Transcript Numerical analysis of nonlinear dynamics Ricardo Alzate Ph.D. Student University of Naples FEDERICO II (SINCRO GROUP)

Numerical analysis of
nonlinear dynamics
Ricardo Alzate Ph.D. Student
University of Naples FEDERICO II (SINCRO GROUP)
Numerical analysis of nonlinear dynamics
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Outline
• Introduction
• Branching behaviour in dynamical systems
• Application and results
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Numerical analysis of nonlinear dynamics
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Introduction
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics
Study of dynamics
Elements for extracting dynamical features:
• Mathematical representation
• Parameters and ranges
• Convenient presentation of results (first insight)
•
Careful quantification and classification of phenomena
• Validation with real world
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Numerical analysis of nonlinear dynamics
Dynamics overview
How to predict more accurately dynamical features on system?
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References
Chronology:
[1]. Seydel R. “Practical bifurcation ans stability analysis: from equilibrium to chaos”. 1994.
[2]. Beyn W. Champneys A. Doedel E. Govaerts W. Kutnetsov Y. and
Sandstede B. “Numerical continuation and computation of normal
forms”. 1999.
[3]. Doedel E. “Lecture notes on numerical analysis of bifurcation problems”. 1997.
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References (2)
[4]. Keller H.B. “Numerical solution of bifurcation and nonlinear eigenvalue problems”. 1977.
[5]. MATCONT manual. 2006. and Kutnetsov Book Ch10.
[6]. LOCA (library of continuation algorithms) manual. 2002.
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics
Why numerics?
Nonlinear systems
Dynamics
- complex behaviour
- closed form solutions not often available
- discontinuities !!!
Computational resources
- availability – technology
- robust/improved numerical methods
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Numerical analysis of nonlinear dynamics
How numerics?
Brute force simulation
- heavy computational cost
- tracing of few branches and just stable cases
- jumps into different attractors (suddenly)
- affected by hysteresis, etc..
Continuation based algorithms
- a priori knowledge for some solution
- a priori knowledge for system interesting regimes
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Numeric bugs
Hysteresis
Branch jump
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Numerical analysis of nonlinear dynamics
Branching behaviour in
dynamical systems
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General statement


sin( f )   0
xA
d
J  f  K  x A tan( f ) 
  y A  d 0  d1  
d
  
2

 cos 2 ( )
cos(

)
cos
(

)
 ( t )
f
f
f 



 
 x1 (t ) 

 
 equilibrium pts
x2 (t )   f ( t )   x2 (t )   f ( x,  )  
equilibrium orbits
 
x3 (t )   t
 x3 (t ) 


x1 (t )   f ( t )
In general, it is possible to study the dependence of dynamics
(solutions) in terms of parameter variation (implicit function theorem).
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics
Implicit function theorem
Establishes conditions for existence over a given interval, for an implicit (vector) function that solves the explicit problem
Given the equation f(y,x) = 0, if:
f(y*,x*) = 0,
f is continuously differentiable on its domain, and
fy(y*,x*) is non singular
Then there is an interval x1 < x* < x2 about x*, in which a vector function y = F(x) is defined by 0 = f(y,x) with the
following properties holding for all x with x1<x<x2 :
-
f(F(x),x) = 0,
F(x) is unique with y* = F(x*),
F(x) is continuously differentiable, and
fy(y,x)dy/dx + fx(y,x) = 0 .
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Implicit function theorem (2)
x2  y 2  1  0  g ( y, x)  y 2  x2 1
1

2 2
 f1 ( x)  1  x 
y  f ( x)  
1
2
 f ( x)   1  x 2
 
 2
 dg
J 
 dx
dg 
  2 x 2 f ( x) 

dy 
2


0  g ( f ( x), x)    1  x 2    x 2  1


1
2
Then, singularity condition on gy(f(x),x) excludes x = ±1 as part of
function domain in order to apply the theorem.
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Branch tracing
The goal is to detect changes in dynamical features depending on
parameter variation:
f ( y,  ) 
df ( y,  ) f ( y,  ) dy f ( y,  )


d
y d 

Then, by conditions of IFT:
f ( y,  )
df ( y,  )
dy

0

f ( y,  )
dy
d
y
Behaviour evolution as function of λ, not defined for singularities on fy(y,λ) (system having zero eigenvalues)
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Branch tracing (2)
In general, there are two main ending point type for a codimension-one
branch namely turning points and single bifurcation points.
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Numerical analysis of nonlinear dynamics
Parameterization
In order to avoid numerical divergence closing to turning points:
- Convenient change of parameter,
- Defining a new measure along the branch, e.g. the arclength
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Arclength
Augmented system with additional constraints:
 y  y(s)
f ( y,  )  0  
   ( s)
s    2   y 2
 dy   d  
  
 1
 ds   ds 
2
2
df
dy
d
 0  fy
 f
ds
ds
ds
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Tangent predictor
Tangential projection of solution:
y j 1  y j  h j v j
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Tangent predictor (2)
Tangent unity vector:
J ( y j )v j  0
J(y j) 
F
y
y y j
 J  j  0
 j 1 T  v   
 (v ) 
1
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v j 1 , v j  1
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Root finding
Newton-Raphson method for location of equilibria:
f ( y0  dy)  f ( y0 )  f y0dy  h.o.t.  f ( y0 )  f y0dy
 f ( y0 )
0  f ( y )  f dy  dy 
f y0
0
0
y
f ( y1 )
y  y  dy  f ( y )  0?  y  y 
...
1
fy
1
0
1
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Root finding (2)
In general:
 fYj 1dY j 1   f (Y j 1 )
f ( y,  )  f (Y )  
j
j 1
j 1
Y

Y

dY

 rank ( fYj 1 )  n  1
i.e. nonsingularity of Jacobian at solution
F
j 1
 f (Y ) 
  j 1 
 g (Y ) 
Allowing implementation of method.
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Correction
Additional relation gj(y) defines an intersection of the curve f(y) with
some surface near predicted solution (ideally containing it):
- Natural continuation:
g j ( y)  yio  yioj 1
 f (Y ) 
F 
0
j 1 
Y

Y
 io io 
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Correction (2)
- Pseudo-arclength continuation:
g j ( y )  y  y j 1 , v j
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Correction (3)
- Moore-Penrose continuation (MATCONT):
g kj ( y )  y  Y k 1 ,V k
 J (Y k 1 )V k  0
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Moore-Penrose
Pseudo-inverse matrix:
A nx  n1  A( n1)x n  AT  AAT 
1
Y  Y  Y  Yf   f Y
1
0
Y Y  f
1
0
0
Y
 f Y   f
0 
Y
0
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0
  Y    f  f Y 
0
Y  nx  n 1 
0 
Y
0
Numerical analysis of nonlinear dynamics
Step size control
Basic and effective approach (there are many !!!):
- Step size decreasing and correction repeat if non converging
- Slightly increase for step size if quick conversion
- Keep step size if iterations are moderated
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Numerical analysis of nonlinear dynamics
Test functions
Detection of stability changes between continued solutions:
- In general are developed as
smooth functions zero valued
at bifurcations, i.e.
 ( y0 , 0 )  0  ( y j ,  j ) ( y j 1,  j 1 )  0
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Test functions (2)
Usual chooses:
 : max 1,2 ,...,n 
 : det f y  y0 , 0 
 f ( y,  ) 
F (Y ) : 


(
y
,

)


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Branch switching
When there is a single bifurcation point, there are more than one
trajectories for the which (y0,λ0) is an equilibrium:
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Branch switching (2)
How to track such new trajectory?
f  y  s  ,   s   0 
df
dy
d
 f y0
 f 0
0
ds
ds
ds
0

v

range
f


y
dy

  0 v   1h  
0
ds
 h  null  f y 
2
- Algebraic branching equation (Keller 1977 !!!)  b  ac  0



a :  T f yy0 hh


2
2
a 1  2b 1 0  c 0  0   b :  T  f yy0 v  f y0  h

T
0
0
0
c :   f yy vv  2 f y v  f  

0
 null    range  f y 
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An algorithm
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Numerical analysis of nonlinear dynamics
Application and results
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Continuation of periodic orbits
P. Piiroinen – National University of Ireland (Galway):
- Single branch continuation
- Extrapolation prediction based
- Parameterization by orbit period
- Step size increasing if fast converging
- Step size reducing if non converging
- Newton-Raphson correction based
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Continuation of periodic orbits (2)
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Numerical analysis of nonlinear dynamics
Tracing a perioud-doubling
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Eigenvalue evolution
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On unit circle
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Sudden chaotic window
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On set – brute force
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On set – continued
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Conjectures
How to explain such particularly regular cascade?
- development of local maps
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Open tasks
- Improvement of numerical approximation for map
- Theoretical prediction (or validation): A. Nordmark (2003)
p( x)  a  bx  cx2  dx3  ex4  f x
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Conclusion
A general description about numerical techniques for branching analysis
of systems has been developed, with promising results for a particular
application on the cam-follower impacting model.
By the way, is not possible to think about a standard or universal
procedure given inherent singularities of systems, then researcher skills
constitute a valuable feature for success purposes.
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...?
http://wpage.unina.it/r.alzate
Grazie e arrivederci !!!
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