time-shift in partial differential equation of density function
$begingroup$
I am reading the following paper:
http://www.nacad.ufrj.br/~amit/lyapdual.pdf (see Appendix A. Lemma A.1. proof)
My understanding of a few notations are:
$rho_t(z)$: the density function around initial point $z$ at time $t$
$phi_t(z)$: the solution $x(t)$ of $dot{x}(t)=f(x(t))$ with $x(0)=z$. This is also called phase flow.
Note that $$rho_t(z)=rho(phi_t(z))|(partialphi(t)/partial z)(z)|$$
this is coming from the evolution of density with time. The term $|(partialphi(t)/partial z)(z)|$ is actually the Jacobian. Also note that $$dot{phi}_t(z) = f(z) = f$$
My question is
- How can we change $t=tau$ to $h=0$.
- Why did the author add $rho_h$, the $h$ parameter?
derivatives partial-derivative dynamical-systems
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
$endgroup$
add a comment |
$begingroup$
I am reading the following paper:
http://www.nacad.ufrj.br/~amit/lyapdual.pdf (see Appendix A. Lemma A.1. proof)
My understanding of a few notations are:
$rho_t(z)$: the density function around initial point $z$ at time $t$
$phi_t(z)$: the solution $x(t)$ of $dot{x}(t)=f(x(t))$ with $x(0)=z$. This is also called phase flow.
Note that $$rho_t(z)=rho(phi_t(z))|(partialphi(t)/partial z)(z)|$$
this is coming from the evolution of density with time. The term $|(partialphi(t)/partial z)(z)|$ is actually the Jacobian. Also note that $$dot{phi}_t(z) = f(z) = f$$
My question is
- How can we change $t=tau$ to $h=0$.
- Why did the author add $rho_h$, the $h$ parameter?
derivatives partial-derivative dynamical-systems
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
$endgroup$
add a comment |
$begingroup$
I am reading the following paper:
http://www.nacad.ufrj.br/~amit/lyapdual.pdf (see Appendix A. Lemma A.1. proof)
My understanding of a few notations are:
$rho_t(z)$: the density function around initial point $z$ at time $t$
$phi_t(z)$: the solution $x(t)$ of $dot{x}(t)=f(x(t))$ with $x(0)=z$. This is also called phase flow.
Note that $$rho_t(z)=rho(phi_t(z))|(partialphi(t)/partial z)(z)|$$
this is coming from the evolution of density with time. The term $|(partialphi(t)/partial z)(z)|$ is actually the Jacobian. Also note that $$dot{phi}_t(z) = f(z) = f$$
My question is
- How can we change $t=tau$ to $h=0$.
- Why did the author add $rho_h$, the $h$ parameter?
derivatives partial-derivative dynamical-systems
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
$endgroup$
I am reading the following paper:
http://www.nacad.ufrj.br/~amit/lyapdual.pdf (see Appendix A. Lemma A.1. proof)
My understanding of a few notations are:
$rho_t(z)$: the density function around initial point $z$ at time $t$
$phi_t(z)$: the solution $x(t)$ of $dot{x}(t)=f(x(t))$ with $x(0)=z$. This is also called phase flow.
Note that $$rho_t(z)=rho(phi_t(z))|(partialphi(t)/partial z)(z)|$$
this is coming from the evolution of density with time. The term $|(partialphi(t)/partial z)(z)|$ is actually the Jacobian. Also note that $$dot{phi}_t(z) = f(z) = f$$
My question is
- How can we change $t=tau$ to $h=0$.
- Why did the author add $rho_h$, the $h$ parameter?
derivatives partial-derivative dynamical-systems
derivatives partial-derivative dynamical-systems
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
edited Dec 30 '18 at 4:45
sleeve chen
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
asked Dec 30 '18 at 0:12
sleeve chensleeve chen
3,14041853
3,14041853
add a comment |
add a comment |
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