## 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.

$$^ 1$$

## 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.

## 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.

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

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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.