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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



Introduction



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



Dynamics overviewHow to predict more accurately dynamical features on system?



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.



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.



Why numerics?

Nonlinear systems

Dynamics

- complex behaviour

- closed form solutions not often available

- discontinuities !!!

Computational resources

- availability – technology

- robust/improved numerical methods



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



Numeric bugs

Hysteresis

Branch jump



Branching behaviour in dynamical systems



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



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 .



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



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



Branch tracing (2)

In general, there are two main ending point type for a codimension-one branch namely turning points and single bifurcation points.



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



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



Tangent predictor

Tangential projection of solution:

1j j j jy y h v



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



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



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



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



Correction (2)

1( ) ,j j jg y y y v

- Pseudo-arclength continuation:



Correction (3)

1( ) ,j k kkg y y Y V

- Moore-Penrose continuation (MATCONT):

1( ) 0k kJ Y V



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



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



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



Test functions (2)

1 2: max , ,..., n

0 0: det ,yf y

Usual chooses:

( , )( ) :

( , )

f yF Y

y



Branch switching

When there is a single bifurcation point, there are more than one trajectories for the which (y0,λ0) is an equilibrium:



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



An algorithm



Application and results



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



Continuation of periodic orbits (2)



Tracing a perioud-doubling



Eigenvalue evolution



On unit circle



Sudden chaotic window



On set – brute force



On set – continued



Conjectures

How to explain such particularly regular cascade?

- development of local maps



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



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.



...?http://wpage.unina.it/r.alzate

Grazie e arrivederci !!!

http://wpage.unina.it/r.alzate

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