The answer is a clear no.
First off, non linear ODEs may have no solution for some initial conditions or on the contrary have several solutions.
For unicity and existence it is needed that the derivative be continuous what may not be obvious at a simple glance for systems of several non linear ODEs.
Once the unicity and existence warranted, in chaotic systems there are always control parameters. A given non linear ODE system will have simple or complex but non chaotic solutions for a range of control parameters and it may have chaotic solutions for another range of control parameters.
The logistic equation even if it is not an ODE (X(n+1) = µ.Xn.(1-Xn)) will show chaotic behaviour only for some values of µ.
There is no known general and simple rule allowing to know whether for a given non linear ODE system, there exists a set of values of the control parameters for which the solutions are chaotic.
However it is slightly easier from a physical point of view. Chaotic orbits in physics have 2 properties - exponential divergence (called sensibility to initial conditions) and dissipation which makes sure that the orbits don't explode to infinity.
This is topologically seen like "stretching" and "folding" in the phase space.
So if you take a forced system which will be dissipative AND its equations of motion will be non linear, then you will have a good chance to find chaotic regimes.
The case of Hamiltonian chaos (gravitational N body problem) is different as it doesn't involve stretching and folding but orbit instabilities on a torus.
Beyond these 2 cases where chaotic solutions may be looked for, there is of course the domain of PDEs which gives rise to spatio temporal chaotic solutions (as opposed to the mere temporal chaos for the case of ODEs).
Even if the question didn't concern the PDE, the answer is obviously no for them too even if chaotic behaviour is more frequent in spatio temporal than in the temporal domain.