L ECTURE 11: D YNAMICAL S YSTEMS 10 T EACHER : G IANNI A. D I C ARO L - - PowerPoint PPT Presentation

LECTURE 11: DYNAMICAL SYSTEMS 10

TEACHER: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S19

2

LIMIT CYCLES

So far … Something new: limit cycles / orbital stability Unstable equilibrium Periodic orbit: ! " + $ = !(") (-limit set of points

3

LIMIT CYCLES

§ A limit cycle is an isolated closed trajectory: neighboring trajectories are not close, they are spiral either away or to the cycle § If all neighboring trajectories approach the limit cycle: stable, unstable

• therwise, half-stable in mixed scenarios

§ In a linear system closed orbits are not isolated à No limit cycles

4

LIMIT CYCLE EXAMPLE

r=1

A solution component !(#) starting outside unit circle ends to the circle (! oscillates with amplitude 1

% ̇ ' = '(1 − '+) ̇ , = 1

' ≥ 0

§ Radial and angular dynamics are uncoupled, such that they can be analyzed separately § The motion in , is a rotation with constant angular velocity § Treating ̇ ' = '(1 − '+) as a vector field on the line, we observe that there are two critical points, (0) and (1) § The phase space (', ̇ ') shows the functional relation: (0) is an unstable fixed point, (1) is stable, since the trajectories from either sides go back to ' = 1

5

VAN DER POL OSCILLATOR

!"" + ! − %(1 − !()!"

Harmonic

• scillator

Nonlinear damping Positive (regular) damping for ! > 1 Negative (reinforcing) damping for ! < 1 Oscillations are large: it forces them to decay Oscillations are small: it pumps them back à System settles into a self-sustained

• scillation where the energy

dissipated over one cycle balances the energy pumped in à Unique limit cycle for each value of , > -

Two different initial conditions converge to the same limit cycle

. = ! 0 = !′ ." = 0 0" = −. + % 1 − .( 0

6

VAN DER POL OSCILLATOR

!"" + ! − %(1 − !()!" Numeric integration. Analytic solution is difficult

% = 5

7

CONDITIONS OF EXISTENCE OF LIMIT CYCLES

We need a few preliminary results, in the form of the next two theorems, formulated for a two dimensional system:

§ Theorem (Closed trajectories and critical points): Let the functions !

" and ! # have continuous first partial derivatives in a domain $

• f the phase plane.

A closed trajectory of the system must necessarily enclose at least one critical point (i.e., an equilibrium) (note: a closed trajectory necessarily lies in a bounded region) If the trajectory encloses only one critical point, the critical point cannot be a saddle point % ̇ '" = !

"('", '#)

̇ '# = !

#('", '#)

, '", '# = (!

", ! #)

Exclusion version: if a given region contains no critical points, or only saddle points, then there can be no closed trajectory lying entirely in the region.

Under which conditions do closed orbits / limit cycles exist?

8

CONDITIONS OF EXISTENCE OF LIMIT CYCLES

§ Theorem (Dulac’s criterion) : Let the functions !

" and ! # have continuous first partial derivatives in a simply

connected domain $ of the phase plane, if there’s exist a continuously differentiable scalar function ℎ('", '#) such that div ℎ- =

/01

2

/32 + /01

4

/34

'", '# has the same sign throughout $, then there is no

closed trajectory of the system lying entirely in $ § If sign changes nothing can be said Not a simply connected domain § Simply connected domains: § A simply connected domain is a domain with no holes § In a simply connected domain, any path between two points can be continuously shrink to a point without leaving the set § Given two paths with the same end points, they can be continuously transformed one into the other while staying the in the domain

9

PROOF OF THE THEOREM (ONLY FOR FUN)

§ The proof is based on the following fundamental theorem in calculus § In general it’s hard (the same as for Lyapounov functions) to identify an ℎ function § Theorem (Green’s theorem): if " is a sufficiently smooth simple closed curve, enclosing a bounded region #, and $ and % are two continuous functions that have continuous first partial derivatives in an open region & containing # then: where " is traversed counterclockwise for the line integration Let’s suppose that " is a periodic solution, and $=ℎ'

(, % = ℎ' + such that $ , + %. has the same sign in

&. This implies that the double integral must be ≠ 0. The line integral can be written as ∮

3 ℎ( ̇

6(, ̇ 6+) 8 9 :ℓ which is zero, because " is a solution and the vector ( ̇ 6(, ̇ 6+) is always tangent to it à We get a contradiction.

10

POINCARE’-BENDIXSON THEOREM (1901)

§ Theorem (Poincare’- Bendixson) Suppose that: § ! is a closed, bounded subset of the phase plane § ̇ # = % # is a continuously differentiable vector field on an open set containing ! § ! does not contain any critical points § There exists a trajectory & that is confined in !, in the sense that it starts in ! and stays in ! for all future time Then, either & is a closed orbit, or it spirals toward a closed orbit as ' → ∞, in either case ! contains a closed orbit / periodic solution (or, limit cycle) Remark: If ! contains a closed orbit, then, because of the previous theorem, it must contain a critical point * ⟹ ! cannot be simply connected, it must have a hole

11

POINCARE’-BENDIXSON THEOREM

§ How do we verify the conditions of the theorem in practice? ü ! is a closed, bounded subset of the phase plane ü ̇ # = % # is a continuously differentiable vector field on an open set containing ! ü ! does not contain any critical points v There exists a trajectory & that is confined in !, in the sense that is starts in ! and stays in ! for all future time: Difficult one! 1. Construct a trapping region !: a closed connected set such that the vector field points inward on the boundary of ! à All trajectories are confined in ! 2. If ! can also be arranged to not include any critical point, the theorem guarantees the presence of a closed

• rbit

12

CHECKING P-B CONDITIONS

§ It’s difficult, in general

! ̇ # = # 1 − #' + )# cos - ̇

• = 1

# ≥ 0 For this system we saw that, for )=0, # = 1 is a limit cycle. Is the cycle still present for ) > 0, but small? § In this case, we know where to look to verify the conditions of the theorem: let’s find an annular region around the circle # = 1: 0 < #345 ≤ # ≤ #

378, that

plays the role of trapping region, finding #345 and #

378 such that ̇

# < 0 on the outer circle, and ̇ # > 0 on the inner one § Condition of no fixed points in the annular region is verified since ̇

• > 0

v v

§ For # = #345, ̇ # must be > 0: # 1 − #' + )# cos - > 0, observing that cos - ≥ −1, it’s sufficient to consider 1 − #' + ) > 0 → #345 < 1 − ), ) < 1 § A similar reasoning holds for #

378: # 378 >

1 + ) § The range should be chosen as tight as possible § Since all the conditions of the theorem as satisfied, a limit cycle exists for the selected #345, #

378

13

CHECKING P-B CONDITIONS FOR VAN DER POL

§ A failing example: !"" + ! − %(1 − !()!" Van Der Pol Critical point: origin, the linearized system has eigenvalues (% ± %( − 4)/2 à (0,0) is unstable spiral for 0 < % < 2 à (0,0) is an unstable node for % ≥ 2 § Closed trajectories? The first theorem says that if they exist, they must enclose the origin, the only critical point. From the second theorem, with ℎ constant,

• bserving that 23

4

25 + 23

6

27

= %(1 − 9(), if there are closed trajectories, they are not in the strip 9 < 1, where the sign of the sum is positive § Neither the application of the P-B theorem is conclusive / easy → … 9" = ; ;" = −9 + % 1 − 9( ; 9 = ! ; = !′ →

14

CHECKING P-B CONDITIONS FOR VAN DER POL

15

POINCARE’-BENDIXON THEOREM: NO CHAOS IN 2D!

§ Only apply to two-dimensional systems! § It says that second-order (two-dimensional) dynamical systems are overall “well-behaved” and the dynamical possibilities are limited: if a trajectory is confined to a closed, bounded region that contains no equilibrium points, then the trajectory must eventually approach a closed orbit, nothing more complicated that this can happen § A trajectory will either diverge, or settle down to a fixed point or a periodic

• rbit / limit cycle, that are the attractors of system’s dynamics

§ What about higher dimensional systems, for ! ≥ #? § Trajectory may wonder around forever in a bounded region without settling down to a fixed point or a closed orbit! § In some cases the trajectories are attracted to a complex geometric objects called strange attractor, a fractal set on which the motion is aperiodic and sensitive to tiny changes in the initial conditions § à Hard to predict the behavior in the long run à Deterministic chaos

16

STRANGE ATTRACTORS, NEXT …

Strange attractor Fractal dimension: coastline length changes with the length of the ruler