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Phase Space

-from now on we shall assume that differential equations are autonomous:
|x' = |f (|x)
|x = (x1, .... , xn) ⊂ U ∈ R^n
-here U is a domain in R^n where the vector function |f(|x) is assumed to be continuous and differentiable
-the domain U is called the phase space
-phase space is shows x' plotted against x


Extended Phase Space

|x' = |f (|x)
|x = (x1, .... , xn) ⊂ U ∈ R^n
-the domain UxR where R stays for time t or xo is called the extended phase space
-i.e. a 2-D space where x is plotted against t


Vector Field

|x' = |f (|x)
|x = (x1, .... , xn) ⊂ U ∈ R^n
-a system of differential equations (above) defines a vector field on U
-namely with any point |x we associate a vector |f(|x)
-a set of these vectors is called a vector field



-let |x(t) be a solution of a differential equation
-suppose we consider a system of three equations and thus:
|x(t) ∈ R^3
-in this 3-D space the point |x(t) is moving along a smooth curve as t is changing
-thus solutions of a differential equation are curves in the phase space given parametrically
-these curves are called trajectories


Trajectory, Vector Field and Tangents

-the vector field is tangent to a trajectory in every point
-let |x(t) be a trajectory
-the equation of a tangent line to the trajectory at the point |x1 = |x(t1) is:
|X(u) = |x1 + (u-t1)*|x'(t1)
= |x1 + (u-t1) * |f(|x1)
-from the general vector form of a line v = |a + n|b
where |a is a vector from the origin to a point on the line, |b is a vector along the line and n is a constant we see that:
|f(|x1) is the directional vector of the straight line tangent to the trajectory at |x1
-conversely any curve which is tangent to the vector field |f(|x) at each point is a trajectory of the differential equation:
|x' = |f(|x)


Integral Curve

-let |x(t) be a solution of a differential equation
-its graph in extended phase space is called an integral curve


Slope Field

-integral curves can be seen as trajectories of the extended system
-namely the system:
|x' = |f(|x) , |x = (x1, ... , xn)
-we extend by one 'dependent variable, x0(t) and one equation xo'=1 with initial condition xo(0)=0
-then we can draw a vector field in the extended phase space (xo,|x) ∈ RxU
-this vector field is called the slope field


Integral Curves and Slope Field

-the trajectories in the extended phase space are integral curves of the original system
-an integral curve is tangent to the slope field at each point
-the trajectory passing through the point (to,xo) is the integral curve corresponding to the initial condition |x(to)=xo


Phase Portrait

-a phase portrait is a set of trajectories with indication of the direction (due to the vector field)


Initial Value Problem and Uniqueness Theorem

-let |f(|x,t) be continuous and differentiable for
|t-to| ≤ α , | |x-|xo | ≤ β
-and, | |f(|x,t) | ≤ M ,
Σ |∂fi(|x,t)/∂xj| ≤ L < ∞
in that interval, where the sum is between 1 and N for both i and j
-let 𝛿 = min(α, β/M)
-then the initial value problem:
|x' = |f(|x,t) , |x(to) = |xo
-has a solution for |t-to|≤𝛿


Fixed Point

-if, |x' = |f(|x) , xϵU
-the point xoϵU is a critical point/fixed point/equilibrium/stationary point of vector field |f(|x) IF
|f(xo) = 0
-at a fixed point, given |x(o)=|xo
-then |x' evaluated at x=x0:
|f(|xo) = 0
|x(t) = |xo for all t


List the methods for solving first order equations and determining stability

-three methods:
1) construct an explicit solution (if possible)
2) use qualitative or geometric methods
3) use numerical methods


Solving First Order Differential Equations Using a Vector Field Approach

-for some differential equations it can be difficult to draw or even consider simple questions about behaviour
-an alternative approach is to treat the differential equation as a vector field, a set of vectors |x' defined at each point |x of phase space
-think of t as time, x as the position of an imaginary particle and x' as the velocity of that particle
-then differential equation x'=f(x) represents a vector field along the line giving a velocity vector x' for each x



-a fixed point is unstable if flow in phase space is away from the point in both directions
-denoted by an empty circle



-a fixed point is stable if the flow in phase space is towards the point on both sides
-denoted by a solid circle



-a point xo is semi stable if the flow in phase space is towards the point on one side and away from the point on the other side
-denoted by a half filled circle where the filled side is on the side that flow is towards the point


Identifying Fixed Points

-zoom in on the fixed point and look at the local behaviour using the coefficients of a Taylor series
f(x) = f(xo) + (x-xo)f'(xo) + (x-xo)²/2! * f''(xo) + ...
-for a fixed point f(xo) is 0 so go to the second coefficient
-if f'(xo) < 0 the point is stable, if f'(xo) > 0 the point is unstable or if f'(xo)=0 then go onto the third coefficient
-if f''(xo) < 0 the point is semi-stable with negative on both sides, if f''(xo) > 0 the point is semi stable with positive on both sides or if f''(xo)=0 then go onto the fourth coefficient etc.


How to sketch long polynomials

-e.g. trying to sketch
x' = f(x) = (x-4)²*(x-2)^3 * (x-1)*x² * (x+1)^5
-draw an x axis line with the fixed points labelled, in this case 4, 3, 2, 1, 0, -1
-then considering each point in turn find out whether the coefficient is positive or negative and sketch the local behaviour
e.g. for x=4, local behaviour follow (x-4)² and the coefficient is found by subbing x=4 into the rest of f(x)
(4-2)^3 * (4-1)*x² * (4+1)^5 > 0
-continue for the rest of the points, as you get further along you can identify the shape by inspection rather than calculating the coefficient since each segment has to connect


Linear Stability Analysis

-let xo be a fixed point
-the slope f'(xo) at the fixed point determines stability
-if f'(xo)<0 then the fixed point is stable
-if f'(xo)>0 then the fixed point is unstable
-the magnitude of f'(xo) determines how stable since 1/|f'(xo)| is the characteristic time scale which determines the time required for x(t) to vary significantly in the vicinity of xo
-if f'(xo)=0 then nothing is known in general



-solutions that reach infinity in finite time


Impossibility of Oscillations

-all trajectories either approach a fixed point or diverge to ±∞ because trajectories are forced to increase/decrease monotonically or remain constant
-i.e. the phase point never changes direction


Phase Fluid

-the imaginary fluid flowing along the real line in phase space
-flows to the right when f(x)>0 and flows to the left when f(x)<0


Phase Point

-to find a solution to x'=f(x), starting from arbitrary initial condition xo, place an imaginary particle, a phase point, at xo and watch how it is carried along by the flow
-as time passes the particle moves along the x axis according to some function x(t), the trajectory based at xo represents the solution of the differential equation starting from initial condition xo


Steps to Drawing a Diagram in Extended Phase Space

1) starting with x'=f(x) sketch a graph of phase space with f(x) on the y axis and x on the x axis
2) mark on the directions of flow, flow is to the right for f(x)>0 and to the left for f(x)<0
3) mark on fixed points and determine their stability
4) in extended phase space (x against t) draw horizontal lines at every fixed point
5) draw lines in between the fixed points showing whether lines tend towards or diverge from fixed points, this depends on their stability