**Flows on the Line**

- Graphically depicted differential equations can help give intuition regarding the behavior of their solutions.
- For a one-dimensional system, dx/dt is plotted against x as in the example below.

- While traversing x, its value changes according to ẋ since ẋ represents the rate of change of x. When ẋ > 0, x flows to the right. When ẋ < 0, x flows to the left. The arrows on the diagram show the direction of flow.
- The points where ẋ = 0 are called fixed points.
- If the arrows to either side point towards a fixed point, then it is a stable fixed point and is shown as a filled circle.
- If the arrows to either side point away from a fixed point, then it is an unstable fixed point and is shown as an empty circle.
- If the arrows to either side point in the same direction, then it is a half-stable fixed point and is shown as a half-filled circle. In this case, x will flow towards the fixed point if on the side with the arrow in the direction of the fixed point and away from the fixed point if on the side with the arrow directing away from the fixed point. If x is on this type of fixed point, then it will stay there unless perturbed.

** ****Bifurcations**

- Bifurcations occur when a changing parameter causes fixed points to be created, destroyed, or to undergo changes in their stability.
- The saddle-node bifurcation is exemplified by the system with the parameter r below. Any system with the same key properties could also have a saddle node bifurcation.

- When r<0, there is a stable fixed point and an unstable fixed point.
- When r=0, there is a half-stable fixed point.
- When r>0, there are no fixed points.

- When a fixed point must exist for all values of a parameter to properly model a system, transcritical bifurcations may occur.
- Transcritical bifurcations are exemplified by the formula below. Any system with the same key properties could also have a saddle transcritical bifurcation.

- When r<0, there is a stable fixed point at x=0 and an unstable fixed point at x=r.
- When r=0, there is a half-stable fixed point at x=0.
- When r>0, there is a stable fixed point at x=r and an unstable fixed point at x=0.
- In transcritical bifurcations, the stabilities of the fixed points “switch” as r changes sign.

**Bifurcation Diagrams**

- In bifurcation diagrams, the value of r is plotted on the horizontal axis and the value of x is plotted on the vertical axis.
- Fixed points are plotted over the diagram since they exist at multiple values of r. Solid lines represent stable fixed points, while dashed lines represent unstable fixed points.
- Below is an example of a bifurcation diagram for a transcritical bifurcation. Note how the diagram shows the exchange of stabilities between the fixed points as r changes sign.

**Phase Plane**

- In phase plane analysis, x(t) and y(t) from a system of differential equations are plotted against each other on the x,y plane with a vector field computed by dy/dx at each point (x,y) to reveal useful properties of the system.
- Note that there are also other versions of the phase plane for other types of differential equations.
- The system of differential equations described by this type of phase plane is autonomous; the right-hand side does not depend explicitly on t.

- Since the equations do not depend explicitly on t, they can be divided to give a formula for the slopes of each vector in the phase plane’s vector field.

- Any trajectory in the vector field defines a solution to the system given an initial condition defined by a starting point (x(0),y(0)) = (x
_{0},y_{0}) on the plot. - Equilibrium solutions occur at a critical points where dx/dt = 0 and dy/dt = 0. The set of all critical points is called the critical point set.
- Critical points are visible as points where the [f,g] vector has zero magnitude.

- Solutions may flow in various ways near their critical points.
- Nullclines are the curves where either dx/dt = 0 or dy/dt = 0 and thus the flow is governed solely by the other variable.
- Using eigenvalues and eigenvectors, the behavior of phase plane solutions can be predicted. To do this, write the system in matrix form as below and then compute the eigenvalues and eigenvectors of the coefficient matrix.

- The table and diagram below describe how to use eigenvalues and eigenvectors to determine the behavior of phase plane solutions.

- The phase plane is a useful tool for analyzing second order nonlinear differential equations and getting an intuitive understanding of their solutions.
- To convert a second order differential equation into a system of two first order differential equations, follow the process given in the example below. This example describes the motion of a pendulum of length L by its angle ϴ(t). Here, g represents gravitational acceleration.

- To linearize nonlinear systems, the Jacobian matrix is often used.

- Note that, in some cases, quadratic terms must also be linearized using Taylor expansions and other methods. The cases in which this is
__not__necessary include fixed points with saddles, nodes, or spirals. However, it is necessary for centers and several more exotic types of behavior. - After applying the Jacobian matrix to the system and evaluating at the critical points, the eigenvalues of the resulting matrices can be found. These eigenvalues can then be used to predict trajectory behavior as in the table above.
- Consider using the Jacobian on the nonlinear pendulum equation described previously.

Graphical plots were adapted from

Nonlinear Dynamics and Chaos(Strogatz) andFundamentals of Differential Equations and Boundary Value Problems(Nagle, Saff, and Snider)