ODEs that are linear involve only state variables, derivatives of the state variables, constants times those things, and constants by themselves No powers, no products, and no transcendentals The first equation has a power in it, so it is nonlinear The second equation has only state variables, and constants times state variables, and derivatives of state variables in it So that is a linear ODE The third one has a product of a state variable and a derivative of a state variable, which makes it nonlinear And the fourth one has a transcendental function in it, making it, too, nonlinear Nonlinearity is a necessary condition for chaos; that is, all chaotic systems are nonlinear It is not a sufficient condition; that is, there are some nonlinear systems that are not chaotic What that means is that that is the arrow that goes where the question marks are If nonlinearity were a necessary and sufficient condition for chaos, you would use a bidirectional arrow And if all nonlinear systems were necessarily chaotic, youd use the other arrow