Flows on the Line Flashcards
Phase Space
Definition
-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
Definition
|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
Trajectories
Definition
-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
Definition
- let |x(t) be a solution of a differential equation
- its graph in extended phase space is called an integral curve
Slope Field
Definition
-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
Definition
-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
Definition
-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
-so:
|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
Description
- 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
Unstable
Definition
- a fixed point is unstable if flow in phase space is away from the point in both directions
- denoted by an empty circle
Stable
Definition
- a fixed point is stable if the flow in phase space is towards the point on both sides
- denoted by a solid circle
