15-382 COLLECTIVE INTELLIGENCE – S19
LECTURE 11: DYNAMICAL SYSTEMS 10
TEACHER: GIANNI A. DI CARO LIMIT CYCLES
So far …
Unstable equilibrium Periodic orbit: ! " + $ = !(") (-limit set of points
Something new: limit cycles / orbital stability 2 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 otherwise, half-stable in mixed scenarios § In a linear system closed orbits are not isolated à No limit cycles 3 LIMIT CYCLE EXAMPLE
'̇ = '(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
A solution component !(#) starting outside unit circle ends to the circle (! oscillates with amplitude 1 r=1
. = ! !"" + ! − %(1 − !()!" 0 = !′ ." = 0 0" = −. + % 1 − .( 0 Harmonic Nonlinear oscillator damping
Positive (regular) Negative (reinforcing) damping for ! > 1 damping for ! < 1
Oscillations are large: it Oscillations are small: it à System settles into a self-sustained forces them to decay pumps them back oscillation 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 5 VAN DER POL OSCILLATOR
!"" + ! − %(1 − !()!"
% = 5
Numeric integration. Analytic solution is difficult 6 CONDITIONS OF EXISTENCE OF LIMIT CYCLES
Under which conditions do closed orbits / limit cycles exist?
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 $ of 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.
7 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 /012 /014 div ℎ- = + , has the same sign throughout $, then there is no /32 /34 '" '# closed trajectory of the system lying entirely in $ § If sign changes nothing can be said
§ 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
Not a simply connected domain
8 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(6̇(, 6̇+) 8 9 :ℓ which is zero, because " is a solution and the vector (6̇(, 6̇+) is always tangent to it à We get a contradiction. 9 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
10 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 orbit 11 CHECKING P-B CONDITIONS
§ It’s difficult, in general For this system we saw that, for )=0, # = 1 is a limit cycle. Is the cycle still #̇ = # 1 − #' + )# cos - ! # ≥ 0 present for ) > 0, but small? -̇ = 1 § 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 ' § For # = #345, #̇ must be > 0: # 1 − # + )# cos - > 0, observing that cos - ≥ −1, it’s sufficient to consider 1 − #' + ) > 0
→ #345 < 1 − ), ) < 1
v § 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 12 CHECKING P-B CONDITIONS FOR VAN DER POL
§ A failing example: Van Der Pol 9 = ! 9" = ; !"" + ! − %(1 − !()!" → ; = !′ ;" = −9 + % 1 − 9( ;
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, 23 23 observing that 4 + 6 = %(1 − 9(), if there are closed trajectories, they 25 27 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 → …
13 CHECKING P-B CONDITIONS FOR VAN DER POL
14 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 orbit / 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 15 STRANGE ATTRACTORS, NEXT …
Strange attractor
Fractal dimension: coastline length changes with the length of the ruler
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