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Numerical analysis of nonlinear dynamics. Ricardo Alzate Ph.D. Student University of Naples FEDERICO II (SINCRO GROUP). Outline. Introduction Branching behaviour in dynamical systems Application and results. Introduction. Study of dynamics. Elements for extracting dynamical features: - PowerPoint PPT Presentation
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Numerical analysis of nonlinear dynamics
Ricardo Alzate Ph.D. StudentUniversity of Naples FEDERICO II (SINCRO GROUP)
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 2/45
Outline
• Introduction• Branching behaviour in dynamical systems • Application and results
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 4/45
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
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 5/45
Dynamics overviewHow to predict more accurately dynamical features on system?
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 6/45
References
Chronology:
[1]. Seydel R. “Practical bifurcation ans stability analysis: from equili-
brium 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 pro-
blems”. 1997.
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 7/45
References (2)
[4]. Keller H.B. “Numerical solution of bifurcation and nonlinear eigen-
value 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 8/45
Why numerics?
Nonlinear systems
Dynamics
- complex behaviour
- closed form solutions not often available
- discontinuities !!!
Computational resources
- availability – technology
- robust/improved numerical methods
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 9/45
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
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 10/45
Numeric bugs
Hysteresis
Branch jump
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 11/45
Branching behaviour in dynamical systems
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 12/45
General statement
1 1
2 2
33
( ) ( ) ( )
( ) ( ) ( ) ( , )
( )( )
f
f
x t t x tequilibrium pts
x t t x t f xequilibrium orbits
x t tx t
In general, it is possible to study the dependence of dynamics (solutions) in terms of parameter variation (implicit function theorem).
0 1 2 2
sin( ) 0tan( )
( )cos( ) cos ( ) cos ( )fA
f A f Af f f
xdJ K x y d d d
t
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 13/45
Implicit function theorem
Establishes conditions for existence over a given interval, for an im-
plicit (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 .
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 14/45
Implicit function theorem (2)
212 220 ( ( ), ) 1 1g f x x x x
12 2
1
12 2
2
( ) 1( )
( ) 1
f x xy f x
f x x
2 2 2 21 0 ( , ) 1x y g y x y x
Then, singularity condition on gy(f(x),x) excludes x = ±1 as part of function domain in order to apply the theorem.
2 2 ( )dg dg
J x f xdx dy
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 15/45
Branch tracing
( , ) ( , ) ( , )( , )
df y f y dy f yf y
d y d
The goal is to detect changes in dynamical features depending on parameter variation:
Then, by conditions of IFT:
Behaviour evolution as function of λ, not defined for singularities on fy(y,λ) (system having zero eigenvalues)
( , )( , )
0( , )
f ydf y dy
f ydy dy
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 16/45
Branch tracing (2)
In general, there are two main ending point type for a codimension-one branch namely turning points and single bifurcation points.
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 17/45
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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Numerical analysis of nonlinear dynamics 18/45
Arclength
2 2s y
( )( , ) 0
( )
y y sf y
s
0 y
df dy df f
ds ds ds
Augmented system with additional constraints:
2 2
1dy d
ds ds
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 19/45
Tangent predictor
Tangential projection of solution:
1j j j jy y h v
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 20/45
Tangent predictor (2)
Tangent unity vector:
( )j
j
y y
FJ y
y
( ) 0j jJ y v
1
0
( ) 1j
j T
Jv
v
1, 1j jv v
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 21/45
Root finding
0 0 0 0 0( ) ( ) . . . ( )y yf y dy f y f dy h o t f y f dy
00 0
0
( )0 ( ) y
y
f yf y f dy dy
f
Newton-Raphson method for location of equilibria:
11 0 1 2 1
1
( )( ) 0? ...
y
f yy y dy f y y y
f
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 22/45
Root finding (2)
1 1 1
1 1
( )( , ) ( )
j j jY
j j j
f dY f Yf y f Y
Y Y dY
1( ) 1jYrank f n
In general:
i.e. nonsingularity of Jacobian at solution
Allowing implementation of method.
11
( )
( )j
j
f YF
g Y
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 23/45
Correction
1
( )0j
io io
f YF
Y Y
Additional relation gj(y) defines an intersection of the curve f(y) with some surface near predicted solution (ideally containing it):
- Natural continuation:
1( )j jio iog y y y
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 24/45
Correction (2)
1( ) ,j j jg y y y v
- Pseudo-arclength continuation:
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Numerical analysis of nonlinear dynamics 25/45
Correction (3)
1( ) ,j k kkg y y Y V
- Moore-Penrose continuation (MATCONT):
1( ) 0k kJ Y V
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 26/45
Moore-Penrose
1( 1)1 xx
T Tn nn n
A A A AA
1 0 0 0 0 0Y YY Y Y Yf f Y Y f f Y
Pseudo-inverse matrix:
1 0 0 0 0
1xY Y n n
Y Y f f Y f
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 27/45
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
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 28/45
Test functionsDetection of stability changes between continued solutions:
- In general are developed as smooth functions zero valued at bifurcations, i.e.
0 0 1 1( , ) 0 ( , ) ( , ) 0j j j jy y y
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 29/45
Test functions (2)
1 2: max , ,..., n
0 0: det ,yf y
Usual chooses:
( , )( ) :
( , )
f yF Y
y
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 30/45
Branch switching
When there is a single bifurcation point, there are more than one trajectories for the which (y0,λ0) is an equilibrium:
R. Alzate - UN Manizales, 2007
Numerical analysis of nonlinear dynamics 31/45
Branch switching (2)
0 0, 0 0y
df dy df y s s f f
ds ds ds
0
0 1 0
y
y
v range fdyv h
ds h null f
0
2 2 0 01 1 0 0
0 0 0
0
:
2 0 :
: 2
Tyy
Tyy y
Tyy y
y
a f hh
a b c b f v f h
c f vv f v f
null range f
How to track such new trajectory?
- Algebraic branching equation (Keller 1977 !!!)2 0b ac
R. Alzate - UN Manizales, 2007
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Application and results
R. Alzate - UN Manizales, 2007
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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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Tracing a perioud-doubling
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Conjectures
How to explain such particularly regular cascade?
- development of local maps
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Numerical analysis of nonlinear dynamics 43/45
Open tasks
- Improvement of numerical approximation for map
- Theoretical prediction (or validation): A. Nordmark (2003)
2 3 4( )p x a bx cx dx ex 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 !!!