spaces. As observed by S.-E. A. Mohammed in 1986, these equations do not necessarily induce a stochastic flow. Consequently, they do not induce a random dynamical system (RDS) on a deterministic path space. Together with M. Ghani Varzaneh and M. Scheutzow, we observed that these equations still induce RDS if we allow the spaces themselves to be random. More precisely, we were able to show that SDDEs induce RDS acting on a particular field of Banach spaces. It turns out that these RDS still have a rich structure that, for example, allows one to deduce the existence of invariant manifolds. In this talk, we explain why it is necessary to study random dynamical systems on fields of Banach spaces when analyzing the dynamics of SDDEs, and discuss our latest results on this topic.

A new class of generalized ordinary differential equations

Ana M Sanz Universidad de Valladolid Spain Co-Author(s):    Sylvia Novo and Rafael Obaya

Abstract: In this talk, we present a new class of generalized ordinary differential equations. Our motivation arises from the fact that solutions of limiting equations associated with certain Carath\`{e}odory ODEs may lose the property of absolute continuity, while remaining continuous and of bounded variation on compact intervals. Several approaches can be employed to address these irregular limiting equations. We propose a novel method, based on the introduction of a class of generalized ODEs given by parametric b-measures, aimed at preserving the aforementioned properties of solutions for the equations in the hull, as well as ensuring the compactness of the hull. This framework allows us to apply techniques from nonautonomous dynamical systems to the study of precompact families of Carath\`{e}odory ODEs.

Existence, uniqueness, and stability of monotone traveling waves for repulsion chemotaxis systems with logistic type source

Wenxian Shen Auburn University USA Co-Author(s):

Abstract: This talk is concerned with the existence, uniqueness, and stability of traveling wave solutions to a repulsion parabolic-elliptic chemotaxis system with logistic type source. When the chemotaxis sensitivity coefficient $\chi=0$, this system reduces to the so called Fisher-KPP equation, which generates a monotone dynamical system, possesses a minimal wave speed $c^*$, and admits a unique monotone stable traveling wave solution connecting the positive constant solution and the zero solution for any speed $c\ge c^*$. When $\chi\not = 0$, the system does not generate a monotone dynamical system, or comparison principle does not hold for the system, and several difficulties appear when studying traveling wave solutions to the system. In this talk, we will show that when $\chic^*$ such that the system admits a unique stable monotone traveling wave solution connecting the positive constant solution and the zero solution for any speed $c>^*_{\chi}$. Though the comparison principle does not hold for the system, it is utilized in some nontrivial way in the proof of the results stated in the above.

$C^1$-theory for smooth non-autonomous monotone dynamical systems

Yi Wang University of Science and Technology of China Peoples Rep of China Co-Author(s):    Jinxiang Yao

Abstract: In this talk, we will report the $C^1$-theory and its recent progress on smooth non-autonomous monotone dynamical systems. This talk is based on a series of joint works with Jinxiang Yao.

Measurably dominated splitting of fields of Banach spaces

Caibin Zeng South China University of Technology Peoples Rep of China Co-Author(s):    Huayan Su

Abstract: In this talk, we establish a quasi-equivalence between measurably contracting cone families and measurably dominated splittings in measurable fields of Banach spaces. Under an integrability condition, we obtain a generalized Krein-Rutman-type theorem for compact, injective linear cocycles on Banach spaces, without requiring the cocycle itself to be compact or integrable.