Interpretation:
To find fixed points, draw nullclines,
Concept Introduction:
Fixed point of a differential equation is a point where
Nullclines are the curves where either
Vector fields in this aspect represent the direction of flow and whether flow is going away from fixed point or coming towards it.
Phase portraits represent the trajectories of the system with respect to the parameters and give qualitative idea about evolution of the system, its fixed points, whether they will attract or repel the flow etc.
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
Nonlinear Dynamics and Chaos
- The question is in the screenshotarrow_forwardConsider R = xi +yj+zk and r = a constant vector A = a1 +a2 í + az k: What is (A. V)R in terms of A? What is V?r-1. Note: You could directly write similar derivatives once you evaluate one of them.arrow_forwardFind a vector parametric equation r(t) for the line through the points P = (4, −3, 4) and Q = (3, −6, 2) for each of the given conditions on the parameter t. (a) If 7(0) = (4, −3, 4) and 7(2) = (3, −6, 2), then r(t) = (b) If 7(6) = P and 7(8) = Q, then r(t) = = (c) If the points P and Q correspond to the parameter values t = 0 and t = -4, respectively, then F(t) =arrow_forward
- A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: d→1 for 0.41 m southwest (that is, at 45° from directly south and from directly west), d→2 for 0.52 m due east, and d→3 for 0.77 m at 60° north of east. Let the positive x direction be east and the positive y direction be north. What are (a) the x component and (b) the y component of d→1? What are (c) the x component and (d) the y component of d→2? What are (e) the x component and (f) the y component of d→3? What are (g) the x component and (h) the y component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, (k) how far and (l) in what direction should it move? Give all angles as positive (counterclockwise) angles relative to the +x-axis.arrow_forwardReproduce the given computer-generated direction field. Then sketch an approximate solution curve that passes through each of the indicated points. dy = 1 - xy dx y 4 2 2 (a) y(0) = 0 y y AAAAAAA AAA A A AA AAA A A AA A AAA A A AAA4 AAA AA AA 4 4 AAA AA AAA A 4 AAA A AAAAA A A A A AAAAA AA AAAA A A A 4 2 AAAAAA4 2 AAAAAA4 4 444444 4 444444 44444 14 Xarrow_forwardIf r(t) is the position vector of a particle in the plane at time t, find the indicated vector. Find the acceleration vector. r(t) = (cos 3t)i + (2 sin t)j %3D O a = (-9 cos 3t)i + (-2 sin t)j O a = (-3 cos 3t)i + (2 sin t)j O a = (9 cos 3t)i + (-2 sin t)j O a = (-9 cos 3t)i + (-4 sin t)jarrow_forward
- Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect.arrow_forwarda. Sketch the graph of r(t) = ti+t2j. Show that r(t) is a smooth vector-valued function but the change of parameter t = 73 produces a vector-valued function that is not smooth, yet has the same graph as r(t). b. Examine how the two vector-valued functions are traced, and see if you can explain what causes the problem.arrow_forwardThe acceleration of an object after t seconds is given by the vector-valued function a(t) . If the object's initial position is ( – 9, – 3, 10), and the object's initial velocity is find a vector-valued function 7 (t) representing the object's position at time t.arrow_forward
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,