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Determining the orbit and omega-limiting set of a first-order autonomous system
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I'm working through Teschl's Ordinary Differential Equations and Dynamical Systems and ran into this problem (Problem 6.7) that I'm having trouble understanding:
Consider a first-order autonomous system in $mathbbR$. Suppose $f(x)$ is differentiable, $f(0)=f(1)=0$, and $f(x)>0$ for $xin (0,1)$. Determine the orbit $gamma(x)$ and $omega_pm$ if $xin [0,1]$.
If I'm understanding orbits/omega-limiting sets correctly, $gamma(x)$ gives the image of the maximal solution $phi_x$. Since the system is autonomous, the interval of validity should be $(-infty, infty)$. So to find $gamma(x)$, we have to say something about $phi_x$, but the only thing we know about $phi_x$ is that $dotphi_x = f(phi_x)$. But while I can say someting about $phi_x$ at its initial condition, I don't know how to determine any properties about the orbit over the entire interval. Any help would be appreciated.
differential-equations dynamical-systems
asked Jul 21 at 17:08
Josh
477
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up vote
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I'm working through Teschl's Ordinary Differential Equations and Dynamical Systems and ran into this problem (Problem 6.7) that I'm having trouble understanding:
Consider a first-order autonomous system in $mathbbR$. Suppose $f(x)$ is differentiable, $f(0)=f(1)=0$, and $f(x)>0$ for $xin (0,1)$. Determine the orbit $gamma(x)$ and $omega_pm$ if $xin [0,1]$.
If I'm understanding orbits/omega-limiting sets correctly, $gamma(x)$ gives the image of the maximal solution $phi_x$. Since the system is autonomous, the interval of validity should be $(-infty, infty)$. So to find $gamma(x)$, we have to say something about $phi_x$, but the only thing we know about $phi_x$ is that $dotphi_x = f(phi_x)$. But while I can say someting about $phi_x$ at its initial condition, I don't know how to determine any properties about the orbit over the entire interval. Any help would be appreciated.
differential-equations dynamical-systems
asked Jul 21 at 17:08
Josh
477
If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm working through Teschl's Ordinary Differential Equations and Dynamical Systems and ran into this problem (Problem 6.7) that I'm having trouble understanding:
Consider a first-order autonomous system in $mathbbR$. Suppose $f(x)$ is differentiable, $f(0)=f(1)=0$, and $f(x)>0$ for $xin (0,1)$. Determine the orbit $gamma(x)$ and $omega_pm$ if $xin [0,1]$.
If I'm understanding orbits/omega-limiting sets correctly, $gamma(x)$ gives the image of the maximal solution $phi_x$. Since the system is autonomous, the interval of validity should be $(-infty, infty)$. So to find $gamma(x)$, we have to say something about $phi_x$, but the only thing we know about $phi_x$ is that $dotphi_x = f(phi_x)$. But while I can say someting about $phi_x$ at its initial condition, I don't know how to determine any properties about the orbit over the entire interval. Any help would be appreciated.
differential-equations dynamical-systems
asked Jul 21 at 17:08
Josh
477
I'm working through Teschl's Ordinary Differential Equations and Dynamical Systems and ran into this problem (Problem 6.7) that I'm having trouble understanding:
Consider a first-order autonomous system in $mathbbR$. Suppose $f(x)$ is differentiable, $f(0)=f(1)=0$, and $f(x)>0$ for $xin (0,1)$. Determine the orbit $gamma(x)$ and $omega_pm$ if $xin [0,1]$.
If I'm understanding orbits/omega-limiting sets correctly, $gamma(x)$ gives the image of the maximal solution $phi_x$. Since the system is autonomous, the interval of validity should be $(-infty, infty)$. So to find $gamma(x)$, we have to say something about $phi_x$, but the only thing we know about $phi_x$ is that $dotphi_x = f(phi_x)$. But while I can say someting about $phi_x$ at its initial condition, I don't know how to determine any properties about the orbit over the entire interval. Any help would be appreciated.
differential-equations dynamical-systems
asked Jul 21 at 17:08
Josh
477
asked Jul 21 at 17:08
Josh
477
asked Jul 21 at 17:08
Josh
477
asked Jul 21 at 17:08
Josh
477
477
If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15
add a comment |Â
If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15
If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15
If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15
add a comment |Â
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If the initial value is between $(0,1)$, can the solution take values not in $(0,1)$? (Hint: uniqueness). Such a solution is bounded, so it is defined on $(-infty,infty)$ (not "since the system is nonautonomous"!). It is (strictly) increasing, so its $omega_pm$-limits sets are just limits as $ttopminfty$. An $omega_pm$-limit set must be invariant. What are invariant singletons?
– user539887
Jul 22 at 7:15