Are there non-chaotic systems which exhibits topological mixing?

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I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties:



  1. Sensitivity to initial conditions/Lyapunov coefficient is positive

  2. Exhibits topological mixing

  3. Dense periodic orbits

It can also be proven that 3+2 implies 1.



We also knew there are systems that satisfy only 1 but are non-chaotic, such as scaling $x_n+1 mapsto kx_n$ where $k neq 0$ because every point diverges to infinty.



We also have systems which satisfy only 3 and thus also non-chaotic



An easy example of a non-chaotic system satisfying 1 and 3 is considering a phase space where there are periodic orbits of different sizes centered round rational points, such that no two periodic orbits intersect each other. Then the property of rationals ensures for arbitrarily small deviations in initial conditions, one ends up in completely different orbits and hence diverging trajectories without a given point ending up in all open sets.



However, is it possible to have topological mixing, such that every open set in the domain can end up intersecting every other open set, but it either does not have dense periodic orbits nor sensitivity to initial conditions. I don't see when one is allowed to have open sets smeared against each other, how can arbitrarily close trajectories not diverge exponentially.




Are there non-chaotic systems satisfy only 2?



Are there non-chaotic systems satisfy 1 and 2?





dynamical-systems chaos-theory




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    I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties:



    1. Sensitivity to initial conditions/Lyapunov coefficient is positive

    2. Exhibits topological mixing

    3. Dense periodic orbits

    It can also be proven that 3+2 implies 1.



    We also knew there are systems that satisfy only 1 but are non-chaotic, such as scaling $x_n+1 mapsto kx_n$ where $k neq 0$ because every point diverges to infinty.



    We also have systems which satisfy only 3 and thus also non-chaotic



    An easy example of a non-chaotic system satisfying 1 and 3 is considering a phase space where there are periodic orbits of different sizes centered round rational points, such that no two periodic orbits intersect each other. Then the property of rationals ensures for arbitrarily small deviations in initial conditions, one ends up in completely different orbits and hence diverging trajectories without a given point ending up in all open sets.



    However, is it possible to have topological mixing, such that every open set in the domain can end up intersecting every other open set, but it either does not have dense periodic orbits nor sensitivity to initial conditions. I don't see when one is allowed to have open sets smeared against each other, how can arbitrarily close trajectories not diverge exponentially.




    Are there non-chaotic systems satisfy only 2?



    Are there non-chaotic systems satisfy 1 and 2?





    dynamical-systems chaos-theory




    share|cite|improve this question























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      I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties:



      1. Sensitivity to initial conditions/Lyapunov coefficient is positive

      2. Exhibits topological mixing

      3. Dense periodic orbits

      It can also be proven that 3+2 implies 1.



      We also knew there are systems that satisfy only 1 but are non-chaotic, such as scaling $x_n+1 mapsto kx_n$ where $k neq 0$ because every point diverges to infinty.



      We also have systems which satisfy only 3 and thus also non-chaotic



      An easy example of a non-chaotic system satisfying 1 and 3 is considering a phase space where there are periodic orbits of different sizes centered round rational points, such that no two periodic orbits intersect each other. Then the property of rationals ensures for arbitrarily small deviations in initial conditions, one ends up in completely different orbits and hence diverging trajectories without a given point ending up in all open sets.



      However, is it possible to have topological mixing, such that every open set in the domain can end up intersecting every other open set, but it either does not have dense periodic orbits nor sensitivity to initial conditions. I don't see when one is allowed to have open sets smeared against each other, how can arbitrarily close trajectories not diverge exponentially.




      Are there non-chaotic systems satisfy only 2?



      Are there non-chaotic systems satisfy 1 and 2?





      dynamical-systems chaos-theory




      share|cite|improve this question













      I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties:



      1. Sensitivity to initial conditions/Lyapunov coefficient is positive

      2. Exhibits topological mixing

      3. Dense periodic orbits

      It can also be proven that 3+2 implies 1.



      We also knew there are systems that satisfy only 1 but are non-chaotic, such as scaling $x_n+1 mapsto kx_n$ where $k neq 0$ because every point diverges to infinty.



      We also have systems which satisfy only 3 and thus also non-chaotic



      An easy example of a non-chaotic system satisfying 1 and 3 is considering a phase space where there are periodic orbits of different sizes centered round rational points, such that no two periodic orbits intersect each other. Then the property of rationals ensures for arbitrarily small deviations in initial conditions, one ends up in completely different orbits and hence diverging trajectories without a given point ending up in all open sets.



      However, is it possible to have topological mixing, such that every open set in the domain can end up intersecting every other open set, but it either does not have dense periodic orbits nor sensitivity to initial conditions. I don't see when one is allowed to have open sets smeared against each other, how can arbitrarily close trajectories not diverge exponentially.




      Are there non-chaotic systems satisfy only 2?



      Are there non-chaotic systems satisfy 1 and 2?






      dynamical-systems chaos-theory





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      share|cite|improve this question




      share|cite|improve this question








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