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<math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n)</math> | <math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n)</math> | ||
| − | When the <math>n</math> ODEs are not all linear, this is a nonlinear system of ODE. | + | When the <math>n</math> ODEs are not all linear, this is a nonlinear system of ODE. Consider an example, |
| + | <math>\frac{dx}{dt}=x(1-2x-3y)</math>, | ||
| − | + | <math>\frac{dy}{dt}=2y(3-x-2y)</math>. | |
| − | <font size="3px"></font> | + | In this tutorial, we will analyse this system in different aspects to build up a basic completed concept. </font> |
| + | |||
| + | |||
| + | '''<font size="4px"> 6.1 Equilibrium Point </font>''' | ||
| + | |||
| + | <font size="3px"> An equilibrium point is a constant solution to a differential equation. Hence, for an ODE system, an equilibrium point is going to be a solution of a pair of constants. Set all of the differential terms equal to <math>0</math> to find the equilibrium point.</font> | ||
Revision as of 20:42, 20 November 2017
Non-Linear Systems of ODEs
6.0 Concept
Consider the system of ODEs in 4.0,
$ \frac{dx_1}{dt}=f_1(t,x_1,x_2,...x_n) $
$ \frac{dx_2}{dt}=f_2(t,x_1,x_2,...x_n) $
...
$ \frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n) $
When the $ n $ ODEs are not all linear, this is a nonlinear system of ODE. Consider an example,
$ \frac{dx}{dt}=x(1-2x-3y) $,
$ \frac{dy}{dt}=2y(3-x-2y) $.
In this tutorial, we will analyse this system in different aspects to build up a basic completed concept.
6.1 Equilibrium Point
An equilibrium point is a constant solution to a differential equation. Hence, for an ODE system, an equilibrium point is going to be a solution of a pair of constants. Set all of the differential terms equal to $ 0 $ to find the equilibrium point.
6.2 Non-Linear Non-Autonomous System
6.3 Exercises
6.4 References
Institute of Natural and Mathematical Science, Massey University. (2017). 160.204 Differential Equations I: Course materials. Auckland, New Zealand.
Robinson, J. C. (2003). An introduction to ordinary differential equations. New York, NY., USA: Cambridge University Press.
