Effect of non-conservative perturbations on homoclinic and heteroclinic orbits
- Series
- CDSNS Colloquium
- Time
- Wednesday, October 16, 2019 - 11:15 for 1 hour (actually 50 minutes)
- Location
- Skiles 05
- Speaker
- Marian Gidea – Yeshiva University
We consider the problem of computing unstable manifolds for equilibrium solutions of parabolic PDEs posed on irregular spatial domains. This new approach is based on the parameterization method, a general functional analytic framework for studying invariant manifolds of dynamical systems. The method leads to an infinitesimal invariance equation describing the unstable manifold. A recursive scheme leads to linear homological equations for the jets of the manifold which are solved using the finite element method. One feature of the method is that we recover the dynamics on the manifold in addition to its embedding. We implement the method for some example problems with polynomial and non-polynomial nonlinearities posed on various non-convex two dimensional domains. We provide numerical support for the accuracy of the computed manifolds using the natural notion of a-posteriori error admitted by the parameterization method. This is joint work with J.D. Mireles-James and Necibe Tuncer.
The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...).
From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013). In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions.
The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.
This is a work in collaboration with Benoît Grébert (Université de Nantes).
In this talk, we consider a quasi-periodically forced system arising from the problem of chemical reactions. For we demonstrate efficient algorithms to calculate the normally hyperbolic invariant manifolds and their stable/unstable manifolds using parameterization method. When a random noise is added, we use similar ideas to give a streamlined proof of the existence of the stochastic invariant manifolds.
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.
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.
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).
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.