Asymptotic behavior means behavior of solutions as time variable goes to infinity. This tacitly assumes that solutions can be extended to infinity.

Oscillatory behavior is only one of possible types of asymptotic behavior, in this case all (or some) solutions have a sequence of zeroes which is unbounded (approaches infinity).

Dear Professor,

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Article Asymptotic and oscillatory behaviour of solutions of a linea...

Asymptotic behavior means behavior of solutions as time variable goes to infinity. This tacitly assumes that solutions can be extended to infinity.

Oscillatory behavior is only one of possible types of asymptotic behavior, in this case all (or some) solutions have a sequence of zeroes which is unbounded (approaches infinity).

I think it is miss leading to put time into the scenario alone. İn practice asymptotic behaviour implies to investigate whether there is any solution, which at large values of independent variable approaches to a well defined finite value smoothly or oscillatory fashion. Especially for the nonlinear systems the long and/or short range stabilities are connected closely to the initial data, which becomes a major headache especially in the numerical solutions..

For Pfaffian DE problem X. dr =0 the integrability necessary and sufficient condition are well understood, which is described by

X. CurlX=0

Here, the knowledge on the physical model of dynamical system associated with PDE is extremely important in foreseen whether one may have oscillatory solution or not. By just looking at the inertial and dissipative terms. If inertial term dominates one has oscillatory solution approaching to finite limiting value having monotonically decreasing in amplitudes (under and over shootings) otherwise on gets monotonically decreasing or increasing solutions which reach to finite limits smoothly at infinity. The nature of the time dependent force especially its amplitude dictates whether system may go to order to order -disorder transition which ends up with the chaotic regime where the solution can be described by the power spectrum supplemented sharp line spectrum. Or the system may be blows up catastrophically.

Hello Kandhasamy Logaarasi

A solution goes to an asymptotic value if as the independent value tends to infinity the dependent variable or its derivative tends to some constant value. That constant value need not be infinite. Example the function y = 2 + 1/x tends to the asymptotic value of 2 as x tends to infinity. An oscillatory function solution will have values that

oscillate about a certain value as x tends to infinity. E.g. y = 2+sin(3x)n oscillates about y = 2. There are functions that both oscillate and decay to some asymptotic value. E.g. the damped vibration equation u = A*exp(-x)* sin(2x) will oscillate and tend to an asymptotic value of zero. Also the Bessel function Jo (x) does the same thing. As special cases the hyperbola (x/a)2 -(y/b)2 = 1 has a derivative dy/dx that tends to an asymptotic value of ± b/a as x tends to infinity. Because dy/dx = (b/a)*x/(x2 -a2)1/2

As sussested by few experts, asymptotic behaviour stands for the behaviour of function as the magnitude of the independent variable(s) is large enough(close to the boundary). Such behaviour may have resemblance with some known (classical) function. Whenever such resemblance exists, corresponding classical function is designated as the asymptotic behavour.

On the othe hand, if the value of the function changes periodically with respect to the independent vatiable (may not zero) throughout the domain, it is regarded as oscillatory.

Dear Dr. Panja, in the Oscillation Theory, only solutions that have unbounded sequence of zeroes are called oscillating/oscillatory. Periodic solutions that do not have zeroes are called weakly oscillating (but their derivatives oscillate in the sense of the standard definition).

I think Yuriy's following statement as far as the boundedness or unboundness doesn't furnish the full picture . There are many oscillatory functions such as Bessel Functions (Jv(x) , Nv(x) etc. ) for real order with real arguments have real definitions and for x goes to + infinity Jv(x) and Nv(x) go to zero. There are bounded oscillatory real functions. Where Jv(x) is also bounded at zero but not Nv(x).for any real order and real value.

I would recommend the Table of Higher Functions by Jahnke, Emde & Lösch for good graphical representations.

Probably the most important question is whether a given function F(x) to be of exponential order as the independent variable goes to infinity namely F/x) =O(exp(bx) ) as x goes to infinity. That is very important property in order to work in the Laplace or Fourier transform spaces.

For example Jv(a x) has a Laplace transform and integration of which from zero to infinity yield 1/a where Rev should be larger then -1 and a is positive.

Dear Tarik,

if you read my explanation carefully, you’ll see that I refer to a sequence of zeroes of a solution as unbounded (in which case solution is called oscillatory), and not to the boundedness of solution itself. Obviously, oscillatory solutions may be bounded or unbounded. By the way, stability or boundedness are two other aspects of asymptotic behavior of solutions to differential equations worth studying, as well as existence of limit cycles and attractors. The concept of asymptotic behavior of solutions to differential equations is rich and multifaceted, oscillatory behavior is just one possibility among many other!

the same as the difference between an undamped oscillator and a strongly damped oscillator.

I do agree with the statement "The concept of asymptotic behavior of solutions to differential equations is rich and multifaceted, oscullatory behavior is just one possibility among many other". My earlier response has no contradiction with this. But regarding the statement "in the Oscillation Theory, only solutions that have unbounded sequence of zeroes are called oscillating/oscillatory" I have some confusion. What will be the nature of the solution of y''(x )+10000 y(x)=0, x in [0, pi] satisfying the conditions y(0) = 0 = y(pi)? is it oscillatory?

All solutions of this equation have finite number of zeroes on a finite interval. In line with the classical definition, this solution is not deemed oscillatory.

Dear Yuriy, unfurtunately you were not too presice in using the terminology such as 'unbounded' in the context of your first statement. It is of course the number of zeros goes to infinity for only certain oscillatory functions but -as you know- there are some functions such as ' the function of parabolic cylinder ' depends upon the order it may have few zeros then approaches to the zero axis asymtotically by staying on the positive sector. That means it is not trivial to characterize the oscillatory functions' boundness just few words .

Oscillatory and non-oscillation theorem for second order DE is presented in Section of 16.61-16.62 (Page 1134-1141). in Table of Integrals, Series and Products by I.S. Gradshteyn and L. M. RYzhik with extreme care and completeness. (Corrected and Enlarged Edition)

I bought that edition of this book in 1981, immeadely appeared in the circulation while I was at Max Planck.

Best Regards

Note: I am talking about on those functions which are not diverging at the infinity

Note: Prof. Rogovchenko is exactly adapted the definition of oscillatory functions as provided by I.S. Gradshteyn and L. M. RYzhik in their monumental book (in page 1134) as I have cited above. But I am not agree with this definition , which completely overlooks the etymological meaning of OSCILLATE - oscillatus, that means swinging back and forth!! that doesn't mean in crossing towards negative domain.

@Logaarasi

Differential equation can have four type of solutions(1) increasing (2)decreasing (3) constant (4) oscillatory . Actual solution depends on equation used.(4) oscillatory can be increasing (as time goes on increasing the amplitude goes on increasing ) or decreasing ( as time goes on increasing the amplitude of motion goes on decreasing) or stable (amplitude remains constant with time) . Asymptotic can be increasing or decreasing . Asymptotic function does not contain sin or cosine function. If possible read my paper in proc.ind.nat.sci.aca A on nonlinear diff equation and solution

Hope it will be helpful.

B.Rath

@Kandhasamy Logaarasi

Read my answer.

B.Rath

@ Tarık Ömer Oğurtanı

Dear Tarik,

My definition of an oscillatory solution is as precise as it should be, cf. https://www.encyclopediaofmath.org/index.php/Oscillating_solution

Best wishes,

Yuriy

• Th

Rogovchenko's answer is quite clear.

Rogovchenko's answer is clear.