A Third-Order Differential Equation with Chaotic Solutions
A Third-Order Differential Equation with Chaotic Solutions
The solution of the following simple third-order ordinary differential equation exhibits chaos [1]:
-x'''=a[x''+x'+x-F(x,x',x'')]
here is the only free parameter. Despite the simplicity of the equation, its solutions are capable of producing a variety of dynamical behaviors, depending on the choice of the nonlinear function , which can depend on all three variables in the system: , , and . Three cases are considered here:
a
F
x
x'
x''
F(x,x',x'')=sign(x),
F(x,x',x'')=
-1 | ifx<0 |
1 | if0≤x<2 |
3 | ifx≥2 |
F(x,x',x'')=
-1 | ifx-x'<0 |
1 | ifx-x'≥0. |