Why must the sum of Lyapunov exponents be negative for dissipative systems?

Local Lyapunov exponents are defined like global Lyapunov exponents but without taking the long-time limit $t\to\infty$, so they retain more spatial/local information about stretching.

Conditions: Comparison is with global Lyapunov exponents; The limit $t\to\infty$ is omitted

Having a positive Lyapunov exponent is presented as a signature of chaos, but the lecturer explicitly says it is not a necessary and sufficient condition for chaos.

Conditions: Discussed in the context of identifying chaos via Lyapunov exponents.; The slide explicitly marks the condition as “not NASC!”.

A negative Lyapunov exponent corresponds to compression of state space along stable manifolds, while a positive Lyapunov exponent corresponds to stretching of state space along unstable manifolds. The lecture uses these signs to distinguish contraction toward a fixed point from expansion away to infinity.

Conditions: Applies to the manifold directions discussed in the clip.; The clip does not treat mixed-sign spectra in detail before ending.

For an $n$-dimensional system, the lecture states there are $n$ Lyapunov exponents $\lambda_i$, matching the number of state-space dimensions.

Conditions: Stated for an $n$-dimensional dynamical system.; The clip does not discuss degeneracies or zero exponents separately.

Lyapunov exponents are ordered by value from largest to smallest, not by absolute value. Thus $\lambda_1$ is the exponent closest to positive infinity and $\lambda_n$ is the exponent closest to negative infinity.

Conditions: n is the dimension of the dynamical system; Ordering is by real value, not magnitude