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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

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!”.

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!”.

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

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

Exponential growth of separation on a bounded attractor is compatible with boundedness because the relevant measurement is local transverse stretching with renormalization along the attractor, not unlimited separation in the ambient space.

Conditions: The trajectory lies on an attractor; Separation is measured transversely with renormalization

Exponential growth of separation on a bounded attractor is compatible with boundedness because the relevant measurement is local transverse stretching with renormalization along the attractor, not unlimited separation in the ambient space.

Conditions: The trajectory lies on an attractor; Separation is measured transversely with renormalization