Numerical analysis of nonlinear dynamics

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

2. Numerical analysis of nonlinear dynamics Outline • Introduction • Branching behaviour in dynamical systems • Application and results

3. Numerical analysis of nonlinear dynamics Introduction

4. 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

5. Numerical analysis of nonlinear dynamics Dynamics overview How to predict more accurately dynamical features on system?

6. Numerical analysis of nonlinear dynamics 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.

7. Numerical analysis of nonlinear dynamics 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.

8. 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

9. 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

10. Numerical analysis of nonlinear dynamics Numeric bugs Hysteresis Branch jump

11. Numerical analysis of nonlinear dynamics Branching behaviour in dynamical systems

12. Numerical analysis of nonlinear dynamics General statement In general, it is possible to study the dependence of dynamics (solutions) in terms of parameter variation (implicit function theorem).

13. Numerical analysis of nonlinear dynamics 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 .

14. Numerical analysis of nonlinear dynamics Implicit function theorem (2) Then, singularity condition on gy(f(x),x) excludes x = ±1 as part of function domain in order to apply the theorem.

15. Numerical analysis of nonlinear dynamics Branch tracing 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)

16. Numerical analysis of nonlinear dynamics Branch tracing (2) In general, there are two main ending point type for a codimension-one branch namely turning points and single bifurcation points.

17. 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

18. Numerical analysis of nonlinear dynamics Arclength Augmented system with additional constraints:

19. Numerical analysis of nonlinear dynamics Tangent predictor Tangential projection of solution:

20. Numerical analysis of nonlinear dynamics Tangent predictor (2) Tangent unity vector:

21. Numerical analysis of nonlinear dynamics Root finding Newton-Raphson method for location of equilibria:

22. Numerical analysis of nonlinear dynamics Root finding (2) In general: i.e. nonsingularity of Jacobian at solution Allowing implementation of method.

23. Numerical analysis of nonlinear dynamics Correction Additional relation gj(y) defines an intersection of the curve f(y) with some surface near predicted solution (ideally containing it): - Natural continuation:

24. Numerical analysis of nonlinear dynamics Correction (2) • Pseudo-arclength continuation:

25. Numerical analysis of nonlinear dynamics Correction (3) - Moore-Penrose continuation (MATCONT):

26. Numerical analysis of nonlinear dynamics Moore-Penrose Pseudo-inverse matrix:

27. 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

28. 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.

29. Numerical analysis of nonlinear dynamics Test functions (2) Usual chooses:

30. Numerical analysis of nonlinear dynamics Branch switching When there is a single bifurcation point, there are more than one trajectories for the which (y0,λ0) is an equilibrium:

31. Numerical analysis of nonlinear dynamics Branch switching (2) How to track such new trajectory? - Algebraic branching equation (Keller 1977 !!!)

32. Numerical analysis of nonlinear dynamics An algorithm

33. Numerical analysis of nonlinear dynamics Application and results

34. Numerical analysis of nonlinear dynamics 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

35. Numerical analysis of nonlinear dynamics Continuation of periodic orbits(2)

36. Numerical analysis of nonlinear dynamics Tracing a perioud-doubling

37. Numerical analysis of nonlinear dynamics Eigenvalue evolution

38. Numerical analysis of nonlinear dynamics On unit circle

39. Numerical analysis of nonlinear dynamics Sudden chaotic window

40. Numerical analysis of nonlinear dynamics On set – brute force

41. Numerical analysis of nonlinear dynamics On set – continued

42. Numerical analysis of nonlinear dynamics Conjectures • How to explain such particularly regular cascade? • development of local maps

43. Numerical analysis of nonlinear dynamics Open tasks - Improvement of numerical approximation for map - Theoretical prediction (or validation): A. Nordmark (2003)

44. Numerical analysis of nonlinear dynamics 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.

45. Numerical analysis of nonlinear dynamics ...? http://wpage.unina.it/r.alzate Grazie e arrivederci !!!