Can you please help with part (b) of the differential equations question attached?
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Transcribed Image Text:Find all of the equilibria for the following difference cquations. Then determine
whether they are (i) locally asymptotically stable or (ii) unstable. Apply Theorems
2.1, 2.3, or 2.4.
(a) X11
ax + x, a + 0
1
(b) x,+1 = x + 3x,
(c) *1 =(x + *)
Transcribed Image Text:Theorem 2.1
Assume f' is continuous on an open interval I containing & and x is a fixed point
of f. Then i is a locally asymptotically stable equilibrium of x, = f(x,) if
IS'(x)| < 1
and unstable if
If'(x)| > 1.
The shorthand notation f'(F) in the theorem means differentiation of f fol-
lowed by cvaluation at , that is,
df(x)
S'(x)
dx
A rigorous proof of Theorem 2.1 is based on the Mean Value Theorem and only
requires that f' and not f" be continuous.
Theorem 2.2
Suppose f" is continuous on an open interval I and the m-cycle,
{X,, f(x1), ....f" '(ã)},
of the difference equation (2.2) is contained in 1. Then the m-cycle is locally
asymptotically stable if
|d[S"(xx)]
< 1
(2.8)
dx
for some k and unstable if
(2.9)
dx
for some k.
The conditions in Theorem 2.2 necd to be verified for only onc of the .
The reason it is necessary to check only one equilibrium value is becausc if
condition (2.8) or condition (2.9) hold for some k, then they hold for all k.
Simplification of the conditions in Theorem 2.2 show that all of the values
X, j = 1,..., k are used to compute (2.8) and (2.9). Consider a 2-cycle. From
the chain rule it follows that
d[f*(x)]
dx
Evaluating at F,, then
= f"(f(x))f"(x) = f'(x2)f"(x)
dx
dx
because for a 2-cycle f()) = 2 and = f(x2). This latter equality illustrates
an alternate method for checking the stability of m-cycles.
Suppose f'(x) - 1, where is an equilibrium point of x,+1 = f(x,) and f" is
continuous on an open interval containing x.
Theorem 2.3
(i) If f"(x) # 0, then is unstable.
(ii) If f"(x) = () and f"(x) > (), then i is unstable.
(iii) If f"(x) = 0 and f" (x) < 0, then is locally asymptotically stable.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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![Theorem 2.1
Assume f' is continuous on an open interval I containing & and x is a fixed point
of f. Then i is a locally asymptotically stable equilibrium of x, = f(x,) if
IS'(x)| < 1
and unstable if
If'(x)| > 1.
The shorthand notation f'(F) in the theorem means differentiation of f fol-
lowed by cvaluation at , that is,
df(x)
S'(x)
dx
A rigorous proof of Theorem 2.1 is based on the Mean Value Theorem and only
requires that f' and not f" be continuous.
Theorem 2.2
Suppose f" is continuous on an open interval I and the m-cycle,
{X,, f(x1), ....f" '(ã)},
of the difference equation (2.2) is contained in 1. Then the m-cycle is locally
asymptotically stable if
|d[S"(xx)]
< 1
(2.8)
dx
for some k and unstable if
(2.9)
dx
for some k.
The conditions in Theorem 2.2 necd to be verified for only onc of the .
The reason it is necessary to check only one equilibrium value is becausc if
condition (2.8) or condition (2.9) hold for some k, then they hold for all k.
Simplification of the conditions in Theorem 2.2 show that all of the values
X, j = 1,..., k are used to compute (2.8) and (2.9). Consider a 2-cycle. From
the chain rule it follows that
d[f*(x)]
dx
Evaluating at F,, then
= f"(f(x))f"(x) = f'(x2)f"(x)
dx
dx
because for a 2-cycle f()) = 2 and = f(x2). This latter equality illustrates
an alternate method for checking the stability of m-cycles.
Suppose f'(x) - 1, where is an equilibrium point of x,+1 = f(x,) and f" is
continuous on an open interval containing x.
Theorem 2.3
(i) If f"(x) # 0, then is unstable.
(ii) If f"(x) = () and f"(x) > (), then i is unstable.
(iii) If f"(x) = 0 and f" (x) < 0, then is locally asymptotically stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15feef40-32fb-4401-88ed-794608a5d767%2F03d93a75-7008-460e-992b-6eea344e691f%2Fheskchi_processed.jpeg&w=3840&q=75)
