Resonance Curves of Multidimensional Chaotic Systems

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• Last week
• Volume
• The volume of a solid object measures how much
  space it occupies.
• The volume of irregularly shaped objects can be
  determined by their displacement of a liquid.

• Density
• The density of a material is the mass per volume

Lecture demo Displacement of water by a
butterfly and a sponge.
density mass / volume
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• Position
• Context Where are you?
• Where are the keys?
• Definition
• The position is the location in a
• coordinate system, usually in two
• or more dimensions.
  Location in a coordinate system
• The position of an object is the
• location of the center of the object.
• The position of a person is a point
• between the eyes.
• Discussion -Center means geometrical center or
  center of mass,
• - Position vs. Time graphs symbol for position
  x, symbol for time t

Lecture demo Pendulum, Motion Detector and
Computer
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Out-of-body experiences with video feedback
- Subject sees video image of itself with 3D
goggles - Two sticks, one strokes person's chest
for two minutes, second stick moves just
under the camera lenses, as if it were
touching the virtual body. - Synchronous
stroking gt people reported the sense of being
outside their own bodies, looking at themselves
from a distance where the camera is located. -
While people were experiencing the illusion, the
experimenter pretended to smash the virtual body
by waving a hammer just below the cameras.
Immediately, the subjects registered a threat
response as measured by sensors on their skin.
They sweated and their pulses raced. Real system
similar virtual system bi-directional
instant. coupling mixed reality
Blanke O et al.Linking OBEs and self processing
to mental own body imagery at the
temporo-parietal junction. J Neurosci 25550-55
(2006).
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Experimental evidence for mixed reality states in
physical systems
Objective Understand synchronization between
virtual and real systems. Approach - Couple a
real dynamical system to its virtual counterpart
with an instantaneous bi-direction coupling. -
Measure the amplitudes of both systems and their
phase difference, and then detect
synchronization.
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Experimental evidence for mixed reality states in
physical inter-reality systems
Results - Experimental evidence for a phase
transition from dual reality states to mixed
reality states. - Phase diagram of the
inter-reality system is in good agreement with
the phase diagram of the simulated inter-reality
system.
Phase diagram of the inter-reality system
amplitude of the coupling versus the frequency
ratio of the real and the virtual system. The
phase boundary between mixed reality states (I)
and dual reality states (II). The solid, dashed,
and dotted lines indicate the critical points in
the experiment, simulation, and analytic theory,
respectively.
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Mixed reality states in physical systems Why are
they important?
- Virtual systems match their real counter parts
with ever-increasing accuracy. - New hardware
for instantaneous bi-directional coupling - In
mixed reality states there is no clear boundary
between the real and the virtual system. Mixed
reality states can be used to analyze and control
real systems with high precision. And then there
is the possibility for time travel by the
virtual system.
Publication The paper "Experimental evidence for
mixed reality states in an inter-reality system"
by Vadas Gintautas and Alfred Hubler, in Phys.
Rev. E 75, 057201 (2007), was selected for the
APS tip sheet http//www.aps.org/about/tipsheets/
tip68.cfm
Photo A. Hubler and V. Gintautas at the
inter-reality system
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• DisplacementContext Where are we
  going?Definition
• Displacement describes the change in
• Position
• Discussion
• Distance is the magnitude of the displacement
• Velocity is the rate of change in position
• Symbol for displacement ?x

Displacement and Distance Traveled
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• VelocityContext Where are we going?Definition
  
• Velocity is the rate of change in position.
• Velocity describes how fast something moves.
• Discussion
• - Symbol for displacement v ?x / ?t
• where ?x is displacement and ?t is time lapsed
• - Units mph, m/s, km/h
• Velocity 0 gt object at rest
• Motion with constant velocity
• frictionless horizontal motion
• sinking object in viscous fluid
• parachute
• motion without push or pull

• Speed is the magnitude
• of the velocity

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• Acceleration
• Context A car accelerates.
• I am speeding up.
• Definition
• Acceleration how fast does fast
• change.
• Acceleration the rate of change of the velocity.
• Discussion
• Symbol a, Units m/s2
• Acceleration 0 gt velocity constant
• Motion with constant acceleration free fall in
  vacuum, rolling
• down an incline, washers on a string, circular
  motion with const. speed

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• Position, velocity, acceleration
• The position of an object is the
• location of the center of the object.
• The position of a person is a point
• between the eyes.
• Velocity is the rate of change in
• position.
• Velocity describes how f ast
• something moves.
• Acceleration describes how fast
• does fast change.
• - Acceleration describes how rapidly an object is
  speeding up.
• - Acceleration the rate of change of the velocity.