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Stability of a fixed point of a discrete dynamical system
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I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me?
I recall the definition given in page 5:
Definition 1.7 Let $(X,F)$ be a dynamical system.
(2)A fixed point $xin X$ is stable if $$forallvarepsilon>0,existsdelta>0,forall yin B_delta(x),forall ninmathbb Z^+,d(F^n(y),x)<varepsilon.$$
where $B_delta(x)$ is the open ball centered at $x$ with radius $delta$, and $F^n$ is the composition of $F$ with itself $n$ times.
The exercise is about determining the fixed points and their stabilities of the following dynamical system: $(I,F_a)$ where $I=[0,1]$, $a>0$ and
beginalign*
F:I&to I\
x&mapsto x+dfracxa+1sin(aln x).
endalign*
The set of fixed points of $F_a$ is $$leftexpleft(kfracpiaright)mid kinmathbbZ^-cup0rightcup0.$$ Let $p_k:=expleft(kfracpiaright).$ We have $F_a'(p_k)=1+(-1)^kdfracaa+1$ so $p_k$ is stable iff $k$ is odd.
I'm stuck with the stability of $0$. $F_a$ is not differentiable at $0$ and $F'_a$ takes values larger than $1$ in any neighborhood of $0$. Numerically, it seems that $0$ is stable, as when I take $x$ close to $0$, $F_a^n(x)$ seems to converge to a fixed point close to it.
So here's my attempt:
0 is the limit point of the other fixed points $(p_k)_kinmathbbZ^-$, with $p_2k$ not stable and $p_2k+1$ stable.
If we pick $varepsilon>0$, $delta>0$ small enough and $xin (0,delta)$, as $F$ is bounded, so is the sequence $(F_a^n(x))_ninmathbb N$. $I$ being compact, $F_a^n(x)$ would have a converging subsequence, which converges to a stable fixed point of $F_a$. But because this point is stable, for big enough $n$, all the $F_a^n(x)$ would stay close to that fixed point.
The problem is, I can't really tell a thing about the first values of $F_a^n(x)$, nor if that stable fixed point the sequence is gets close to will be in $(0,varepsilon)$. I attempted to solve this problem as follows: we have the inequality $$|F_a(x)-x|ledfracxa+1.$$
So, if $0<x<delta$, we get
$$|F_a^n(x)-x|ledfracn(n+1)delta2(a+1).$$
Could you please help me?
general-topology dynamical-systems fixed-point-theorems stability-theory fixedpoints
asked Jul 15 at 14:14
Scientifica
4,60921230
add a comment |Â
up vote
2
down vote
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I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me?
I recall the definition given in page 5:
Definition 1.7 Let $(X,F)$ be a dynamical system.
(2)A fixed point $xin X$ is stable if $$forallvarepsilon>0,existsdelta>0,forall yin B_delta(x),forall ninmathbb Z^+,d(F^n(y),x)<varepsilon.$$
where $B_delta(x)$ is the open ball centered at $x$ with radius $delta$, and $F^n$ is the composition of $F$ with itself $n$ times.
The exercise is about determining the fixed points and their stabilities of the following dynamical system: $(I,F_a)$ where $I=[0,1]$, $a>0$ and
beginalign*
F:I&to I\
x&mapsto x+dfracxa+1sin(aln x).
endalign*
The set of fixed points of $F_a$ is $$leftexpleft(kfracpiaright)mid kinmathbbZ^-cup0rightcup0.$$ Let $p_k:=expleft(kfracpiaright).$ We have $F_a'(p_k)=1+(-1)^kdfracaa+1$ so $p_k$ is stable iff $k$ is odd.
I'm stuck with the stability of $0$. $F_a$ is not differentiable at $0$ and $F'_a$ takes values larger than $1$ in any neighborhood of $0$. Numerically, it seems that $0$ is stable, as when I take $x$ close to $0$, $F_a^n(x)$ seems to converge to a fixed point close to it.
So here's my attempt:
0 is the limit point of the other fixed points $(p_k)_kinmathbbZ^-$, with $p_2k$ not stable and $p_2k+1$ stable.
If we pick $varepsilon>0$, $delta>0$ small enough and $xin (0,delta)$, as $F$ is bounded, so is the sequence $(F_a^n(x))_ninmathbb N$. $I$ being compact, $F_a^n(x)$ would have a converging subsequence, which converges to a stable fixed point of $F_a$. But because this point is stable, for big enough $n$, all the $F_a^n(x)$ would stay close to that fixed point.
The problem is, I can't really tell a thing about the first values of $F_a^n(x)$, nor if that stable fixed point the sequence is gets close to will be in $(0,varepsilon)$. I attempted to solve this problem as follows: we have the inequality $$|F_a(x)-x|ledfracxa+1.$$
So, if $0<x<delta$, we get
$$|F_a^n(x)-x|ledfracn(n+1)delta2(a+1).$$
Could you please help me?
general-topology dynamical-systems fixed-point-theorems stability-theory fixedpoints
asked Jul 15 at 14:14
Scientifica
4,60921230
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me?
I recall the definition given in page 5:
Definition 1.7 Let $(X,F)$ be a dynamical system.
(2)A fixed point $xin X$ is stable if $$forallvarepsilon>0,existsdelta>0,forall yin B_delta(x),forall ninmathbb Z^+,d(F^n(y),x)<varepsilon.$$
where $B_delta(x)$ is the open ball centered at $x$ with radius $delta$, and $F^n$ is the composition of $F$ with itself $n$ times.
The exercise is about determining the fixed points and their stabilities of the following dynamical system: $(I,F_a)$ where $I=[0,1]$, $a>0$ and
beginalign*
F:I&to I\
x&mapsto x+dfracxa+1sin(aln x).
endalign*
The set of fixed points of $F_a$ is $$leftexpleft(kfracpiaright)mid kinmathbbZ^-cup0rightcup0.$$ Let $p_k:=expleft(kfracpiaright).$ We have $F_a'(p_k)=1+(-1)^kdfracaa+1$ so $p_k$ is stable iff $k$ is odd.
I'm stuck with the stability of $0$. $F_a$ is not differentiable at $0$ and $F'_a$ takes values larger than $1$ in any neighborhood of $0$. Numerically, it seems that $0$ is stable, as when I take $x$ close to $0$, $F_a^n(x)$ seems to converge to a fixed point close to it.
So here's my attempt:
0 is the limit point of the other fixed points $(p_k)_kinmathbbZ^-$, with $p_2k$ not stable and $p_2k+1$ stable.
If we pick $varepsilon>0$, $delta>0$ small enough and $xin (0,delta)$, as $F$ is bounded, so is the sequence $(F_a^n(x))_ninmathbb N$. $I$ being compact, $F_a^n(x)$ would have a converging subsequence, which converges to a stable fixed point of $F_a$. But because this point is stable, for big enough $n$, all the $F_a^n(x)$ would stay close to that fixed point.
The problem is, I can't really tell a thing about the first values of $F_a^n(x)$, nor if that stable fixed point the sequence is gets close to will be in $(0,varepsilon)$. I attempted to solve this problem as follows: we have the inequality $$|F_a(x)-x|ledfracxa+1.$$
So, if $0<x<delta$, we get
$$|F_a^n(x)-x|ledfracn(n+1)delta2(a+1).$$
Could you please help me?
general-topology dynamical-systems fixed-point-theorems stability-theory fixedpoints
asked Jul 15 at 14:14
Scientifica
4,60921230
I'm stuck with studying the stability of one fixed point of a discrete dynamical system given in exercise (3) page 44 of Petr Kůrka's Topological and Symbolic Dynamics. Could you please help me?
I recall the definition given in page 5:
Definition 1.7 Let $(X,F)$ be a dynamical system.
(2)A fixed point $xin X$ is stable if $$forallvarepsilon>0,existsdelta>0,forall yin B_delta(x),forall ninmathbb Z^+,d(F^n(y),x)<varepsilon.$$
where $B_delta(x)$ is the open ball centered at $x$ with radius $delta$, and $F^n$ is the composition of $F$ with itself $n$ times.
The exercise is about determining the fixed points and their stabilities of the following dynamical system: $(I,F_a)$ where $I=[0,1]$, $a>0$ and
beginalign*
F:I&to I\
x&mapsto x+dfracxa+1sin(aln x).
endalign*
The set of fixed points of $F_a$ is $$leftexpleft(kfracpiaright)mid kinmathbbZ^-cup0rightcup0.$$ Let $p_k:=expleft(kfracpiaright).$ We have $F_a'(p_k)=1+(-1)^kdfracaa+1$ so $p_k$ is stable iff $k$ is odd.
I'm stuck with the stability of $0$. $F_a$ is not differentiable at $0$ and $F'_a$ takes values larger than $1$ in any neighborhood of $0$. Numerically, it seems that $0$ is stable, as when I take $x$ close to $0$, $F_a^n(x)$ seems to converge to a fixed point close to it.
So here's my attempt:
0 is the limit point of the other fixed points $(p_k)_kinmathbbZ^-$, with $p_2k$ not stable and $p_2k+1$ stable.
If we pick $varepsilon>0$, $delta>0$ small enough and $xin (0,delta)$, as $F$ is bounded, so is the sequence $(F_a^n(x))_ninmathbb N$. $I$ being compact, $F_a^n(x)$ would have a converging subsequence, which converges to a stable fixed point of $F_a$. But because this point is stable, for big enough $n$, all the $F_a^n(x)$ would stay close to that fixed point.
The problem is, I can't really tell a thing about the first values of $F_a^n(x)$, nor if that stable fixed point the sequence is gets close to will be in $(0,varepsilon)$. I attempted to solve this problem as follows: we have the inequality $$|F_a(x)-x|ledfracxa+1.$$
So, if $0<x<delta$, we get
$$|F_a^n(x)-x|ledfracn(n+1)delta2(a+1).$$
Could you please help me?
general-topology dynamical-systems fixed-point-theorems stability-theory fixedpoints
asked Jul 15 at 14:14
Scientifica
4,60921230
asked Jul 15 at 14:14
Scientifica
4,60921230
asked Jul 15 at 14:14
Scientifica
4,60921230
asked Jul 15 at 14:14
Scientifica
4,60921230
4,60921230
add a comment |Â
add a comment |Â
1 Answer
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1
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Fix $varepsilon > 0$. We can take a negative integer $k$ such that
$$
x_0 := expleft(kfracpiaright)
$$
is closer to $0$ than $varepsilon$.
We need to show now that the derivative of the mapping on $(0,1]$ is positive. Indeed,
$$
F_a'(x) = 1 + frac1a+1 sin(a lnx) + fracaa+1cos(a lnx),
$$
$sin(a lnx) ge -1 $, $cos(a lnx) ge -1 $, and they cannot be simultaneously equal to $-1$.
Consequently, $F_a$ preserves orientation. Therefore, for any $x in (0,x_0)$ its iterates $F_a^n(x)$ are contained between the iterates of $0$ and the iterates of $F_a^n(x_0)$, that is, they belong to $(0, x_0)$.
So, a $delta > 0$ required in the definition is just the distance between $0$ and $x_0$.
answered Jul 15 at 21:12
user539887
1,4841313
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Fix $varepsilon > 0$. We can take a negative integer $k$ such that
$$
x_0 := expleft(kfracpiaright)
$$
is closer to $0$ than $varepsilon$.
We need to show now that the derivative of the mapping on $(0,1]$ is positive. Indeed,
$$
F_a'(x) = 1 + frac1a+1 sin(a lnx) + fracaa+1cos(a lnx),
$$
$sin(a lnx) ge -1 $, $cos(a lnx) ge -1 $, and they cannot be simultaneously equal to $-1$.
Consequently, $F_a$ preserves orientation. Therefore, for any $x in (0,x_0)$ its iterates $F_a^n(x)$ are contained between the iterates of $0$ and the iterates of $F_a^n(x_0)$, that is, they belong to $(0, x_0)$.
So, a $delta > 0$ required in the definition is just the distance between $0$ and $x_0$.
answered Jul 15 at 21:12
user539887
1,4841313
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
add a comment |Â
up vote
1
down vote
accepted
Fix $varepsilon > 0$. We can take a negative integer $k$ such that
$$
x_0 := expleft(kfracpiaright)
$$
is closer to $0$ than $varepsilon$.
We need to show now that the derivative of the mapping on $(0,1]$ is positive. Indeed,
$$
F_a'(x) = 1 + frac1a+1 sin(a lnx) + fracaa+1cos(a lnx),
$$
$sin(a lnx) ge -1 $, $cos(a lnx) ge -1 $, and they cannot be simultaneously equal to $-1$.
Consequently, $F_a$ preserves orientation. Therefore, for any $x in (0,x_0)$ its iterates $F_a^n(x)$ are contained between the iterates of $0$ and the iterates of $F_a^n(x_0)$, that is, they belong to $(0, x_0)$.
So, a $delta > 0$ required in the definition is just the distance between $0$ and $x_0$.
answered Jul 15 at 21:12
user539887
1,4841313
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Fix $varepsilon > 0$. We can take a negative integer $k$ such that
$$
x_0 := expleft(kfracpiaright)
$$
is closer to $0$ than $varepsilon$.
We need to show now that the derivative of the mapping on $(0,1]$ is positive. Indeed,
$$
F_a'(x) = 1 + frac1a+1 sin(a lnx) + fracaa+1cos(a lnx),
$$
$sin(a lnx) ge -1 $, $cos(a lnx) ge -1 $, and they cannot be simultaneously equal to $-1$.
Consequently, $F_a$ preserves orientation. Therefore, for any $x in (0,x_0)$ its iterates $F_a^n(x)$ are contained between the iterates of $0$ and the iterates of $F_a^n(x_0)$, that is, they belong to $(0, x_0)$.
So, a $delta > 0$ required in the definition is just the distance between $0$ and $x_0$.
answered Jul 15 at 21:12
user539887
1,4841313
Fix $varepsilon > 0$. We can take a negative integer $k$ such that
$$
x_0 := expleft(kfracpiaright)
$$
is closer to $0$ than $varepsilon$.
We need to show now that the derivative of the mapping on $(0,1]$ is positive. Indeed,
$$
F_a'(x) = 1 + frac1a+1 sin(a lnx) + fracaa+1cos(a lnx),
$$
$sin(a lnx) ge -1 $, $cos(a lnx) ge -1 $, and they cannot be simultaneously equal to $-1$.
Consequently, $F_a$ preserves orientation. Therefore, for any $x in (0,x_0)$ its iterates $F_a^n(x)$ are contained between the iterates of $0$ and the iterates of $F_a^n(x_0)$, that is, they belong to $(0, x_0)$.
So, a $delta > 0$ required in the definition is just the distance between $0$ and $x_0$.
answered Jul 15 at 21:12
user539887
1,4841313
answered Jul 15 at 21:12
user539887
1,4841313
answered Jul 15 at 21:12
user539887
1,4841313
answered Jul 15 at 21:12
user539887
1,4841313
1,4841313
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
add a comment |Â
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
Thank you very much for your answer!
– Scientifica
Jul 17 at 10:55
add a comment |Â
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