Dynamic Systems and Linear Behavior

The equation x(t) = x̄ + δ + & represents the system's response to both δ and & influences.

What is the expression that represents the superposition of disturbances δ and & on the system's response?

What is the fixed point of the system when it is subject to both δ and & influences?

Superposition applies to the ______ to disturbances, not the fixed point.

Match each term with its corresponding definition:

Why does the system converge to the fixed point x̄ despite the combined influence of δ and &?

A fixed percentage growth rate, even if small, is still considered exponential over time.

The variable z is introduced to simplify the analysis by shifting the fixed point to the origin.

The book mentions that we hear regularly about references to modest economic growth or monetary inflation of ______ or ______%.

What is the condition required for the dynamics of z to be expressed as z(t) = VΛ V −1 z(0)?

Which of the following references is NOT cited in the provided text?

What is the main difficulty people seem to have in understanding exponential functions?

Match the following books with their authors:

In the context of linear dynamic systems, the equation ż(t) = A(t)z(t) + Bc (t)u(t) + Bo (t)w(t) represents the system's state change over time, where z(t) is the state vector, u(t) represents the ______ input, and w(t) represents the disturbance input.

Which of the following is NOT a reason why most socio-environmental systems are challenging to control?

A system with long time constants is generally easier to control than a system with short time constants.

What does the term "observability" refer to in the context of system control?

Match the following control challenges with their corresponding explanations:

Which of the following scenarios is an example of a system with bounded inputs?

The state vector z(t) in the system equation ż(t) = A(t)z(t) + Bc (t)u(t) + Bo (t)w(t) represents a single value that describes the system's state.

Give an example of a socio-environmental system that has long time constants.

The state vector, z(t), is defined as a column vector with the position, x(t), in the ______ row and the velocity, v(t), in the ______ row.

One of the ecological disturbances mentioned is ______.

What is the primary focus of resilience in the context of the provided text?

The slowest mode (least negative eigenvalue) determines how quickly a linear system recovers from a disturbance.

What is the mathematical formula used to calculate Resilience(A)?

Match the following terms with their corresponding definitions:

What does a reactivity greater than zero imply about a system?

Which of the following is NOT mentioned as a social disturbance?

What is the primary factor affecting the stability of the system?

The given system is described as stable and normal.

What type of force is acting on the point mass described in the system?

The system's velocity is subject to ______, which pulls it to zero.

Match the eigenvalue with its characteristics:

What is the expected behavior of a non-normal system with negative eigenvalues?

The eigenvectors for θ = π/4 are classified as non-normal.

What is the role of the return force in the system dynamics?

For a real symmetric matrix, what can be concluded about its eigenvalues?

Real-symmetric dynamics can give rise to oscillating or spiral dynamics.

What property do eigenvectors of a real symmetric matrix have?

The matrix B is defined as B = (A + A^T)/______, where A is a random matrix.

When generating a random matrix A, how does it usually compare to its symmetric counterpart B in terms of diagonalizability?

Match the following properties to their corresponding matrix types:

Non-diagonalizability in matrices is a common property.

What does the notation & A represent in the context of matrix A?