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Positively Invariant If ! ", $ ∈ ' for All $ ∈ ' and All " ≥ 0

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Positively Invariant If ! Lecture 5: Asymptotic behaviour Global properties of trajectories: • Suppose ! ", $% is the flow of & $ with ! 0, $% = $% (i.e. $ " = ! ", $% is a solution of $̇ = &($) such that ! 0, $% = $%) • This solution defines a path or trajectory in some set , containing $%: Γ$. = $ ∈ , ∶ $ = ! ", $% , " ∈ ℝ • We want to determine the asymptotic behaviour of this solution Hence define the 2 and 3 limit points of the trajectory Dynamical Systems: Lecture 5 1 Limit points Definition: A point ! ∈ # is called an $ limit point of the trajectory % &, ( if there exists a sequence of times &) , &) → ∞, such that lim % &), ( = ! )→/ this point is denoted $ ( . • Note that we may need to choose the times &) carefully in order to get a limit point (e.g. consider % &, ( = e234 + cos(&)) • An a limit point is defined in the same way, but with &) → −∞. This point is denoted < ( Dynamical Systems: Lecture 5 2 Limit points Definition: !(Γ) and %(Γ) are called the !-limit set and %-limit set respectively. These are the sets of all ! limit points and % limit points for the trajectory Γ. • Hence !(Γ) is the set of points from which the trajectory Γ originates (at & = −∞) and %(Γ) is the set of points to which it tends (at & = ∞) • The set of all limit points is called the limit set of Γ Dynamical Systems: Lecture 5 3 Example limit sequences D Γ " !(Γ) " = !(Γ) A sequence of points leading A carefully chosen sequence of ! to an isolated point !-limit set points to an -limit point when the !-limit set is not an isolated point Dynamical Systems: Lecture 5 4 Equilibrium points • An equilibrium point !∗ is its own # and $ limit point. Conversely, if a trajectory has a unique $ limit point !∗, then !∗ must be an equilibrium point. • Not all $ limit points are equilibrium points (e.g. see the previous slide). If a point % is a limit point and %̇ ≠ 0 then this trajectory is a closed orbit. Note that we had to choose the sequence of points on the trajectory carefully in order to find a limit point and that there are infinitely many points on the $ limit set. Dynamical Systems: Lecture 5 5 Invariance Definition: Let ! ", $ be the flow of % $ on a set &, then a set ' ⊂ & is called positively invariant if ! ", $ ∈ ' for all $ ∈ ' and all " ≥ 0. • All points in ' stay in ' under the action of the flow – the solution cannot ‘escape’ from '. We saw an example of this in the case of the stable and unstable invariant manifolds in earlier lectures. • If a region , is positively invariant, closed and bounded, then the - limit set is not empty (i.e. you have to go somewhere!). The limit set itself is positively invariant (since once there, you stay in the set). Dynamical Systems: Lecture 5 6 A"rac&on Definition: An invariant set ! ⊂ # is attracting if 1. there is some neighbourhood $ of # ! which is positively invariant, and ! 2. all trajectories starting in $ tend to $ ! as % → ∞. $ is called a trapping region of !. Neighbourhood: A set surrounding a point ) so that the distance to all points in the neighbourhood from the point ) is less than some positive number *. Dynamical Systems: Lecture 5 7 A"ractors Definition: An attractor is an invariant attracting set (e.g. a limit cycle or equilibrium point) such that no subset of the invariant set is itself an invariant attracting set. Example: A stable node or focus is the ! limit set of all trajectories passing through points in a neighbourhood of the equilibrium point – the equilibrium point is an attractor. A saddle point on its own cannot be an attractor as trajectories leave the saddle point’s neighbourhood. Dynamical Systems: Lecture 5 8 Example "̇ = −% + " 1 − "( − %( %̇ = " + % 1 − "( − %( In polar co-ords: )̇ = ) 1 − )( *̇ = 1 Here: ) = 0 is an unstable hyperbolic equilibrium point ) = 1 is a limit cycle (since )̇ = 0) Setting ) = 1 + ,) gives ,)̇ ≈ −2,), so ) = 1 is a stable limit cycle. Dynamical Systems: Lecture 5 9 Example ' Global behaviour: !̇ = ! 1 − !+ > 0 if ! > 1 < 0 if 0 < ! < 1 0 & Hence ! = 1 is the $ limit set for all points in the plane except for the origin (which is its own $ limit set). The trapping region % is the whole of the plane minus the origin. Dynamical Systems: Lecture 5 10 Basin of a)rac,on The domain or basin of attraction of an attracting set ! is the union of all trajectories forming a trapping region of !. The domain of attraction of the left hand equilibrium point of the Duffing oscillator Dynamical Systems: Lecture 5 11 La Salle’s Invariance Principle • Let "̇ = $ " , " ∈ ℝ( and let ) ⊂ ℝ( be a positively invariant set (all points starting in ) remain in )). If the boundary of ) is differentiable and ) has a non-empty interior, then ) is a trapping region. • Suppose there exists a +(") that satisfies +̇ (") ≤ 0 within ) and consider the following two sets: 0 = " ∈ ) ∶ +̇ " = 0 2 = union of all positively invariant sets in 0 The Principle: Every trajectory starting at " ∈ ) tends to 2 as A → ∞. Dynamical Systems: Lecture 5 12 Example: Duffing Oscillator "̇ = $ $̇ = " − "& − '$, ' > 0 ,- /- /1 Let + ", $ = − + , then . 2 +̇ = −'$. • Let 3 = { ", $ ∶ + ", $ ≤ 7} for 7 > 0, then is positively invariant since +̇ ≤ 0. • Here 9 = ", $ ∶ $ = 0 and : = { −1,0 , 0,0 , 1,0 }. • LaSalle’s principle implies that all trajectories starting in 3 converge to :, and hence to one of the three equiliBria. Dynamical Systems: Lecture 5 13 The Duffing Oscillator and LaSalle D M -1 0 1 "̇ =0 $̇ ≠0 Dynamical Systems: Lecture 5 14 Types of orbits – Preparation for The Poincaré Bendixson Theorem The Poincaré Bendixson concerns the range of attractors that can exist in the Phase Plane. We will consider possible attractors, but first define some terms: - A homoclinic orbit is a trajectory that joins a saddle point equilibrium point to itself (it moves out on an unstable manifold and comes back on a stable manifold). - A heteroclinic orbit (or heteroclinic connection) joins two different equilibrium points. - A separatrix cycle partitions phase space into two regions with different characteristics and there are many ways to construct such cycles. Dynamical Systems: Lecture 5 15 Homoclinic orbit example Consider the Hamiltonian system: #̇ = % %̇ = # + #' %' #' #, ( #, % = − − 2 2 3 • The solution curves are the level sets of the Hamiltonian (energy is ' constant) and are defined by %' − #' − #, = . , ' • If . = 0, then %' = #' + #,, which goes through a saddle point at , 0 1 #, % = (0,0) (the Jacobian is with eigenvalues 1, -1). 1 0 Dynamical Systems: Lecture 5 16 Homoclinic orbit example 2.5 2 1.5 The trajectory shown in 1 red leaves the origin on 0.5 the local unstable 0 y manifold and returns on -0.5 the local stable manifold. -1 -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 x Dynamical Systems: Lecture 5 17 Heteroclinic example • The undamped simple pendulum "̇ = $ $̇ = −sin" • There are saddle points when the pendulum is pointing upwards – these saddle points are connected by a heteroclinic orbit. • The two heteroclinic orbits in the upper and lower half of the plane define a heteroclinic separatrix cycle. • A number of ‘compound’ separatrix cycles are shown in the notes. Note: At any point where the trajectory appears to cross itself there must be an equilibrium point. Dynamical Systems: Lecture 5 18 Simple pendulum phase portrait 2.5 2 Heteroclinic separatrix cycle 1.5 1 0.5 0 w -0.5 -1 -1.5 -2 -2.5 -4 -3 -2 -1 0 1 2 3 4 q Dynamical Systems: Lecture 5 19 Example compound separatrix cycle Dynamical Systems: Lecture 5 20 Poincaré Bendixson Theorem in the plane • Let ! be a positively invariant region of a vector field in ℝ# containing only a finite number of equilibria. Let $ ∈ ! and consider &($). Then one of the following possibilities holds: i. &($) is an equilibrium. ii. &($) is a closed orbit. ∗ ∗ iii. &($) consists of a finite number of equilibria $), … , $- and orbits ∗ ∗ . with / . = $1 and & . = $2. (Note: this defines a set of heteroclinic connections – consider the pendulum phase portrait.) • If there are only stable equilibria inside !, then there can only be one. If there are no stable equilibria in !, then there is a closed orbit in !. Dynamical Systems: Lecture 5 21 Example applica+on of Poincaré Bendixson Thm. • Consider the Duffing oscillator: "̇ = $ $̇ = " − "& − '$, ' > 0 ,- /- /1 The level sets of + ", $ = − + = 3 are positively invariant . 2 since +̇ (", $) = −'$. ≤ 0. • For 3 > 0, three equilibria lie inside ", $ ∶ + ", $ ≤ 3 : an unstable equilibrium at (0,0), and two stable equilibria at −1,0 , (1,0). • For 3 = 0, the level sets split into two with a common point at (0,0). Hence trajectories leaving the unstable equilibrium point at (0,0) must end up at the stable equilibria. Dynamical Systems: Lecture 5 22 Duffing Oscillator example illustra>ng two types of 0.8 0.6 stable equilibria 0.4 0.2 0 y Saddle 0 < # ≤ 8 -0.2 Foci 0.4 -0.4 0.3 -0.6 -0.80.2 0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x y 0 # > 8 Saddle -0.1 Nodes -0.2 -0.3 Dynamical Systems: Lecture 5 23 -0.4 -1.5 -1 -0.5 0 0.5 1 1.5 x Further example & & "̇# = "# + "& − "#("# + "& ) & & "̇& = −"# + "& − "&("# + "& ) • 1 1 The origin is an equilibrium point, linearisation gives Jacobian −1 1 with eigenvalues at 1 ± , (an unstable spiral). & & & & & & & & • Let - = "# + "& , then -̇ = 2"# + 2"& − 2 "# + "& = 2- − 2- . • -̇ ≤ 0 whenever - ≥ 1 so 2 = ", 4 ∶ - ≤ 6 is invariant for 6 ≥ 1. • As there is only an unstable equilibrium point inside 2, by Poincaré Bendixson 2 must contain a stable limit cycle. Dynamical Systems: Lecture 5 24 Unstable spiral within a stable limit cycle 2 1.5 1 0.5 2 0 x -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 1 Dynamical Systems: Lecture 5 25.
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