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Theorem 2 Assume that yy > dz, then the equilibrium point of Eq. (1) is a global attractor if 8(a – 1) + B. Proof. Assume that p, q E R and let g : [p, q] Eq. (3). Since yy > öz, the function g is increasing in x and z and decreasing in y. Let (ø, v) be a solution for the following system: 3 → p, q] is a function defined by $ = g(4, v, ¢), v = g(4, ¢, Þ). Plugging this into Eq. (1) leads to 0 = g(4, Þ, ø) = ao + b = g(4, 4, 4) = arb + which can be rewritten as ayop – adg? + B¢², ayoıp – ady? + Bb². (8) (9) Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – 6²) = 0. Hence, if d(a – 1) + B, then o = b. Therefore, Theorem B in [21] ensures that the equilibrium point is a global attractor.

Theorem 2 Assume that yy > dz, then the equilibrium point of Eq. (1) is a global attractor if 8(a – 1) + B. Proof. Assume that p, q E R and let g : [p, q] Eq. (3). Since yy > öz, the function g is increasing in x and z and decreasing in y. Let (ø, v) be a solution for the following system: 3 → p, q] is a function defined by $ = g(4, v, ¢), v = g(4, ¢, Þ). Plugging this into Eq. (1) leads to 0 = g(4, Þ, ø) = ao + b = g(4, 4, 4) = arb + which can be rewritten as ayop – adg? + B¢², ayoıp – ady? + Bb². (8) (9) Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – 6²) = 0. Hence, if d(a – 1) + B, then o = b. Therefore, Theorem B in [21] ensures that the equilibrium point is a global attractor.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Bun-1un-5 Un+1 = aun-1+ n = 0, 1, .., (1) yun-3 – Sun-5 Bun-1un-5 Un+1 = QUn-1 – n = 0, 1, ..., (2) Yun-3 + dun-5 Eq. (1). In order to discuss the stability of Eq. (1), we first define the function g : (0, 00)³ - → (0, o0) by Brz g(x, y, z) = = ax + (3) YY – 8z Then, ag(x, y, z) Bz a + YY – 8z' Byrz (4) ag(x, y, z) ду ag(x, y, z) (5) (ry – ôz)2 Byry (6) dz (ry – 8z)2 °
Transcribed Image Text:Bun-1un-5 Un+1 = aun-1+ n = 0, 1, .., (1) yun-3 – Sun-5 Bun-1un-5 Un+1 = QUn-1 – n = 0, 1, ..., (2) Yun-3 + dun-5 Eq. (1). In order to discuss the stability of Eq. (1), we first define the function g : (0, 00)³ - → (0, o0) by Brz g(x, y, z) = = ax + (3) YY – 8z Then, ag(x, y, z) Bz a + YY – 8z' Byrz (4) ag(x, y, z) ду ag(x, y, z) (5) (ry – ôz)2 Byry (6) dz (ry – 8z)2 °
Theorem 2 ASsume that yy > dz, then the equilibrium point of Eq. (1) is a global attractor if 8(a – 1) + B. Proof. 3 Assume that p, q E R and let g : p, q]° p, q] is a function defined by Eq. (3). Since yy > 8z, the function g is increasing in x and z and decreasing in y. Let (ø, v) be a solution for the following system: $ = g(ø, V, ¢), v = g(y, 4, v). Plugging this into Eq. (1) leads to Ø = g($, Þ, ø) = aø + V = g(4, Ø, v) = ar) + which can be rewritten as ayop – adø? + B¢², ayop – adıy? + By?. (9) - Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – 4²) = 0. Hence, if d (a - 1) + B, then $ = p. Therefore, Theorem B in [21] ensures that the equilibrium point is a global attractor.
Transcribed Image Text:Theorem 2 ASsume that yy > dz, then the equilibrium point of Eq. (1) is a global attractor if 8(a – 1) + B. Proof. 3 Assume that p, q E R and let g : p, q]° p, q] is a function defined by Eq. (3). Since yy > 8z, the function g is increasing in x and z and decreasing in y. Let (ø, v) be a solution for the following system: $ = g(ø, V, ¢), v = g(y, 4, v). Plugging this into Eq. (1) leads to Ø = g($, Þ, ø) = aø + V = g(4, Ø, v) = ar) + which can be rewritten as ayop – adø? + B¢², ayop – adıy? + By?. (9) - Subtracting Eq. (9) from Eq. (8) leads to [8(1 – a) + B](4² – 4²) = 0. Hence, if d (a - 1) + B, then $ = p. Therefore, Theorem B in [21] ensures that the equilibrium point is a global attractor.
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