The Jacobian Matrix of the System $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 8 y$

question 30

Multiple Choice

The Jacobian matrix of the system $x ^ { \prime } = x ^ { 3 } - y ^ { 3 } - 8 , y ^ { \prime } = 8 y$ at the critical point $( 2,0 )$

• A) \(A = \left[ \begin{array} { r r }
• 3 x ^ { 2 } & - 3 y ^ { 2 } \\
• 8 & 0
• \end{array} \right]\)
• B) \(A = \left[ \begin{array} { r r }
• 3 x ^ { 2 } & - 3 y ^ { 2 } \\
• 0 & 8
• \end{array} \right]\)
• C) \(A = \left[ \begin{array} { r r }
• 3 x ^ { 2 } & 3 y ^ { 2 } \\
• 0 & 8
• \end{array} \right]\)
• D) \(A = \left[ \begin{array} { c c }
• 12 & 0 \\
• 0 & 8
• \end{array} \right]\)
• E) \(A = \left[ \begin{array} { c c }
• 12 & 0 \\
• 8 & 0
• \end{array} \right]\)

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Definitions:

Absolute Threshold

The minimum intensity of stimulation required to produce a sensory experience or for a stimulus to be detected 50% of the time.

Just Noticeable Difference

The minimum amount by which stimulus intensity must be changed in order to produce a noticeable variation in sensory experience.

Action Potentials

Quick electrical impulses that travel down the axon of a neuron, triggering the release of neurotransmitters at synapses.

Proximal Stimulation

Proximal stimulation refers to the sensory information received directly from stimuli that are in close proximity to the sensory receptors.