Is a cascaded chaotic system still chaotic?












3












$begingroup$


I am curious whether a new system which cascades two individual chaotic systems is always chaotic.



My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that
$${rm range}( S_1 ) = {rm domain}(S_2)$$
and
$${rm range}( S_2 ) = {rm domain}(S_1)$$
then their composite system $S' = S_1circ S_2$ is also chaotic. I guess should use proof by contradiction. However, I donot know whether this claim is indeed true.



For example, a logistic map $L(cdot)$ is defined as
$$x_{i+1}=L(x)=rx_i(1-x_i)$$
and a tent map $T(cdot)$ is defined as
$$x_{i+1}=T(x)=left{begin{array}{lr}{ux_iover c}&, x_i<c\{u(1-x_i)over (1-c)}&, x_igeq cend{array}right.$$



Assume parameters of both maps are carefully chosen to ensure the map chaotic behaviors (both range and domain are $(0,1)$). Is the following system also chaotic?
$$G(c)=Lcirc T(x) = L(,T(x),)$$



Can anyone provide any idea here?






dynamical-systems chaos-theory







share|cite|improve this question











$endgroup$

















    3












    $begingroup$


    I am curious whether a new system which cascades two individual chaotic systems is always chaotic.



    My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that
    $${rm range}( S_1 ) = {rm domain}(S_2)$$
    and
    $${rm range}( S_2 ) = {rm domain}(S_1)$$
    then their composite system $S' = S_1circ S_2$ is also chaotic. I guess should use proof by contradiction. However, I donot know whether this claim is indeed true.



    For example, a logistic map $L(cdot)$ is defined as
    $$x_{i+1}=L(x)=rx_i(1-x_i)$$
    and a tent map $T(cdot)$ is defined as
    $$x_{i+1}=T(x)=left{begin{array}{lr}{ux_iover c}&, x_i<c\{u(1-x_i)over (1-c)}&, x_igeq cend{array}right.$$



    Assume parameters of both maps are carefully chosen to ensure the map chaotic behaviors (both range and domain are $(0,1)$). Is the following system also chaotic?
    $$G(c)=Lcirc T(x) = L(,T(x),)$$



    Can anyone provide any idea here?






    dynamical-systems chaos-theory







    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$


      I am curious whether a new system which cascades two individual chaotic systems is always chaotic.



      My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that
      $${rm range}( S_1 ) = {rm domain}(S_2)$$
      and
      $${rm range}( S_2 ) = {rm domain}(S_1)$$
      then their composite system $S' = S_1circ S_2$ is also chaotic. I guess should use proof by contradiction. However, I donot know whether this claim is indeed true.



      For example, a logistic map $L(cdot)$ is defined as
      $$x_{i+1}=L(x)=rx_i(1-x_i)$$
      and a tent map $T(cdot)$ is defined as
      $$x_{i+1}=T(x)=left{begin{array}{lr}{ux_iover c}&, x_i<c\{u(1-x_i)over (1-c)}&, x_igeq cend{array}right.$$



      Assume parameters of both maps are carefully chosen to ensure the map chaotic behaviors (both range and domain are $(0,1)$). Is the following system also chaotic?
      $$G(c)=Lcirc T(x) = L(,T(x),)$$



      Can anyone provide any idea here?






      dynamical-systems chaos-theory







      share|cite|improve this question











      $endgroup$




      I am curious whether a new system which cascades two individual chaotic systems is always chaotic.



      My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that
      $${rm range}( S_1 ) = {rm domain}(S_2)$$
      and
      $${rm range}( S_2 ) = {rm domain}(S_1)$$
      then their composite system $S' = S_1circ S_2$ is also chaotic. I guess should use proof by contradiction. However, I donot know whether this claim is indeed true.



      For example, a logistic map $L(cdot)$ is defined as
      $$x_{i+1}=L(x)=rx_i(1-x_i)$$
      and a tent map $T(cdot)$ is defined as
      $$x_{i+1}=T(x)=left{begin{array}{lr}{ux_iover c}&, x_i<c\{u(1-x_i)over (1-c)}&, x_igeq cend{array}right.$$



      Assume parameters of both maps are carefully chosen to ensure the map chaotic behaviors (both range and domain are $(0,1)$). Is the following system also chaotic?
      $$G(c)=Lcirc T(x) = L(,T(x),)$$



      Can anyone provide any idea here?






      dynamical-systems chaos-theory




      dynamical-systems chaos-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 9 at 12:49









      Wrzlprmft

      3,06111233




      3,06111233










      asked Apr 15 '14 at 2:39









      pitfallpitfall

      35626




      35626






















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it).



          Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
            $endgroup$
            – pitfall
            Apr 16 '14 at 21:05



















          0












          $begingroup$

          At least empirically, your conjecture does not hold:



          The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).



          I found these values with the following Python script appended below.
          It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
          The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
          On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.



          Appendix: Python Script





          import numpy as np
          
from sympy import Lambda, lambdify
          
from sympy.abc import x
          

          
def logistic(r):
          
    return Lambda(x,r*x*(1-x))
          

          
def lyapunov(f,nsteps=10000,pre=1000):
          
    Map = lambdify(x,f(x))
          
    Der = lambdify(x,f(x).diff(x))
          

          
    y = np.random.uniform(0.01,0.99)
          
    for _ in range(pre):
          
        y = f(y)
          

          
    diffs = np.empty(nsteps)
          
    for i in range(nsteps):
          
        y = f(y)
          
        diffs[i] = Der(y)
          

          
    return np.mean(np.log(np.abs(diffs)))
          

          
while True:
          
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
          
    map_1 = logistic(r_1)
          
    map_2 = logistic(r_2)
          
    composite = Lambda(x,map_1(map_2(x)))
          

          
    for n in [100,1000,10000]:
          
        if (
          
                   lyapunov(composite,n,n) > -0.2
          
                or lyapunov(map_1,n,n) < 0.2
          
                or lyapunov(map_2,n,n) < 0.2
          
            ):
          
            break
          
    else:
          
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
          
```





          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            2 Answers
            2






            active

            oldest

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            active

            oldest

            votes






            active

            oldest

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            1












            $begingroup$

            Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it).



            Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
              $endgroup$
              – pitfall
              Apr 16 '14 at 21:05
















            1












            $begingroup$

            Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it).



            Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
              $endgroup$
              – pitfall
              Apr 16 '14 at 21:05














            1












            1








            1





            $begingroup$

            Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it).



            Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.






            share|cite|improve this answer









            $endgroup$



            Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it).



            Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Apr 16 '14 at 7:27







            user137035















            • 1




              $begingroup$
              Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
              $endgroup$
              – pitfall
              Apr 16 '14 at 21:05














            • 1




              $begingroup$
              Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
              $endgroup$
              – pitfall
              Apr 16 '14 at 21:05








            1




            1




            $begingroup$
            Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
            $endgroup$
            – pitfall
            Apr 16 '14 at 21:05




            $begingroup$
            Thank you for your answer. I guess the fixed point issue is really not a big problem as long as the two used chaotic system have identical fixed points.
            $endgroup$
            – pitfall
            Apr 16 '14 at 21:05











            0












            $begingroup$

            At least empirically, your conjecture does not hold:



            The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).



            I found these values with the following Python script appended below.
            It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
            The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
            On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.



            Appendix: Python Script





            import numpy as np
            
from sympy import Lambda, lambdify
            
from sympy.abc import x
            

            
def logistic(r):
            
    return Lambda(x,r*x*(1-x))
            

            
def lyapunov(f,nsteps=10000,pre=1000):
            
    Map = lambdify(x,f(x))
            
    Der = lambdify(x,f(x).diff(x))
            

            
    y = np.random.uniform(0.01,0.99)
            
    for _ in range(pre):
            
        y = f(y)
            

            
    diffs = np.empty(nsteps)
            
    for i in range(nsteps):
            
        y = f(y)
            
        diffs[i] = Der(y)
            

            
    return np.mean(np.log(np.abs(diffs)))
            

            
while True:
            
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
            
    map_1 = logistic(r_1)
            
    map_2 = logistic(r_2)
            
    composite = Lambda(x,map_1(map_2(x)))
            

            
    for n in [100,1000,10000]:
            
        if (
            
                   lyapunov(composite,n,n) > -0.2
            
                or lyapunov(map_1,n,n) < 0.2
            
                or lyapunov(map_2,n,n) < 0.2
            
            ):
            
            break
            
    else:
            
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
            
```





            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              At least empirically, your conjecture does not hold:



              The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).



              I found these values with the following Python script appended below.
              It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
              The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
              On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.



              Appendix: Python Script





              import numpy as np
              
from sympy import Lambda, lambdify
              
from sympy.abc import x
              

              
def logistic(r):
              
    return Lambda(x,r*x*(1-x))
              

              
def lyapunov(f,nsteps=10000,pre=1000):
              
    Map = lambdify(x,f(x))
              
    Der = lambdify(x,f(x).diff(x))
              

              
    y = np.random.uniform(0.01,0.99)
              
    for _ in range(pre):
              
        y = f(y)
              

              
    diffs = np.empty(nsteps)
              
    for i in range(nsteps):
              
        y = f(y)
              
        diffs[i] = Der(y)
              

              
    return np.mean(np.log(np.abs(diffs)))
              

              
while True:
              
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
              
    map_1 = logistic(r_1)
              
    map_2 = logistic(r_2)
              
    composite = Lambda(x,map_1(map_2(x)))
              

              
    for n in [100,1000,10000]:
              
        if (
              
                   lyapunov(composite,n,n) > -0.2
              
                or lyapunov(map_1,n,n) < 0.2
              
                or lyapunov(map_2,n,n) < 0.2
              
            ):
              
            break
              
    else:
              
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
              
```





              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                At least empirically, your conjecture does not hold:



                The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).



                I found these values with the following Python script appended below.
                It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
                The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
                On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.



                Appendix: Python Script





                import numpy as np
                
from sympy import Lambda, lambdify
                
from sympy.abc import x
                

                
def logistic(r):
                
    return Lambda(x,r*x*(1-x))
                

                
def lyapunov(f,nsteps=10000,pre=1000):
                
    Map = lambdify(x,f(x))
                
    Der = lambdify(x,f(x).diff(x))
                

                
    y = np.random.uniform(0.01,0.99)
                
    for _ in range(pre):
                
        y = f(y)
                

                
    diffs = np.empty(nsteps)
                
    for i in range(nsteps):
                
        y = f(y)
                
        diffs[i] = Der(y)
                

                
    return np.mean(np.log(np.abs(diffs)))
                

                
while True:
                
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
                
    map_1 = logistic(r_1)
                
    map_2 = logistic(r_2)
                
    composite = Lambda(x,map_1(map_2(x)))
                

                
    for n in [100,1000,10000]:
                
        if (
                
                   lyapunov(composite,n,n) > -0.2
                
                or lyapunov(map_1,n,n) < 0.2
                
                or lyapunov(map_2,n,n) < 0.2
                
            ):
                
            break
                
    else:
                
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
                
```





                share|cite|improve this answer









                $endgroup$



                At least empirically, your conjecture does not hold:



                The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).



                I found these values with the following Python script appended below.
                It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
                The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
                On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.



                Appendix: Python Script





                import numpy as np
                
from sympy import Lambda, lambdify
                
from sympy.abc import x
                

                
def logistic(r):
                
    return Lambda(x,r*x*(1-x))
                

                
def lyapunov(f,nsteps=10000,pre=1000):
                
    Map = lambdify(x,f(x))
                
    Der = lambdify(x,f(x).diff(x))
                

                
    y = np.random.uniform(0.01,0.99)
                
    for _ in range(pre):
                
        y = f(y)
                

                
    diffs = np.empty(nsteps)
                
    for i in range(nsteps):
                
        y = f(y)
                
        diffs[i] = Der(y)
                

                
    return np.mean(np.log(np.abs(diffs)))
                

                
while True:
                
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
                
    map_1 = logistic(r_1)
                
    map_2 = logistic(r_2)
                
    composite = Lambda(x,map_1(map_2(x)))
                

                
    for n in [100,1000,10000]:
                
        if (
                
                   lyapunov(composite,n,n) > -0.2
                
                or lyapunov(map_1,n,n) < 0.2
                
                or lyapunov(map_2,n,n) < 0.2
                
            ):
                
            break
                
    else:
                
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
                
```






                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 9 at 11:12









                WrzlprmftWrzlprmft

                3,06111233




                3,06111233






























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                    MongoDB - Not Authorized To Execute Command

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                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
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                    How to fix TextFormField cause rebuild widget in Flutter

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                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I've got the same issue at #9280. I tried all solutions from that topic with/without GlobalKey for Scaffold and Form but the problem wasn't fixed. I tried to build a authentication form with login and signup. But the keyboard doesn't show when I tap the TextFormField at the first time. Then I try to input some letter from physical keyboard, the widget is rebuilt. Like this: login screen Nothing to be log when I run flutter run --verbose, so I logged manually. Every tapping on field, the widget called dispose -> init -> build. [√] Flutter (Channel beta, v1.0.0, on Microsoft Windows [Version 10.0.17763.195], ...
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                    in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

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                    1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
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