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DISORDER & CHAOS TEST 3 questions and answers with solutions 2025
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DISORDER & CHAOS TEST 3 questions and answers with solutions 2025
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DISORDER & CHAOS TEST 3 questionsand answers with solutions 2025
1D Flows - meaning of derivatives - ANSWER 1D flows only have fixed points due to having one state
variable and the non-crossing theorem; dx/dt=0 (rate never changes), dx/dt=c (system is changing at a
constant rate), dx/dt=+-kx (exponential growth and decay)
Be able to identify fixed points and whether they are attracting or repelling on a graph - ANSWER Fixed
points at dx/dt=0, positive derivative = arrow to the right, negative derivative = arrow to the left, if
arrows point together -> attractor, if arrows point away from each other -> repeller
Dissipative, conservative, and driven systems - ANSWER Dissipative: J<1, more energy leaves the system
than goes in over time; Conservative (Hamiltonian): J=1, there is no change in energy, this is an isolated
system; Driven: J>1, the phase space is expanding, more energy is going in than is leaving
Properties of attractors - ANSWER 1. A has zero volume in the embedding phase space 2. A is invariant
under the action of the flow 3. A is contained in a region of phase space known as a Basin of Attraction, B
4. If A is chaotic ( a STRANGE ATTRACTOR) then A displays Sensitivity to Initial Conditions
Linear v. Nonlinear equations - ANSWER Linear: an equation between two variables that is a straight line;
Non-linear: an equation with multiple variables, graph is a curved line
Identifying finite and infinite fixed points - ANSWER Finite fixed points: where dx/dt=0. Infinite fixed
points: when the graph goes to negative or positive infinity
Non-crossing rule - ANSWER a trajectory cannot cross itself without going into a cycle
Analytical v. numerical solutions - ANSWER Analytical solutions - the solution is an equation/function.
Numerical solutions - the solution is a number.
Trajectories v. Orbits - ANSWER Trajectory: specification of a single time evolution where time is
continuous, X(t); Orbits: A specification of a single time evolution where time is discrete, Xn
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