Well the obvious issues are that the argument of the square root for the initial condition should be real, otherwise the equation describes a complex function; furthermore the argument of the square root should be non-zero, in order that the initial condition be finite and m and b5 should be of same sign, otherwise the denominator of the RHS can vanish. Beyond that, if a periodic solution is sought, it's sufficient to set u(t)=Aexp(i*ωt) and obtain an equation for the frequency ω. The relations between the coefficients that ensure that the equation for ω has roots that are real are the conditions that ensure that a periodic solution exists.

Incidentally, since this is a second order equation, it's useful to define v = du/dt and study the system of the functions (u(t),v(t)).

for best determining the BC you should look at the identity of the Governing Dif. Eq. (if it is so)

for further details u can email me. also, i recommend you to study "Global Error Minimization method for solving strongly nonlinear

oscillator differential equations" by dr yadolla farzaneh.

good luck

I tried your problem with Mathematica with random parameters from the interval {-1,1}. I used DSolve function.  Indeed if the first initial condition is imaginary the solution is a complex function. Sometimes it produces a periodic function which decay to a certain value. Sometimes it is a function which goes to infinity. Sometimes Mathematica reports "singularity or stiff system suspected" and stops in certain point. See attached Mathematica notebook.

In addition to mr Ryszard Walentynski's answer, I must say there are terms called " secular terms" when we are dealing with nonlinear dif. eq.s which we should consider them 0 since they're causing the solution to tend to infinity.

I suggest these books : nonlinear oscillation by Alihasan Nayfeh

Solution with small coeff. b3, b4, b6, b5 can be approximated with linear ODE

uddot=-( b1*u+b2*udot) /m, the solution of which can be analyzed analytically by the corresponding characteristic algebraic equation. Even in this linear case (depending on b1, b2, and m) the solution can be periodic /with or without damping/, or non-periodic /in a case of the critical damping/.

So I would recommend to start with a periodic choice a-priori for the linear ODE. The periodic solution (for the simplified linear ODE) can be futher damped with non-linear terms b3*u^2*udot+b4*udot^3, the term b6*u^3 gives the behaviour of the system /softer or stiffer/ compared to the linear ODE.

The periodicity of the linear ODE does not depend on initial conditions, but only on b1, b2, and m.

A lot of parameters but a straightforward phase portrait problem.

Introduce v = \dot u.

Divide two equations. Obtain du/dv.

Find the values of parameters when the solution always returns from all infinities. It is caged now. Should be periodic.

If you cannot ensure that then you should find the regions defined by du/dv = 0 and du/dv = \infty and check to which of them your initial condition belongs.

That's fun.

In the region of small u and \dot u this is just a dumped oscillator provided the signs of the coefficients are right. The desired periodic solutions are not possible. The attractor is a converging focus. The non-linear terms must stand for pumping energy into it. So, the initial conditions must be in the range of displacement where the energy supply exceeds damping. 

Cheers

Igor> A lot of parameters

On first count there are 7 continuous parameters. It is going to take a very long time  to scan through a 7-dimensional parameters space of phase portraits. Actually, by scaling the t and u variables, and taking into account that the right hand side only depends on ratio of parameters you may bring the parameter space down to 4 continuous dimensions, multiplied a set of discrete choices.

I tried another approach, based on the requirement that the solution should be periodic. This often requires the system to have a constant of motion. A weaker condition would be that the solution is only asymptotically periodic, which may only require the existence of a periodic attractor. Se the attached file for more.

Note: I have corrected some errors in the pdf-file.

Check the existence of a periodic solution of second-order system by constructing the Lyapunov function, wherein if the equilibrium positions have a positive real part then the limit mode of the system is a cycle.

My advices are the following. 1. If the nonlinear terms are smaller that the linear ones (pliase, compare values of all coefficients and possible amplitudes of variables and velocities!) you can introduce the small parameter before the nonlinear terms; then you can construct the possible periodic solution of the limit sicle type by some variant of the small parameter method. But I can see here the dissipation term. If the coefficient is positive, I have some doubt in existing of the periodic solution. I recommend to use in this case the multiple scale method introducing the small parameter before the dissipation term too.    2. It seems to me that your solution must be real; the initial condition shows that some coefficients are negative. In this case you must analyze all equilibrium positions. You will obtain either the single trivial equilibtium, or few of them. In the last case some of these equilibria will be unstable, and you can obtain some separatrix type trajectories. If the dissipation is absent, these trajectories separate regions with different behavior of the system solutions.   

Hi there,

In addition, I also suggest to take a look at the Bendixson's criterion and Poincaré-Bendixson theorem for the existence of periodic solutions in 2D flows. You should write your differential equation in the format:

\dot u= v

\dot v= f(u, v)

It might be possible then to decide from your choice of parameters if there is a periodic solution.  You need to find an annular region in which the flow is outward everywhere in the inner boundary of the annular region and inward in the outer boundary of the annular region. You might be able to construct this region. There are examples of this for the van der Pol oscillator on how to do this that might help you.

Just as a starter see what happens at the lines u=0 and v=0 for ex.

I also suggest you plot the 2d vector field corresponding to this flow and observe if you have then a limit cycle attractor based on what I just commented.

Suggested books:

Strogatz, "Nonlinear Dynamics and Chaos"

Verhulst, "Nonlinear Differential Equations and Dynamical Systems"

Guckenheimer and Holmes book.

Best,

Adriano.

It strange because your equation is second order, so either goes to fixed point or limit cycle (excluding unbounded solutions ), but you must wait enough to system reach stable attractor. you probably see transient behavior. About cases which doesn't produce anything, it's probably because of term in denominator goes to zero and problem becomes singular. you can check this case easily.