fa.functional analysis – Definition of Lyapunov exponents for compact operators

There is the following well-known result from Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exhibitors:

Let $ H $ be a $ mathbb R $-Hilbert space, $ A_n in mathfrak L (H) $ be compact and $ B_n: = A_n cdots A_1 $ for $ n in mathbb N $. Let $ | B_n |: = sqrt {B_n ^ ast B_n} $ and $ sigma_k (B_n) $ denote the $ k $e greatest singular value of $ B_n $ for $ k, n in mathbb N $. Yes $$ limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 tag1 $$ and $$ frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for all} k in mathbb N tag2, $$ then

  1. $$ | B_n | ^ { frac1n} xrightarrow {n to infty} B $$ for some non-negative and self-adhesive compacts $ B in mathfrak L (H) $.
  2. $$ frac { ln sigma_k (B_n)} n xrightarrow {n to infty} Lambda_k: = left. begin {cases} gamma ^ {(k)} – gamma ^ {(k -1)} & text {, if} gamma ^ {(i)}> – infty \ – infty & text {, otherwise} end {cases} right } tag2 $$ for everyone $ k in mathbb N $.

question 1: I have seen this result in many textbooks, but I have wondered why it is stated this way. First of all, isn't it $ (2) $ clearly equivalent to $$ frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3 $$ for some people $ lambda_i $ for everyone $ k in mathbb N $ which in turn is equivalent to $$ sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4 $$ for some people $ mu_i ge0 $ for everyone $ k in mathbb N $? $ (4) $ seems to be much more intuitive than $ (3) $, since no $ lambda_i $, But $ mu_i = e ^ { lambda_i} $ are precisely Lyapunov's exponents of the limit operator $ B $. Am I missing something? The definition of $ Lambda_i $ (which is equal to $ lambda_i $) seems odd to me.

question 2: What is the interpretation of $ B $? Usually I look at a discrete dynamic system $ x_n = B_nx_0 $. What is $ B $ (or $ Bx $) tells us about the asymptotic behavior / evolution of orbits?

fa.functional analysis – Lyapunov indices of an operator product

The deterministic part of the proof of the multiplicative ergodic theorem can be proved using proposition 1.3 in the Lyapunov paper indices of a product of random matrices.$ ^ 1 $

They consider a sequence of matrices $ A_n in mathbb R ^ {d times d} $. I would like to generalize this result to $ A_n in mathfrak L (H) $ for some people $ mathbb R $-Hilbert space $ H $. In section 7, the authors provide some guidance on how such a generalization might be presented, but I have trouble following their arguments (and their notation).

I don't need complete generality. I'm ready to assume that $ A_n $ is compact. Note that if $ A in mathfrak L (H) $, then $ A ^ ast A $ is non-negative and self-supporting and therefore $$ | A |: = sqrt {A ^ ast A} $$ is well defined. So, with the compactness assumption, we know that $ | A_n | $ has a nice spectral decomposition. This could simplify the way we have to state and prove the generalization of Proposition 1.3.


$ ^ 1 $
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Lyapunov functions for ordinary differential equations

I am trying to solve exercise 3.4 (among others) of "ordinary differential equations: qualitative theory" by Barreira e Vall. My goal is to cover my handicap on this theory in order to follow my course in applied mathematics. But I have a hard time guessing the functions of Lyapunov (and noting the phases of the portrait).
For example, is there a Lyapunov function for the zero solution of the system below?
$$ left { begin {matrix} x & # 39; & = & -x + x ^ 2 + y ^ 2 \ y & # 39; & = & 2x-3y + y ^ 3 end {matrix} right. $$
I would like a hint on how to solve these kinds of problems, so that I can solve the other elements. It would also be great to have the portrait phase of the above equation, as I also have trouble solving these kinds of problems.

Thanks in advance,

actual analysis – Lyapunov function unbound radially

Example

$ V (x) = frac {1} {2} (x_ {1} ^ 2 + x_ {2} ^ 2) $

Is it okay to test whether the function is radial with no limit of this type or if it is a better way?

Check each coordinate to infinity

$ (x_ {1}, 0): lim_ {| x_ {2} | to infty} V (x) = frac {1} {2} (x_ {1} ^ 2) rightarrow infty $

$ (0, x_ {2}): lim_ {| x_ {2} | to infty} V (x) = frac {1} {2} (x_ {2} ^ 2) rightarrow infty $

$ x_ {1} = x_ {2} $

$ (x_ {1}, x_ {1}): lim_ {| x_ {1} | to infty} V (x) = (x_ {1} ^ 2) rightarrow infty $

For it $ V (x) rightarrow infty $ when $ || x || rightarrow infty $

Do I have to specify something more in my calculation?

Find direct Lyapunov

Show that the origin is globally asymptotically stable.

x '= – x + y ^ 2
y = – – y

I can prove that V '(x) must be negative, which I can prove. However, I do not understand how to get V (x). Can any one put me in the right direction on how to calculate V (x) for that.
Thank you in advance.

Differential equations – Is it possible to find the attractor of Rössler using only a set of Lyapunov exponents?

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Real Analysis – How is the Dominated Convergence Theorem Applied in the Proof of the Lyapunov Criterion?

Let $$ Gamma (f, g): = frac12f & g; g; ; ; text {for} f, g in C ^ 1 ( mathbb R), $$ $ mu $ to be a measure of probability on $ ( mathbb R, mathcal B ( mathbb R)) $ with a continuously differentiable and positive density $ varrho $ regarding the Lebesgue measure $ lambda $, $$ mathcal E (f, g): = int Gamma (f, g) : { rmd} mu $$ and $$ Af: = frac12f & # 39; + frac12 ( ln varrho) & # 39; f & # 39; ; ; ; text {for} f in C ^ 2 ( mathbb R). $$ (For the simplicity of the notation, $ Gamma (f): = Gamma (f, g) $ and $ mathcal E (f): = mathcal E (f, f) $.)

Now let $ v in C ^ 2 ( mathbb R) $ with $ v ge1 $. Suppose we know that $$ – mu left ( frac {Av} v | g | ^ 2 right) the math E (g) tag1 ; ; ; text {for all} g in C_c ^ infty ( mathbb R). $$ In addition, suppose there is a $ ( zeta_k) _ {k in mathbb N} subseteq C_c ^ infty ( mathbb R) $ with $$ 0 the zeta_k the zeta_ {k + 1} ; ; ; text {for all} k in mathbb N tag2 $$ and $$ zeta_k xrightarrow {k to infty} 1 ; ; ; mu text {- almost surely} tag3 $$ as good as $$ Gamma ( zeta_k) the frac1k ; ; ; text {for all} k in mathbb N. tag4 $$

Now let $ f in C_c ^ infty ( mathbb R) $ and $ m in mathbb R $. How can we show $ (1) $ for $ g: = f-m $?

The idea is that $ g_k: = zeta_kg $ belongs to $ C_c ^ infty ( mathbb R) $ (contrary to $ g $) and so $$ – mu left ( frac {Av} v | g_k | ^ 2 right) the math E (g_k) tag5 ; ; ; text {for all} g in C_c ^ infty ( mathbb R) ; ; ; text {for all} k in mathbb N. $$ Now, $$ | Gamma (g_k) | le | Gamma (g) | +2 | g | sqrt { Gamma (g)} + g ^ 2 tag6 $$ and $$ | Gamma (g_k) – Gamma (g) | le | zeta_k ^ 2-1 || Gamma (g) | + frac2 { sqrt k} | g_k | sqrt { Gamma (g)} + frac1kg ^ 2 xrightarrow {k to infty} 0 tag7 $$ and so $$ mathcal E (g_k) xrightarrow {k to infty} mathcal E (g) tag8 $$ by the dominated convergence theorem.

The left side of $ (5) $ seems to be more complicated. Clearly, $ frac {Av} v | g_k | ^ 2 xrightarrow {k to infty} frac {Av} v | g | ^ 2 $but $ frac {Av} v | g_k | ^ 2 $ does not seem to be dominated by an integrable function.