The first question is Stable and unstable manifolds play no role in the shape of a chaotic attractor.
This is false; the stable and unstable manifolds are actually exactly what cause the shape of a chaotic attractor
Question 2 is that Stable manifolds have the effect of compressing state space.
This is absolutely true
In contrast, the unstable manifolds have the effect of spreading out state space, not compressing it
So question 3 is false
For question 4: in general, an attracting fixed point in an n-dimensional nonlinear dynamical system does live at the intersection of n stable manifolds
So this question is true
As an example not as a proof, merely an example consider the fixed points in the dampened pendulum
The attracting fixed points lie at the intersection of two stable manifolds
And the non-attracting, unstable fixed points lie at the intersection of a stable and an unstable manifold
Again, this is not a proof that this holds in general, merely an example to help you understand the question better
However, this does hold in general
For question 5, were given the following periodic orbit of a nonlinear dynamical system, and asked what we can say about the dimension of that system
We can actually say something
For example, we can say that the state space has to be more than one dimension, because its moving in two directions
We can also say that the state space cannot be in two dimensions, because of this intersection
For the deterministic dynamical systems we are studying, an intersection like this cannot occur
If it could, then at this intersection point in phase space, a trajectory would have two directions to go, which would violate determinism
So we know, because this intersection cannot occur, the state space must have at least three dimensions
It could, however, have many more
We can also not say that the periodic orbit is an attractor
It could be that this is an unstable periodic orbit, for example, that does not attract anything
To know that this is an attractor or not, wed have to find a set of initial conditions that are attracted to this periodic orbit
The amount of information given is not enough to conclude this
The only thing we can conclude is that the state space has at least three dimensions
Question 6 asks if there is at least one stable manifold associated with every point on an attracting periodic orbit in a nonlinear dynamical system
This is definitely true
One way of seeing this is, because its an attracting periodic orbit, then for every point along the periodic orbit there needs to be a direction of shrinking
Said differently, if at some point along the periodic orbit all manifolds were unstable, then at this point the trajectories would be ejected from the periodic orbit, and this would not be an attracting periodic orbit of the dynamical system
For question 7, recall the Lorenz system
This system has three state variables and three parameters, not two, so this question is false
For question 8, the Lorenz system does not model a spring-loaded pendulum; its a truncated version of the Navier-Stokes equation, and models a chunk of fluid heated from below
So this question is false
Question 9 asks if there are two stable fixed points in the dynamics of the Lorenz system for some values of the systems parameters
This question is true
For example, consider this picture, which has r less than approximately 25, a = 16, and b = 4
Question 10 asks if theres a chaotic attractor in the dynamics of the Lorenz system for some values of the systems parameters
This is absolutely true, and why the Lorenz system is so famous
For example, see this attractor, which is a chaotic attractor of the Lorenz equations
The parameters used were rho = 28, sigma = 10, and beta = 8/3
Note that sometimes rho, sigma, and beta are called r, a, and b
This can be somewhat confusing, but its good to start getting in the practice of switching between the names of parameters
Question 11 says that Nonlinear dynamical systems can have lots of attractors in different regions of their state space, but only one type at a time
So for example, all the attractors for a particular parameter set are fixed points, or periodic orbits, or chaotic attractors
This is not the case; this question is false
In fact, nonlinear dynamical systems can have an arbitrary number of all types all over the state space, and theres no way to know a priori how many or what types or where they are
This is one of the things that makes studying nonlinear dynamical systems so interesting and rich
Number 12 states that The basins of attraction of different attractors in a dynamical system can overlap
This is false: if this were the case, then initial conditions lying in the intersection of these basins of attraction could go to two different places
This is not the kind of dynamical system were studying
Question 13 asks if Lorenz was the first person to recognize chaos
While some people argue it was actually Poincare, Lorenz is generally recognized as the first person to recognize chaos, in his seminal paper Deterministic Nonperiodic Flow
So this question is true
Question 14 states that Lorenz was the first person to use the term chaos for this kind of behavior
This is false
Yorke and Li were actually the first people to use the term chaos in their paper Period 3 Implies Chaos