What is the difference between nonlinearity and chaos, or..

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Nonlinearity and chaos are distinct concepts, though chaos can emerge from nonlinear systems. Not all nonlinear equations lead to chaotic behavior, highlighting that the relationship is conditional. A chaotic system is characterized by exponential divergence of nearby orbits and confinement within a compact space. Linear systems cannot exhibit both traits, as exponential divergence would result in unbounded solutions. Understanding these differences is crucial for analyzing complex systems.
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What is the difference between nonlinearity and chaos, or is there a difference?
 
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Not all nonlinear systems of eqs are chaotic.
 
You're asking the question the wrong way round. A more logical question would be "under what conditions are they the same?"

As it stands, the shortest answer would be "they're not the same thing". Chaos is a different phenomenon to nonlinearity, but sometimes arises from it.
 
What is sure, is that chaotic systems are non-linear.

The definition of a chaotic system is:
- exponential divergence of initially close orbits (Liapunov instability)
- the state of the system remains in a compact space

Well, you cannot satisfy both with a linear system: in the case there is exponential divergence, at least one solution will become unbounded.
 
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