15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 11: D YNAMICAL S YSTEMS 10 T EACHER : G IANNI A. D I C ARO
L IMIT CYCLES So far … Unstable equilibrium Periodic orbit: ! " + $ = !(") ( -limit set of points Something new: limit cycles / orbital stability 2
L IMIT 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
̇ L IMIT CYCLE EXAMPLE ' = '(1 − ' + ) % ̇ ' ≥ 0 , = 1 § Radial and angular dynamics are uncoupled, such that they can be analyzed separately The motion in , is a rotation with constant angular velocity § ' = '(1 − ' + ) as a vector field on the line, we observe Treating ̇ § 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 4
V AN D ER P OL O SCILLATOR . = ! 0 = !′ ! "" + ! − %(1 − ! ( )! " . " = 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
V AN D ER P OL O SCILLATOR ! "" + ! − %(1 − ! ( )! " % = 5 Numeric integration. Analytic solution is difficult 6
̇ C ONDITIONS 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 i n the region. 7
C ONDITIONS 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 /01 /3 2 + /01 2 4 div ℎ- = ' " , ' # has the same sign throughout $ , then there is no /3 4 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
P ROOF 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. 9
̇ P OINCARE ’-B ENDIXSON T HEOREM (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
̇ P OINCARE ’-B ENDIXSON T HEOREM How do we verify the conditions of the theorem in practice? § ü ! is a closed, bounded subset of the phase plane # = % # is a continuously differentiable vector field on an open set ü 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
̇ C HECKING 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 - ! ̇ present for ) > 0, but small? # ≥ 0 - = 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 § # must be > 0 : # 1 − # ' + For # = # 345 , ̇ § )# cos - > 0 , observing that cos - ≥ −1, it’s sufficient to consider 1 − # ' + ) > 0 → # 345 < 1 − ) , ) < 1 A similar reasoning holds for # 378 : # 378 > 1 + ) v v § 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
C HECKING P-B CONDITIONS FOR V AN D ER P OL A failing example: § Van Der Pol 9 " = ; 9 = ! ! "" + ! − %(1 − ! ( )! " → ; " = −9 + % 1 − 9 ( ; ; = !′ % ( − 4)/2 Critical point: origin, the linearized system has eigenvalues (% ± 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 observing that 23 6 = %(1 − 9 ( ) , if there are closed trajectories, they 4 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
C HECKING P-B CONDITIONS FOR V AN D ER P OL 14
P OINCARE ’-B ENDIXON T HEOREM : N O 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
S TRANGE ATTRACTORS , NEXT … Strange attractor Fractal dimension: coastline length changes with the length of the ruler 16
Recommend
More recommend
