Nonlinear oscillator with velocity dependent frequency












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In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






mp.mathematical-physics differential-equations







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    $begingroup$


    In a physical problem I need to investigate the following nonlinear differential equation
    $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






    mp.mathematical-physics differential-equations







    share|cite|improve this question









    $endgroup$















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      5








      5


      1



      $begingroup$


      In a physical problem I need to investigate the following nonlinear differential equation
      $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






      mp.mathematical-physics differential-equations







      share|cite|improve this question









      $endgroup$




      In a physical problem I need to investigate the following nonlinear differential equation
      $$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$ where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found https://www.sciencedirect.com/science/article/pii/S0022460X00929894






      mp.mathematical-physics differential-equations




      mp.mathematical-physics differential-equations






      share|cite|improve this question













      share|cite|improve this question











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      asked 5 hours ago









      Zurab SilagadzeZurab Silagadze

      10.8k2569




      10.8k2569






















          1 Answer
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          This type of ODE,
          $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
          is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



          A particular case with nice properties is
          $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
          see Design of nonlinear isochronous oscillators.






          share|cite|improve this answer









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            $begingroup$

            This type of ODE,
            $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
            is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



            A particular case with nice properties is
            $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
            see Design of nonlinear isochronous oscillators.






            share|cite|improve this answer









            $endgroup$


















              7












              $begingroup$

              This type of ODE,
              $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
              is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



              A particular case with nice properties is
              $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
              see Design of nonlinear isochronous oscillators.






              share|cite|improve this answer









              $endgroup$
















                7












                7








                7





                $begingroup$

                This type of ODE,
                $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
                is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



                A particular case with nice properties is
                $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
                see Design of nonlinear isochronous oscillators.






                share|cite|improve this answer









                $endgroup$



                This type of ODE,
                $$ddot{x}+f(x)dot{x}^2+g(x)=0$$
                is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).



                A particular case with nice properties is
                $$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
                see Design of nonlinear isochronous oscillators.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 4 hours ago









                Carlo BeenakkerCarlo Beenakker

                74.9k9170277




                74.9k9170277






























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