nonlinear dynamics and resonances: a surveyevents.iitgn.ac.in/2019/now2019/content/8... · the...
TRANSCRIPT
![Page 1: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/1.jpg)
Nonlinear Dynamics and Resonances: a surveyO.V.Gendelman
Technion – Israel Institute of Technology
![Page 2: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/2.jpg)
Technion
1924 Einstein Tree
![Page 3: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/3.jpg)
Technion
![Page 4: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/4.jpg)
Faculty of Mechanical Engineering
- Biomechanics- Optomechaincs- Robotics- Dynamical Systems- Control Theory- Hydrodynamics and
Microfluidics- CAD/CAM
![Page 5: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/5.jpg)
Resonance in Linear Systems
![Page 6: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/6.jpg)
Resonance in Linear Systems –No Damping
00, = =00, 1.1 = =
![Page 7: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/7.jpg)
Beatings
1 1 1 2
2 2 2 1
( ) 0
( ) 0
u u u u
u u u u
+ + − =
+ + − =
![Page 8: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/8.jpg)
Nonlinear oscillator (weak nonlinearity)
3( cos )u u u u t + = − + +
- Frequency close to natural- Co-existence of the response
regimes- Hystheresis
![Page 9: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/9.jpg)
Co-existence of responses
![Page 10: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/10.jpg)
Simple Hystheresis3( cos(1 cos ) )u u u u t t + = − + + +
Frequency versus time
![Page 11: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/11.jpg)
Resonance – idea (not definition!)SMALL REASONS (FORCING, COUPLING) LEAD TO LARGE CONSEQUENCES
![Page 12: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/12.jpg)
Why we see only one side of the moon?
![Page 13: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/13.jpg)
Why we see only one side of the moon?
Equal frequencies of rotation – a variant of the resonance.
![Page 14: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/14.jpg)
3
3( ) sin 2 0
2
GMC B A
r + − =
![Page 15: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/15.jpg)
Capture into the resonance
qL = − L – Phase on the Kepler orbitq – rational number
02sin =+ QC
![Page 16: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/16.jpg)
Capture into the resonance
qL = − L – Phase on the Kepler orbitq – rational number
02sin =+ QC
PENDULUM!
![Page 17: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/17.jpg)
Pendulum –most famous oscillatory system
17
The first dynamical system - pendulum
Galileo Galilei (1564-1642) The famous lamp(Pisa cathedral)
![Page 18: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/18.jpg)
Pendulum-oscillatory regimes
18
The first dynamical system – pendulumLinear – weakly nonlinear –strongly nonlinear
Frequency of oscillations depends on the energy
![Page 19: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/19.jpg)
Strong nonlinearity versus weak nonlinearity
19
The first dynamical system – pendulumLinear – weakly nonlinear –strongly nonlinearFrequency of oscillations depends on the energy
0 1.98E =
0 2.02E =
![Page 20: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/20.jpg)
Capture into the resonance –account of tidal forces
baQC −=+ 2sin
No damping
Q a Q a
![Page 21: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/21.jpg)
Capture into the resonance –account of tidal forces
baQC −=+ 2sin With the damping
![Page 22: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/22.jpg)
Balthazar Van der Pol (1889 –1959)
- Limit cycle oscillations (electric circuits)
- Relaxation oscillations (1929)
- Chaos in driven autoresonant systems
- Oscillations in biological systems
- Heart pacemakers
![Page 23: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/23.jpg)
Van der Pol oscillator (modern version)
![Page 24: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/24.jpg)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − −Characteristics of the tunneling diode (negative damping)
![Page 25: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/25.jpg)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − −Characteristics of the tunneling diode (negative damping)
0
1( ( ) )
1
V V E WC
W VL
= − + −
=
![Page 26: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/26.jpg)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − − 0
1( ( ) )
1
V V E WC
W VL
= − + −
=
21 1( 3 ) 0V V V V
C LC − − + =
![Page 27: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/27.jpg)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − − 0
1( ( ) )
1
V V E WC
W VL
= − + −
=
21 1( 3 ) 0V V V V
C LC − − + =
Rescaling:
3, t ,
t Lx V
CLC
= → =
2(1 ) 0x x x x− − + =Canonic form of Van der Pol (VdP)
Equation
![Page 28: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/28.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε
Time series for ε=0.1
Phase portrait
![Page 29: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/29.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε - analysis
2
3
( ) exp( )
1( ) 0
2 4
( )exp( ( ))
1( )
2 4
x ix t it
N t i t
N N N
+ =
− − =
=
= −
![Page 30: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/30.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε - analysis
2
3
( ) exp( )
1( ) 0
2 4
( )exp( ( ))
1( )
2 4
x ix t it
N t i t
N N N
+ =
− − =
=
= −
Solution
02
0
2
0
2( ) , (0)
41 exp( )
N t N NN
tN
= =−
+ −
![Page 31: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/31.jpg)
• AND WHAT ABOUT LARGE FORCING AND LARGE DAMPING?
![Page 32: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/32.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – simulation
ε=10
![Page 33: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/33.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – phase portrait
![Page 34: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/34.jpg)
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – Lienard variables
3
3
/ 3 /
( / 3 )
/
y x x x
x x x y
y x
= − + +
= − +
= −
![Page 35: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/35.jpg)
VdP equation
2(1 ) 0x x x x− − + =
2 2(1 ) 0x x x x − − + =
The case of large ε – reduction of the VdP
equation
1/ 1
/d d
tdt d
=
= =
![Page 36: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/36.jpg)
VdP equation
2(1 ) 0x x x x− − + =
2 2(1 ) 0x x x x − − + =
The case of large ε – reduction of the VdP
equation
1/ 1
/d d
tdt d
=
= =
Small parameter multiplies the term with the highest derivative!
Singularly perturbed problem.
![Page 37: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/37.jpg)
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
Lienard system, ε>>1
Equation for x – “fast” dynamics
The system “tries” to nullify the velocity in x direction and does it fast!
![Page 38: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/38.jpg)
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
Lienard system, ε>>1
Equation for x – “fast” dynamics
Equation for y – “slow” dynamics
Slow evolution of x and y provided that the fast evolution is absent
![Page 39: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/39.jpg)
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − −
= −
3( / 3 )
0
x x x y
y y const
= − −
= =
Fast dynamics (principal approximation)
![Page 40: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/40.jpg)
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
3( / 3 )
0
x x x y
y y const
= − +
= =
Fast dynamics (principal approximation)
The system is attracted to the “slow” manifold
3 / 3y x x= − +
![Page 41: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/41.jpg)
Van der Pol model
3 / 3y x x= − +
Slow manifold
Stable branches
Unstable branch
![Page 42: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/42.jpg)
Van der Pol model
3 / 3y x x= − +
Slow manifold
Stable branches
Unstable branch
Fast motion
![Page 43: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/43.jpg)
Van der Pol model2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
3
2
2
/ 3( 1)
1/
y x x x xx x x
xy x
= − = − − = −
−= −
Lienard system, ε>>1
“Slow” motion:
![Page 44: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/44.jpg)
Van der Pol model
“Slow” motion
![Page 45: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/45.jpg)
Van der Pol model
Complete cycle of the relaxation oscillation!
![Page 46: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/46.jpg)
Van der Pol model
Complete cycle of the relaxation oscillation!
Let us compute the period.
![Page 47: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/47.jpg)
Van der Pol model
2
2
1
12 ( ) (3 2ln 2)
1slow
xx T x dx
x x = − = − = −
−
Complete cycle of the relaxation oscillation!
Slow part:
![Page 48: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/48.jpg)
Van der Pol model
(3 2ln 2)slowT = −
3( / 3 )
0
x x x y
y y const
= − +
= =
23
3
1
( / 3 ) 2(1/ )
/ 3 2 / 32 / 3fast
x x x y dxT O
x xy
−
= − + = =
− + +=
Complete cycle of the relaxation oscillation!
Slow part:
Fast part:
For jump: y=±2/3
![Page 49: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/49.jpg)
Van der Pol model
(3 2ln 2)slowT = −
3( / 3 )
0
x x x y
y y const
= − +
= =
23
3
1
( / 3 ) 2(1/ )
/ 3 2 / 32 / 3fast
x x x y dxT O
x xy
−
= − + = =
− + +=
Complete cycle of the relaxation oscillation!
Slow part:
Fast part:
For jump: y=±2/3
Are there any corrections of lower order?
![Page 50: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/50.jpg)
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
![Page 51: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/51.jpg)
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
Approximation breaks down – infinite slow velocity.
![Page 52: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/52.jpg)
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
Approximation breaks down – infinite slow velocity.
Matching is required
![Page 53: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/53.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Matching in physical plane
![Page 54: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/54.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
![Page 55: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/55.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
2
0 0 0
2
0 0
0
( 1) 0
ln ( 1)
0, 0, 0
x x x
x x A
x A
− + =
= − − +
=
![Page 56: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/56.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
2
0 0 0
2
0 0
0
0
( 1) 0
ln ( 1)
0, 0, 0
0, ~ 1 2
Singularity!
x x x
x x A
x A
x
− + =
= − − +
=
→ − + −
Near x=1 separate approximation is required
![Page 57: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/57.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
4/3 2/3
2
2
2
, 1 ( )
2 1 0
, const
x v
d v dvv
d d
dvv B B
d
= = +
+ + =
+ + = =
![Page 58: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/58.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
4/3 2/3
2
2
2
, 1 ( )
2 1 0
, const
x v
d v dvv
d d
dvv B B
d
= = +
+ + =
+ + = =
/ ,
( ) (Ai( ) Bi( ))
, const
v z z
z B C B
C
=
= − + −
=
![Page 59: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/59.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
Matching with Zone 1
, ~
0
( ) Ai ( ) / Ai( )
v
B C
v
→ − −
= =
= − − −
![Page 60: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/60.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
Matching with Zone 1
, ~
0
( ) Ai ( ) / Ai( )
v
B C
v
→ − −
= =
= − − −
Everything is OK until singularity…
1
( ) 0, 2,33811
In the vicinity:
( ) ~ ( ) simple pole
Ai
v
−
− = =
− −
![Page 61: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/61.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 3
4/3 2
22
2
3
( 1) 0
/ 3
d w dww
d d
dww w D
d
= +
+ − =
= − +
![Page 62: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/62.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 3
4/3 2
22
2
3
( 1) 0
/ 3
d w dww
d d
dww w D
d
= +
+ − =
= − +
Matching with Zone 2
4/3
4/3
1 1 2 / 3ln( )
1 3 1
, 2 / 3 (exp( ))
w
w w
w O
+ +− = +
− −
→ = − − + −
![Page 63: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/63.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =2
0 0
4/3
4/3
ln ( 1)
3 / 2 ln 2 3 / 2
3 2ln 2 3
x x A
A
T
= − − +
= − +
= − +
Zone 1
Matching with Zone 1 (lower branch) and completing the half – cycle.
4/3 4/3( , ) ( 2 / 3, )x = − −
![Page 64: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/64.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
1/3(3 2ln 2) 3 , 2.3381T −= − + =
Expression for the period (initial variables)
![Page 65: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/65.jpg)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
1/3(3 2ln 2) 3 , 2.3381T −= − + =
1/3 1
1
4/3
2(3 2ln 2) 3 ln
3
{ln 2 ln 3 3 1 ln 2ln Ai ( )}
( ln ), 2.3381, 0.17235
T
b
O b
− −
−
−
= − + − +
− + − − − − +
+ = =
Expression for the period (initial variables)
After much more painstaking efforts one can obtain:
![Page 66: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/66.jpg)
Phenomenology - 1Concept of fast – slow partition (boundary layer)
![Page 67: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/67.jpg)
Phenomenology - 2Concept of fast – slow partition (boundary layer)
Slow process Fast process
![Page 68: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/68.jpg)
Phenomenology - 3
![Page 69: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/69.jpg)
Synchronization
![Page 70: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/70.jpg)
Synchronization – simple model
),(
),(
21222
21111
Fdt
d
Fdt
d
+=
+=
1 2 1 2 1 2
1 2 1 2
( , ) ( );
( ),
i iF F
dF F F F
dt
= − = −
= − + = −
![Page 71: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/71.jpg)
Synchronization – simple model
The first term of Fourier series
1 2 1 0sin( )d
Fdt
= − + +
Possibility of Stationary Solution
1
~
21
1 −
F
![Page 72: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/72.jpg)
Earthquake
L'Aquila Earthquake, Italy, 200972
Simple Procedure for Seismic Analysis of Liquid-Storage Tanks, Malhotra, Structural Engineering International, 2000
Tanks with liquids
![Page 73: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/73.jpg)
Liquid SloshingWhat is it?
73
Sloshing noun any motion of the free liquid
surface inside its container. It is caused by
any disturbance to partially filled liquid
containers.
(Liquid sloshing dynamics- Theory and Applications, R.A. Ibrahim, 2005)
Sloshing/convective portion
Static portion
![Page 74: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/74.jpg)
Liquid SloshingWhat is it?
74
![Page 75: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/75.jpg)
1. Infinitely many degrees of freedom
2. Substantially nonlinear vibrations
How to explain the dynamics?
"כעש
תה
ר ייא
בד
ו"
75
http://www.mscsoftware.com/en/product/dytran
![Page 76: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/76.jpg)
Reduced-0rder modeling
( ) ( )2 42 2 2
1 2
1 1 1 1 1
2 2 2 2 4tt ttL T V Mu mv k u k v u k v u= − = + − − − − −
( )21
2t tD c v u= −
![Page 77: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/77.jpg)
Analytic treatmentGeneral structure of slow invariant manifold
Possibility of relaxation oscillations? Analysis of slow motion is required.
![Page 78: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/78.jpg)
Numeric experimentsSteady – state and strongly quasiperiodic responses coexist for different IC
A=0.225, λ=0.2, ε=0.05. y1(0)=0.29,dy1/dt(0)=0.25, y2(0)=0, dy2/dt(0)=-0.15
A=0.225, λ=0.2, ε=0.05. y1(0)=0, dy1/dt(0)=0, y2(0)=0, dy2/dt(0)=0.
![Page 79: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/79.jpg)
Numeric experimentsWeakly quasiperiodic response and SMR coexist for different IC
A=0.24, λ=0.2, ε=0.05. y1(0)=0.29, dy1/dt(0)=0.25, y2(0)=0, dy2/dt(0)=-0.15
A=0.24, λ=0.2, ε=0.05. y1(0)=0, dy1/dt(0)=0, y2(0)=0, dy2/dt(0)=0.
![Page 80: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/80.jpg)
Successive captures into the resonance
Captured
Jumps out
![Page 81: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/81.jpg)
Experimentsl and numeric results (full-scale sloshing!)
a) b)
c) d)
![Page 82: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/82.jpg)
Experimental and numeric results (full-scale sloshing1)
![Page 83: Nonlinear Dynamics and Resonances: a surveyevents.iitgn.ac.in/2019/NOW2019/content/8... · the resonance. 3 3 0 2 GM A r T\ Capture into the resonance JT qL L –Phase on the Kepler](https://reader033.vdocuments.us/reader033/viewer/2022060312/5f0af8287e708231d42e3a5d/html5/thumbnails/83.jpg)
83