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Reading Question 11. 1 What is the state of a dynamical system?
TipAnswer
A dynamical system is, naturally, a system. Like all systems, it comprises inter-related quantities. The state of a dynamical system is the current value of the quantities comprised by the system. (In Section 5.1 the state quantities are those named in the boxes that are connected by arrows.) The word “dynamic” reflects that the state can change over time.
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Reading Question 11. 2 What is the input to the dynamical rule?
TipAnswer
The input to the dynamical rule is the state of the system, that is, the set of quantities that the state comprises.
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Reading Question 11. 3 What does the output from the dynamical rule tell about the state? What does the word “instantaneous” have to do with it?
TipAnswer
The output from the dynamical rule is the instantaneous rate of change for each of the quantities in the state. Thus, the dynamical rule tells how the state is changing over time. The word “instantaneous” signifies that the rate of change applies only for the instant at which the system state is at the corresponding input to the dynamical rule. In the brief time interval after that instant, the state changes slightly as described by the rate of change.
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Reading Question 11. 4 Explain the relationship between the state space and the trajectory?
TipAnswer
As described in Section 2.7, a space consists of the possible values for a set of quantities. Each point in the space corresponds to a specific value for each quantity in the set. The cartesian coordinate plane is an example of a space appropriate for a set of two quantities. Each point in the space corresponds to a particular value for each of those quantities.
The state of a dynamical system is the values of the set of quantities that make up the state. The state space contains all possible values for the state quantities. Each point in the state space is a particular value for the state. If the state consists of two quantities, the state space will be a cartesian coordinate plane. Any given state will correspond to a single point in the state space.
The trajectory of the dynamical system is a path through state space. The path consists of the sequence of instantaneous states that the system occupies as time goes by. The dynamical rule tells the instantaneous direction of motion from one point on the path to the next.
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Reading Question 11. 5 The figure shows three aspects of a dynamical system. Match each to the appropriate color.
the flow field dsp-a
a trajectory dsp-b
a time series dsp-c
Figure RQ11- 1: Aspects of a dynamical system
Figure RQ11- 2: Aspects of a dynamical system
Explain what you see in the figure that tells you that the flow field, trajectory, and time series are consistent with one another.
