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Series: CDSNS Colloquium

(Please note the unusual day)

New and proposed missions for approaching moons, and particularly icy moons, increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. The search for approach trajectories within highly nonlinear, chaotic regimes where multi-body effects dominate becomes increasingly complex, especially when landing, orbiting, or flyby scenarios must be considered in the analysis. In the case of icy moons, approach trajectories must also be tied into the broader tour which includes flybys of other moons. The tour endgame typically includes the last several flybys, or resonances, before the final approach to the moon, and these resonances further constrain the type of approach that may be used.

In this seminar, new methods for approaching moons by traversing the chaotic regions near the Lagrange point gateways will be discussed for several examples. The emphasis will be on landing trajectories approaching Europa including a global analysis of trajectories approaching any point on the surface and analyses for specific landing scenarios across a range of different energies. The constraints on the approach from the tour within the context of the endgame strategy will be given for a variety of different moons and scenarios. Specific approaches using quasiperiodic or Lissajous orbits will be shown, and general landing and orbit insertion trajectories will be placed into context relative to the invariant manifolds of unstable periodic and quasiperiodic orbits. These methods will be discussed and applied for the specific example of the Europa Lander mission concept. The Europa Lander mission concept is particularly challenging in that it requires the redesign of the approach scenario after the spacecraft has launched to accommodate landing at a wide range of potential locations on the surface. The final location would be selected based on reconnaissance from the Europa Clipper data once Europa Lander is in route. Taken as a whole, these methods will provide avenues to find both fundamentally new approach pathways and reduce cost to enable new missions.

Series: CDSNS Colloquium

Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T-periodic behavior is reduced with increasing angle of incline β, since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T-periodic solutions lose stability to period doubling solutions for β ≥ 0, which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.

Series: CDSNS Colloquium

We present a topological mechanism of diffusion in a priori chaotic systems. The method leads to a proof of diffusion for an explicit range of perturbation parameters. The assumptions of our theorem can be verified using interval arithmetic numerics, leading to computer assisted proofs. As an example of application we prove diffusion in the Neptune-Triton planar elliptic restricted three body problem. Joint work with Marian Gidea.

Series: CDSNS Colloquium

It is anticipated that the invariant statistics of many of smooth dynamical systems with a `chaotic’ asymptotic character are given by invariant measures with the SRB property- a geometric property of invariant measures which, roughly, means that the invariant measure is smooth along unstable directions. However, actually verifying the existence of SRB measures for concrete systems is extremely challenging: indeed, SRB measures need not exist, even for systems exhibiting asymptotic hyperbolicity (e.g., the figure eight attractor).

The study of asymptotic properties for dynamical systems in the presence of noise is considerably simpler. One manifestation of this principle is the theorem of Ledrappier and Young ’89, where it was proved that under very mild conditions, stationary measures for a random dynamical system with a positive Lyapunov exponent are automatically random SRB measures (that is, satisfy the random analogue of the SRB property). I will talk today about a new proof of this result in a joint work with Lai-Sang Young. This new proof has the benefit of being (1) conceptually lucid and to-the-point (the original proof is somewhat indirect) and (2) potentially easily adapted to more general settings, e.g., to appropriate infinite-dimensional random dynamics, such as time-t solutions to certain classes SPDE (this generalization is an ongoing work, joint with LSY).

Series: CDSNS Colloquium

One of the characteristics observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. We introduce a model of network growth based on this notion of specialization and show that as a network is specialized its topology becomes increasingly modular, hierarchical, and sparser, each of which are properties observed in real networks. This model is also highly flexible in that a network can be specialized over any subset of its components. By selecting these components in various ways we find that a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions and clustering coefficients. This growth model also maintains the basic spectral properties of a network, i.e. the eigenvalues and eigenvectors associated with the network's adjacency network. This allows us in turn to show that a network maintains certain dynamic properties as the network's topology becomes increasingly complex due to specialization.

Series: CDSNS Colloquium

The real world is inherently noisy, and so it is natural to consider the random perturbations of deterministic dynamical systems and seek to understand the corresponding asymptotic behavior, i.e., the phenomena that can be observed under long-term iteration. In this talk, we will study the random perturbations of a family of circle maps $f_a$. We will obtain, a checkable, finite-time criterion on the parameter a for random perturbation of $f_a$ to exhibit a unique, and thus ergodic, stationary measure.

Series: CDSNS Colloquium

We will discuss the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L^4 is irreducible over rationals. We consider a volume-preserving C^1-small perturbation f of L. We show that if the Lyapunov exponents of f with respect to the volume are the same as the Lyapunov exponents of L, then f is C^1+ conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle. This is joint work with Andrey Gogolev and Victoria Sadovskaya

Series: CDSNS Colloquium

We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,f) with values in GL(d,R), or more generally in the group of invertible bounded linear operators on a Banach space. We discuss approximation of the Lyapunov exponents of A in terms of its periodic data, i.e. its return values along the periodic orbits of f. For a GL(d,R)-valued cocycle A, its Lyapunov exponents with respect to any ergodic f-invariant measure can be approximated by its Lyapunov exponents at periodic orbits of f. In the infinite-dimensional case, the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the return values of A at periodic points of f. Similar results are obtained in the non-uniformly hyperbolic setting, i.e. for hyperbolic invariant measures. This is joint work with B. Kalinin.

Series: CDSNS Colloquium

The fluctuations of ergodic sums by the means of global and local specifications on periodic points will be discussed. Results include a Lindeberg-type central limit theorems in both setups of specification. As an application, it is shown that averaging over randomly chosen periodic orbits converges to the integral with respect to the measure of maximal entropy as the period approaches infinity. The results also suggest to decompose the variances of ergodic sums according to global and local sources.

Series: CDSNS Colloquium

In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.