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UID:dd8996d49533788a0b1eaa484ba59432
CATEGORIES:Applied and Computational Math Seminar
CREATED:20231115T143356
SUMMARY:Methods of Geometric Control in Hamiltonian Dynamics
LOCATION:Hill 705
DESCRIPTION:We consider an integrable Hamiltonian system subject to a small, time-perio
 dic perturbation. We assume that the perturbed system has a normally hyperb
 olic invariant manifold (NHIM) whose stable and unstable manifolds intersec
 t transversally. Associated to each transverse intersection one can define 
 a scattering map, which gives the future asymptotic of a homoclinic orbit a
 s a function of its past asymptotic.  We assume that we have a system of su
 ch scattering maps. We provide results on the geometric controllability of 
 the system. We show that, under explicit conditions on the scattering maps 
 and on the inner dynamics (restricted to the NHIM),  for any two points on 
 the NHIM, there is an orbit of the Hamiltonian flow that goes from near the
  first point to near the second point.  Also, for any path on the NHIM, the
 re is an orbit of the Hamiltonian flow that shadows that path.  The upshot 
 is that we use the perturbation as a controller.\n
X-ALT-DESC;FMTTYPE=text/html:<p><span style="border: 0px; font-style: normal; font-weight: 400; font-siz
 e: 12pt; line-height: inherit; font-family: Calibri, Helvetica, sans-serif;
  margin: 0px; padding: 0px; vertical-align: baseline; letter-spacing: norma
 l; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; wi
 dows: 2; word-spacing: 0px; white-space: normal;">We consider an integrable
  Hamiltonian system subject to a small, time-periodic perturbation.&nbsp;</
 span><span style="border: 0px; font-style: normal; font-weight: 400; font-s
 ize: 12pt; line-height: inherit; font-family: Calibri, Helvetica, sans-seri
 f; margin: 0px; padding: 0px; vertical-align: baseline; letter-spacing: nor
 mal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; 
 widows: 2; word-spacing: 0px; white-space: normal;">We assume that the pert
 urbed system has a normally hyperbolic invariant manifold (NHIM) whose stab
 le and unstable manifolds intersect transversally. Associated to each trans
 verse intersection one can define a scattering map, which gives the future 
 asymptotic of a homoclinic orbit as a function of its past asymptotic. &nbs
 p;We assume that we have a system of such scattering maps.&nbsp;</span><spa
 n style="border: 0px; font-style: normal; font-weight: 400; font-size: 12pt
 ; line-height: inherit; font-family: Calibri, Helvetica, sans-serif; margin
 : 0px; padding: 0px; vertical-align: baseline; letter-spacing: normal; orph
 ans: 2; text-align: left; text-indent: 0px; text-transform: none; widows: 2
 ; word-spacing: 0px; white-space: normal;">We provide results on the geomet
 ric controllability of the system. We show that, under explicit conditions 
 on the scattering maps and on the inner dynamics (restricted to the NHIM), 
 &nbsp;for any two points on the NHIM, there is an orbit of the Hamiltonian 
 flow that goes from near the first point to near the second point. &nbsp;Al
 so, for any path on the NHIM, there is an orbit of the Hamiltonian flow tha
 t shadows that path. &nbsp;The upshot is that we use the perturbation as a 
 controller.</span></p>
CONTACT:Marian Gidea (Yeshiva University)
DTSTAMP:20260829T124405
DTSTART;TZID=America/New_York:20231121T110000
DTEND;TZID=America/New_York:20231121T120000
SEQUENCE:0
TRANSP:OPAQUE
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