Lecture 5: Asymptotic behaviour - GitHub Pages

Lecture 5: Asymptotic behaviourGlobal 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 solutionHence define the 2 and 3 limit points of the trajectory

1Dynamical Systems: Lecture 5

Limit pointsDefinition: 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 pointsDefinition: !(Γ) and %(Γ) are called the !-limit set and %-limit setrespectively. 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


Example limit sequences


A sequence of points leading to an isolated point !-limit set

A carefully chosen sequence of points to an !-limit point when the !-limit set is not an isolated point

"!(Γ)

" = !(Γ)

ΓD

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.


InvarianceDefinition: 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).


A"rac&onDefinition: An invariant set ! ⊂ # is attracting if1. 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 *.

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.


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.


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.


Example

&

'

0

Global behaviour:!̇ = ! 1 − !+ > 0 if ! > 1

< 0 if 0 < ! < 1

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 Duffingoscillator

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 = " ∈ ) ∶ +̇ " = 02 = union of all positively invariant sets in 0

The Principle: Every trajectory starting at " ∈ ) tends to 2 as A → ∞.


Example: Duffing Oscillator"̇ = $$̇ = " − "& − '$, ' > 0

Let + ", $ = ,-. −

/-. +

/12 , then

+̇ = −'$.

• 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.


The Duffing Oscillator and LaSalle


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M

D

"̇ =0$̇ ≠0

Types of orbits – Preparation for The PoincaréBendixson 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.


Homoclinic orbit exampleConsider 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

#, % = (0,0) (the Jacobian is 0 11 0 with eigenvalues 1, -1).



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The trajectory shown in red leaves the origin on the local unstable manifold and returns on the local stable manifold.

Homoclinic orbit example

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.



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Heteroclinic separatrix cycle

Simple pendulum phase portrait

Example compound separatrix cycle


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 !.


Example applica+on of Poincaré Bendixson Thm.• Consider the Duffing oscillator:

"̇ = $$̇ = " − "& − '$, ' > 0

The level sets of + ", $ = ,-. −

/-. +

/12 = 3 are positively invariant

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.



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0 < # ≤ 8Foci

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Duffing Oscillator example illustra>ng two types of stable equilibria

Saddle

Saddle

Further example"̇# = "# + "& − "#("#& + "&&)"̇& = −"# + "& − "&("#& + "&&)

• The origin is an equilibrium point, linearisation gives Jacobian 1 1−1 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 PoincaréBendixson 2 must contain a stable limit cycle.



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Unstable spiral within a stable limit cycle

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Lecture 5: Asymptotic behaviour - GitHub Pages