Yes, it does.

My take is that, without nonlinearity, folding is missing.

One of the main mechanisms behind classical chaos is the so-called stretch and fold. It can be visualized as a blob of initial conditions being stretched and then folded over itself by the mapping: stretching leads to a divergence of close trajectories (the hallmark of chaos), while folding keeps them bounded (and dense).

A linear system may be able to produce stretching, but this alone corresponds a trivial behavior, divergence. But, in the spaces we're considering, a linear system cannot produce folding:

[none of the] orbits of a linear operator in finite dimensions [...] are dense in the space

Notice, though, that topology can provide the folding that a linear transformation alone can't, see, e.g., Arnold's cat map, which is a linear map on a torus.

You ask about the condition 3, but we don't need to address it directly:

if $f$ has a dense set of periodic points and is transitive, then $f$ must have sensitive dependence on initial conditions. Hence only the first two conditions of the definition of chaos need to be verified when showing that a particular function $f$ is chaotic.

As for differential equations, any linear ODE, $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}$, can be solved. More pictorially, the understanding of maps can be somewhat transfered to flows by means of Poincaré maps and, especially, Poincaré recurrence plots. See, e.g., Carroll's A review of return maps for Rössler and the complex Lorenz or Crutchfield's slides from the lecture Example Dynamical Systems.

Yes, it does.

My take is that, without nonlinearity, folding is missing.

One of the main mechanisms behind classical chaos is the so-called stretch and fold. It can be visualized as a blob of initial conditions being stretched and then folded over itself by the mapping: stretching leads to a divergence of close trajectories (the hallmark of chaos), while folding keeps them bounded (and dense).

A linear system may be able to produce stretching, but this alone corresponds to a trivial behavior, divergence. But, in the spaces we're considering, a linear system cannot produce folding:

[none of the] orbits of a linear operator in finite dimensions [...] are dense in the space

Why can't linear functions produce folding?

• The reason is that folding means that an interval is mapped on itself twice, and that requires a non-monotonic function (and linear functions are monotonic), such as the logistic map's parabola.

Notice, though, that topology can provide the folding that a linear transformation alone can't, see, e.g., Arnold's cat map, which is a linear map on a torus.

You ask about the condition 3, but we don't need to address it directly:

if $f$ has a dense set of periodic points and is transitive, then $f$ must have sensitive dependence on initial conditions. Hence only the first two conditions of the definition of chaos need to be verified when showing that a particular function $f$ is chaotic.

As for differential equations, any linear ODE, $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}$, can be solved. More pictorially, the understanding of maps can be somewhat transfered to flows by means of Poincaré maps and, especially, Poincaré recurrence plots. See, e.g., Carroll's A review of return maps for Rössler and the complex Lorenz or Crutchfield's slides from the lecture Example Dynamical Systems.