15-382 COLLECTIVE INTELLIGENCE – S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE § Damped pendulum One cp in the region between two separatrix Saddle Asymptotically Separatrix unstable Basin of attraction Asymptotically stable spiral (or node) § Undamped Closed orbits pendulum (periodic) Fixed point (any period) Center: the linearization approach doesn’t allow to say much about stability 2 GEOMETRIES 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 have in the phase space? Limit cycles § 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 LYAPUNOV DIRECT METHOD: POTENTIAL FUNCTIONS �̇ = � � , �: ℝ) → ℝ) � �(�) = Potential energy of the system when in state �, �: ℝ) → ℝ � �(�) �1 �0 § Time rate of chanGe of � �(�) alonG a solution trajectory �(�), we need to take the derivative of � with respect to �. UsinG the chain rule: �� �� ��0 �� ��) �� �� = + ⋯ + = �0 �0,… , �) + ⋯ + �) �0, … , �) �� ��0 �� ��) �� ��0 ��) Solutions do not appear, only the system itself! 4 LYAPUNOV FUNCTIONS § �̇ = � � , �: ℝ) → ℝ) § �9 equilibrium point of the system § A function �: ℝ) → ℝ continuously differentiable is called a Lyapunov function for �9 if for some neighborhood � of �9 the following hold: 1. � �9 = 0, and � � > 0 for all � ≠ �9 in � 2. �̇ � ≤ 0 for all � in � § If �̇ � < 0, it’s called a strict Lyapunov function � �(�) § � �(�) = Energy of the system when in state � 1. �9 is a the bottom of the graph of the Lyapunov function 2. Solutions can’t move up, but can � 1 only move down the side of the potential hole or stay level �0 5 LYAPUNOV STABILITY THEOREM § Theorem (Sufficient conditions for stability): Let �9 be an (isolated) equilibrium point of the system �̇ = � � . If there exists a Lyapunov function for �9, then �9 is stable. If there exists a strict Lyapunov function for �9, then �9 is asymptotically stable � �(�) § Any set � on which � is a strict Lyapunov function for �9 is a subset of the basin �(�9) § If there exists a strict Lyapunov function, then there are no closed orbits in the reGion �1 of its basin of attraction �0 § Definition: Let �9 be an asymptotically stable equilibrium of �̇ = � � . Then the basin of attraction of �9, denoted �(�9), is the set of initial conditions 9 �C such that limG→H � �C,� = � 6 HOW DO WE DEFINE LYAPUNOV FUNCTIONS? § Physical systems: Use the energy function of the system itself PQ For a damped pendulum (� = θ, � = ) PG § Other systems: Guess! �9 = (0,0) For � = 0, �, � > 0 → �̇ < 0, � > 0 ⇒ (0,0) is asymptotically stable 7 LIMIT CYCLES So far … Unstable equilibrium Periodic orbit: � � + � = �(�) �-limit set of points Something new: limit cycles / orbital stability 8 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 9 LIMIT CYCLE EXAMPLE �̇ = �(1 − �1) T �̇ = 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 − �1) as 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 10 VAN DER POL OSCILLATOR �ZZ + � − �(1 − �1)�Z Harmonic Nonlinear oscillator damping Positive (regular) Negative (reinforcing) damping for � > 1 damping for � < 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 � > � Two different initial conditions converge to the same limit cycle 11 VAN DER POL OSCILLATOR �ZZ + � − �(1 − �1)�Z � = 5 Numeric integration. Analytic solution is difficult 12 CONDITIONS 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: �̇ = � (� ,� ) T 0 0 0 1 �̇1 = �1 (�0,�1) § Theorem (Closed trajectories and critical points): Let the functions �0 and �1 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 (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 in the reGion. 13 CONDITIONS OF EXISTENCE OF LIMIT CYCLES § Theorem (Existence of closed trajectories): Let the functions �0 and �1 have continuous first partial derivatives in a simply connected domain � of the phase plane. _`a _`c If + , has the same sign throuGhout �, then there is no closed _ba _bc �0 �1 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 14 PROOF OF THE THEOREM (ONLY FOR FUN) § Theorem (Existence of closed trajectories): Let the functions �0 and �1 have continuous first partial derivatives in a simply connected domain � of the phase plane. _`a _`c If + , has the same sign throuGhout �, then there is no closed _ba _bc �0 �1 trajectory of the system lyinG entirely in � § The proof is based on Green’s theorem, a fundamental theorem in calculus: if � is a sufficiently smooth simple closed curve, and if � and � are two continuous functions and have continuous first partial derivatives, then: where � is traversed counterclockwise and � is the reGion enclosed by C. Let’s suppose that � is a periodic solution and �=�0, � = �1, such that �b + �i has the same sign in �. This implies that the double inteGral must be ≠ 0. The line inteGral can be written as ∮m (�̇ 0, �̇ 1) k � �ℓ which is zero, because � is a solution and the vector (�̇ 0,�̇ 1) is always tanGent to it à We Get a contradiction. 15 POINCARE’-BENDIXSON THEOREM § 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 is 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 (and, possibly, a 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 16 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 exits 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 17 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 − �1 + �� cos� T � ≥ 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 < �uv) ≤ � ≤ �uwb, that plays the role of trapping region, finding �uv) and �uwb such that �̇ < 0 on the outer circle, and �̇ > 0 on the inner one § The conditions of no fixed points in the annulus region is verified since �̇ > 0 1 § For � = �uv), �̇ must be > 0: � 1 − � + �� cos � > 0, observing that cos� ≥ −1, it’s sufficient 1 to consider 1 − � + � > 0 → �uv) < 1 − �, � < 1 v § A similar reasoning holds for �uwb: �uwb > 1 + � § The should be chosen as tight as possible § Since all the conditions of the theorem as satisfied, a limit cycle exists for the selected � , � uv) uwb 18 CHECKING P-B CONDITIONS FOR VAN DER POL § A failing example �ZZ + � − �(1 − �1)�Z Van Der Pol Critical point: origin, the linearized system has eigenvalues (� ± �1 − 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.