L ECTURE 8: D YNAMICAL S YSTEMS 7 I NSTRUCTOR : G IANNI A. D I C ARO - - PowerPoint PPT Presentation
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
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§ Damped pendulum Separatrix Basin of attraction Saddle Asymptotically unstable Asymptotically stable spiral (or node)
One cp in the region between two separatrix
§ Undamped pendulum Closed orbits (periodic) Fixed point (any period) Center: the linearization approach doesn’t allow to say much about stability
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§ Question 1: The linearization approach for studying the stability of critical points is a purely 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 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? Lyapounov functions Limit cycles
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𝒚̇ = 𝒈 𝒚 , 𝒈:ℝ) → ℝ) 𝑊 𝒚(𝑢) = Potential energy of the system when in state 𝒚, 𝑊:ℝ) → ℝ
𝑦0 𝑦1 𝑊 𝒚(𝑢)
§ 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 𝜖𝑦0 𝑒𝑦0 𝑒𝑢 + ⋯+ 𝜖𝑊 𝜖𝑦) 𝑒𝑦) 𝑒𝑢 = 𝜖𝑊 𝜖𝑦0 𝑔
0 𝑦0,… ,𝑦) + ⋯ + 𝜖𝑊
𝜖𝑦) 𝑔
) 𝑦0,…, 𝑦)
Solutions do not appear, only the system itself!
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§ 𝒚̇ = 𝒈 𝒚 , 𝒈: ℝ) → ℝ) § 𝒚9 equilibrium point of the system § A function 𝑊:ℝ) → ℝ continuously differentiable is called a Lyapunov function for 𝒚9 if for some neighborhood 𝐸 of 𝒚9 the following hold: 1. 𝑊 𝒚9 = 0, and 𝑊 𝒚 > 0 for all 𝒚 ≠ 𝒚9 in 𝐸 2. 𝑊̇ 𝒚 ≤ 0 for all 𝒚 in 𝐸 § If 𝑊̇ 𝒚 < 0, it’s called a strict Lyapunov function § 𝑊 𝒚(𝑢) = Energy of the system when in state 𝒚 1. 𝒚9 is a the bottom of the graph
2. Solutions can’t move up, but can
potential hole or stay level
𝑦0 𝑦1 𝑊 𝒚(𝑢)
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§ Theorem (Sufficient conditions for stability): Let 𝒚9 be an (isolated) equilibrium point of the system 𝒚̇ = 𝒈 𝒚 . If there exists a Lyapunov function for 𝒚9, then 𝒚9 is stable. If there exists a strict Lyapunov function for 𝒚9, then 𝒚9 is asymptotically stable
𝑦0 𝑦1 𝑊 𝒚(𝑢)
§ Definition: Let 𝒚9 be an asymptotically stable equilibrium of 𝒚̇ = 𝒈 𝒚 . Then the basin of attraction of 𝒚9, denoted 𝐶(𝒚9), is the set of initial conditions 𝒚C such that limG→H 𝑮 𝒚C,𝑢 = 𝒚9 § Any set 𝐸 on which 𝑊 is a strict Lyapunov function for 𝒚9 is a subset of the basin 𝐶(𝒚9) § If there exists a strict Lyapunovfunction, then there are no closed orbits in the region
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§ Physical systems: Use the energy function of the system itself For 𝑐 = 0, 𝑏, 𝑑 > 0 → 𝑊̇ < 0,𝑊 > 0 ⇒ (0,0) is asymptotically stable 𝒚9 = (0,0) § Other systems: Guess! For a damped pendulum (𝑦 = θ, 𝑧 =
PQ PG)
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So far … Something new: limit cycles / orbital stability Unstable equilibrium Periodic orbit: 𝒚 𝑢 + 𝑈 = 𝒚(𝑢) 𝝏-limit set of points
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§ 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
§ In a linear system closed orbits are not isolated
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r=1
A solution component 𝑦(𝑢) starting outside unit circle ends to the circle (𝑦 oscillates with amplitude 1
T𝑠̇ = 𝑠(1 − 𝑠1) 𝜄̇ = 1
𝑠 ≥ 0
§ 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 − 𝑠1) 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
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𝑣ZZ + 𝑣 − 𝜈(1 − 𝑣1)𝑣Z
Harmonic
Nonlinear damping Positive (regular) damping for 𝑣 > 1 Negative (reinforcing) damping for 𝑣 < 1 Oscillations are large: it forces them to decay Oscillations are small: it pumps them back
à System settles into a self- 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
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𝑣ZZ + 𝑣 − 𝜈(1 − 𝑣1)𝑣Z Numeric integration. Analytic solution is difficult
𝜈 = 5
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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 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.
T𝑦̇0 = 𝑔
0(𝑦0,𝑦1)
𝑦̇1 = 𝑔
1(𝑦0,𝑦1)
Exclusion version: if a given region contains no critical points,
can be no closed trajectory lying entirely in the region. Under which conditions do close orbits / limit cycles exist?
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§ 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. If
_`
a
_ba + _`
c
_bc
𝑦0, 𝑦1 has the same sign throughout 𝐸, then there is no closed
trajectory of the system lying entirely in 𝐸 § 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
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§ 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. If
_`
a
_ba + _`
c
_bc
𝑦0, 𝑦1 has the same sign throughout 𝑆, then there is no closed
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 ∮ (𝑦̇0,𝑦̇1) k 𝒐
m
𝑒ℓ which is zero, because 𝐷 is a solution and the vector (𝑦̇0,𝑦̇1) is always tangent to it à We get a contradiction.
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§ Theorem (Poincare’- Bendixson) Suppose that:
containing 𝑆
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
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§ How do we verify the conditions of the theorem in practice? ü 𝑆 is a closed, bounded subset of the phase plane ü 𝒚̇ = 𝒈 𝒚 is a continuously differentiable vector field on an open set containing 𝑆 ü 𝑆 does not contain any critical points v There exits a trajectory 𝐷 that is confined in 𝑆, in the sense that is starts in 𝑆 and stays in 𝑆 for all future time: Difficult one! 1. Construct a trapping region 𝑆: a closed connected set such that the vector field points inward on the boundary of 𝑆 à All trajectories are confined in 𝑆 2. If 𝑆 can also be arranged to not include any critical point, the theorem guarantees the presence of a closed orbit
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§ It’s difficult, in general
T𝑠̇ = 𝑠 1 − 𝑠1 + 𝜈𝑠 cos𝜄 𝜄̇ = 1
𝑠 ≥ 0 For this system we saw that, for 𝜈=0, 𝑠 = 1 is a limit cycle. Is the cycle still present for 𝜈 > 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 𝑠 = 1: 0 < 𝑠uv) ≤ 𝑠 ≤ 𝑠
uwb,
that plays the role of trapping region, finding 𝑠uv) and 𝑠
uwb such that
𝑠̇ < 0 on the outer circle, and 𝑠̇ > 0 on the inner one § The conditions of no fixed points in the annulus region is verified since 𝜄̇ > 0
v v
§ For 𝑠 = 𝑠uv), 𝑠̇ must be > 0: 𝑠 1 − 𝑠1 + 𝜈𝑠 cos 𝜄 > 0, observing that cos𝜄 ≥ −1, it’s sufficient to consider 1 − 𝑠1 + 𝜈 > 0 → 𝑠uv) < 1 − 𝜈, 𝜈 < 1 § A similar reasoning holds for 𝑠
uwb: 𝑠 uwb >
1 + 𝜈 § The should be chosen as tight as possible § Since all the conditions of the theorem as satisfied, a limit cycle exists for the selected 𝑠uv), 𝑠
uwb
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§ A failing example 𝑣ZZ + 𝑣 − 𝜈(1 − 𝑣1)𝑣Z Van Der Pol Critical point: origin, the linearized system has eigenvalues (𝜈 ± 𝜈1 − 4)/2 à (0,0) is unstable spiral for 0 < 𝜈 < 2 à (0,0) is an unstable node for 𝜈 ≥ 2 § Closed trajectories? The first theorem says that if they exist, they must enclose the origin, the only critical point. From the second theorem,
_`
{
_b + _`
|
_i
= 𝜈(1 − 𝑦1), if there are closed trajectories, they are not in the strip 𝑦 < 1, where the sign of the sum is positive § Neither the application of the P-B theorem is conclusive / easy → …
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§ 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 𝒐 ≥ 𝟒? § 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
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Strange attractor Fractal dimension: coastline length changes with the length of the ruler