Critical Points in a system of differential equations

Thread starter rreaves
Start date Dec 15, 2010
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Critical points Differential Differential equations Points System

In summary, critical points in a system of differential equations are points where the slope or rate of change is equal to zero. They help us understand the behavior of the system and can be determined by solving the equations and setting the derivatives to zero. A system can have multiple critical points, which can be real or complex, and they affect the stability of the system. If all critical points are stable, the system is stable, but if there is at least one unstable critical point, the system is considered unstable. Semi-stable critical points can also exist, resulting in a semi-stable system.

Dec 15, 2010

rreaves

i am told to investagate the nature of the critical points of the system:

x'=e^y
y'=(e^y)*cos(x)

i am not sure where to begin because x' is always non-zero.

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Dec 15, 2010

HallsofIvy

Science Advisor

Homework Helper

Yes, exactly. And a "critcal point" for a system is a point where both x' and y' are 0 (or undefined). So what does that tell you about the critical points for this system?

Related to Critical Points in a system of differential equations

1. What are critical points in a system of differential equations?

Critical points in a system of differential equations are the points where the slope or rate of change of the system is equal to zero. These points are also known as equilibrium points or stationary points.

2. How are critical points important in understanding a system of differential equations?

Critical points help us understand the behavior of a system of differential equations. They can tell us if the system is stable, unstable, or semi-stable. They also provide valuable information about the long-term behavior of the system.

3. How are critical points determined in a system of differential equations?

To determine critical points, we need to solve the system of differential equations and set the derivatives to zero. This will give us a set of equations that we can then solve for the values of the variables at the critical points.

4. Can a system of differential equations have multiple critical points?

Yes, a system of differential equations can have multiple critical points. These points can be real or complex depending on the nature of the system. It is important to analyze all critical points to fully understand the behavior of the system.

5. How do critical points affect the stability of a system of differential equations?

The stability of a system of differential equations is determined by the nature of its critical points. If all critical points are stable, the system is considered stable. If there is at least one unstable critical point, the system is considered unstable. Semi-stable critical points can also exist, resulting in a semi-stable system.