15-382 C OLLECTIVE I NTELLIGENCE – S18 L ECTURE 8: D YNAMICAL S YSTEMS 7 I NSTRUCTOR : G IANNI A. D I C ARO
G EOMETRIES IN THE PHASE SPACE § Damped pendulum One cp in the region between two separatrix Saddle Asymptotically Separatrix unstable Basin of Asymptotically attraction stable spiral (or node) Closed orbits § Undamped (periodic) pendulum Fixed point (any period) Center: the linearization approach doesn’t allow to say much about stability 2
G EOMETRIES IN THE PHASE SPACE … Question 1: The linearization approach for § Lyapounov studying the stability of critical points is a purely functions local approach. Going more global, what about the basin of attraction of a critical point? Question 2: When the linearization approach § fails as a method to study the stability of a critical point, can we rely on something else? Question 3: Are critical points and well § separated closed orbits all the geometries we can Limit cycles have in the phase space? Question 4: Does the dimensionality of the phase § space impact on the possible geometries and limiting behavior of the orbits? Question 5: Are critical points and closed orbits § the only forms of attractors in the dynamics of the phase space? Is chaos related to this? 3
L YAPUNOV DIRECT METHOD : P OTENTIAL FUNCTIONS 𝒈:ℝ ) → ℝ ) 𝒚̇ = 𝒈 𝒚 , 𝑊 𝒚(𝑢) = Potential energy of the system when in state 𝒚 , 𝑊:ℝ ) → ℝ 𝑊 𝒚(𝑢) 𝑦 1 𝑦 0 Time rate of change of 𝑊 𝒚(𝑢) along a solution trajectory 𝒚(𝑢) , we need § to take the derivative of 𝑊 with respect to 𝑢. Using the chain rule: 𝑒𝑊 𝑒𝑢 = 𝜖𝑊 𝑒𝑦 0 𝑒𝑢 + ⋯+ 𝜖𝑊 𝑒𝑦 ) 𝑒𝑢 = 𝜖𝑊 0 𝑦 0 ,… ,𝑦 ) + ⋯ + 𝜖𝑊 𝑔 𝑔 ) 𝑦 0 ,…, 𝑦 ) 𝜖𝑦 0 𝜖𝑦 ) 𝜖𝑦 0 𝜖𝑦 ) Solutions do not appear, only the system itself! 4
L YAPUNOV FUNCTIONS 𝒚̇ = 𝒈 𝒚 , 𝒈: ℝ ) → ℝ ) § 𝒚 9 equilibrium point of the system § A function 𝑊:ℝ ) → ℝ continuously differentiable is called a Lyapunov § function for 𝒚 9 if for some neighborhood 𝐸 of 𝒚 9 the following hold: 𝑊 𝒚 9 = 0 , and 𝑊 𝒚 > 0 for all 𝒚 ≠ 𝒚 9 in 𝐸 1. 𝑊̇ 𝒚 ≤ 0 for all 𝒚 in 𝐸 2. If 𝑊̇ 𝒚 < 0 , it’s called a strict Lyapunov function § 𝑊 𝒚(𝑢) 𝑊 𝒚(𝑢) = Energy of the system § when in state 𝒚 𝒚 9 is a the bottom of the graph 1. of the Lyapunov function 2. Solutions can’t move up, but can 𝑦 1 only move down the side of the potential hole or stay level 𝑦 0 5
L YAPUNOV STABILITY THEOREM Theorem (Sufficient conditions for stability): § Let 𝒚 9 be an ( isolated ) equilibrium point of the system 𝒚̇ = 𝒈 𝒚 . If there exists a Lyapunov function for 𝒚 9 , then 𝒚 9 is stable. If there exists a strict Lyapunov function for 𝒚 9 , then 𝒚 9 is asymptotically stable 𝑊 𝒚(𝑢) Any set 𝐸 on which 𝑊 is a strict Lyapunov § function for 𝒚 9 is a subset of the basin 𝐶(𝒚 9 ) If there exists a strict Lyapunovfunction, § then there are no closed orbits in the region 𝑦 1 of its basin of attraction 𝑦 0 Definition: Let 𝒚 9 be an asymptotically stable equilibrium of 𝒚̇ = 𝒈 𝒚 . Then § the basin of attraction of 𝒚 9 , denoted 𝐶(𝒚 9 ) , is the set of initial conditions 𝒚 C such that lim G→H 𝑮 𝒚 C ,𝑢 = 𝒚 9 6
H OW DO WE DEFINE L YAPUNOV FUNCTIONS ? Physical systems: Use the energy function of the system itself § PQ For a damped pendulum ( 𝑦 = θ, 𝑧 = PG ) § Other systems: Guess! 𝒚 9 = (0,0) For 𝑐 = 0, 𝑏, 𝑑 > 0 → 𝑊̇ < 0,𝑊 > 0 ⇒ (0,0) is asymptotically stable 7
L IMIT CYCLES So far … Unstable equilibrium Periodic orbit: 𝒚 𝑢 + 𝑈 = 𝒚(𝑢) 𝝏 -limit set of points Something new: limit cycles / orbital stability 8
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 9
L IMIT CYCLE EXAMPLE T𝑠̇ = 𝑠(1 − 𝑠 1 ) 𝑠 ≥ 0 𝜄̇ = 1 § Radial and angular dynamics are uncoupled, such that they can be analyzed separately § The motion in 𝜄 is a rotation with constant angular velocity Treating 𝑠̇ = 𝑠(1 − 𝑠 1 ) as vector field on the line, we observe § that there are two critical points, (0) and (1) The phase space ( 𝑠, 𝑠̇) shows the functional relation: (0) is an unstable fixed § point, (1) is stable, since the trajectories from either sides go back to 𝑠 = 1 A solution component 𝑦(𝑢) starting outside unit circle ends to the circle ( 𝑦 oscillates with amplitude 1 r=1 10
V AN D ER P OL O SCILLATOR 𝑣 ZZ + 𝑣 − 𝜈(1 − 𝑣 1 )𝑣 Z Harmonic Nonlinear oscillator damping Positive (regular) Negative (reinforcing) damping for 𝑣 > 1 damping for 𝑣 < 1 Oscillations are large: Oscillations are small: à System settles into a self- it forces them to decay it pumps them back sustained oscillation where the energy dissipated over one cycle balances the energy pumped in à Unique limit cycle for each value of 𝝂 > 𝟏 Two different initial conditions converge to the same limit cycle 11
V AN D ER P OL O SCILLATOR 𝑣 ZZ + 𝑣 − 𝜈(1 − 𝑣 1 )𝑣 Z 𝜈 = 5 Numeric integration. Analytic solution is difficult 12
C ONDITIONS OF EXISTENCE OF LIMIT CYCLES Under which conditions do close orbits / limit cycles exist? We need a few preliminary results, in the form of the next two theorems, formulated for a two dimensional system: T𝑦̇ 0 = 𝑔 0 (𝑦 0 ,𝑦 1 ) 𝑦̇ 1 = 𝑔 1 (𝑦 0 ,𝑦 1 ) § Theorem (Closed trajectories and critical points): Let the functions 𝑔 0 and 𝑔 1 have continuous first partial derivatives in a domain 𝐸 of the phase plane. A closed trajectory of the system must necessarily enclose at least one critical (equilibrium) point . If it 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. 13
C ONDITIONS OF EXISTENCE OF LIMIT CYCLES § Theorem (Existence of closed trajectories): Let the functions 𝑔 0 and 𝑔 1 have continuous first partial derivatives in a simply connected domain 𝐸 of the phase plane. _` _` a c If _b a + 𝑦 0 , 𝑦 1 has the same sign throughout 𝐸 , then there is no closed _b c trajectory of the system lying entirely in 𝐸 § 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 14
P ROOF OF THE THEOREM ( ONLY FOR FUN ) § Theorem (Existence of closed trajectories): Let the functions 𝑔 0 and 𝑔 1 have continuous first partial derivatives in a simply connected domain 𝑆 of the phase plane. _` _` a c If _b a + 𝑦 0 , 𝑦 1 has the same sign throughout 𝑆 , then there is no closed _b c trajectory of the system lying entirely in 𝑆 § The proof is based on Green’s theorem , a fundamental theorem in calculus: if 𝐷 is a sufficiently smooth simple closed curve, and if 𝐺 and 𝐻 are two continuous functions and have continuous first partial derivatives, then: where 𝐷 is traversed counterclockwise and 𝐵 is the region enclosed by C . Let’s suppose that 𝐷 is a periodic solution and 𝐺 = 𝑔 0 ,𝐻 = 𝑔 1 , such that 𝐺 b + 𝐺 i has the same sign in 𝑆. This implies that the double integral must be ≠ 0. The line integral can be written as ∮ (𝑦̇ 0 ,𝑦̇ 1 ) k 𝒐 𝑒ℓ which is m zero, because 𝐷 is a solution and the vector (𝑦̇ 0 ,𝑦̇ 1 ) is always tangent to it à We get a contradiction. 15
P OINCARE ’-B ENDIXSON T HEOREM § Theorem (Poincare’- Bendixson) Suppose that: 𝑆 is a closed, bounded subset of the phase plane • 𝒚̇ = 𝒈 𝒚 is a continuously differentiable vector field on an open set • containing 𝑆 𝑆 does not contain any critical points • There exists a trajectory 𝐷 that is confined in 𝑆 , in the sense that is • starts in 𝑆 and stays in 𝑆 for all future time Then, either 𝐷 is a closed orbit, or it spirals toward a closed orbit as 𝑢 → ∞ , in either case 𝑆 contains a closed orbit / periodic solution (and, possibly, a limit cycle) Remark: If 𝑆 contains a closed orbit, then, because of the previous theorem, it must contain a critical point 𝑄 ⟹ 𝑆 cannot be simply connected, it must have a hole 16
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