A STABILITY THEORY FOR PERTURBED DIFFERENTIAL EQUATIONS

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is X′=f(t,X) and the associated perturbed differential equation is Y′=f(t,Y)

the two solutions.Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.KEY WORDS AND PHRASES.Liapunov functions, asymptotic behavior of solutions, asymptotic equivalence.AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.34DI0, 34D20.I. INTRODUCTION.One of the paramount uses of stability theory is to determine which stability properties of a particular syst of differential equations are preserved under small perturbations.This problem has been studie in nnerous ways.The method introduced in the present paper, however, is felt to be an essentially new ap- proach for dealing with this situation.It is similar to some work done by Lak- shmikantham 3 in a different context.Instead of considering explicitly which stability properties are preserved under perturbations, a theory based on the actual behavior of the solutions of the perturbed differential equation with respect to those of the original differential equation is developed.The results so obtained are given in terms of the existence of an extended form of Liapunov function.
We note that a similar concept, that of aeymptotic equivalence, was introduced by Brauer I though the general approach oth he and his successors used is different from the one employed here.oreover, the notion of asymptotic equivalence is merely one of the nmerous possibilities which can be considered in terms of the present approach.Some analogous results for the discrete case involving the behavior of solu- tions of difference equations have previously been done by the author 2 2. DEFINITIONS AND BASIS ONGEFTS.
We will consider the differential equation X'(t) f(t,X(t)).
(2.1) f(t,X) here represents a function with values in Em, an arbitrary m-dimensional vector space, and defined on some region D in I x E m which contains the axis { X O, t I }, where I is the set of non-negative real numbers.For simplicity, we may take for D the semi-infinlte cylinder D Dto R { (t,X) I x E m t% t o O, IIX; R Here, XI denotes any m-dimensional norm of the vector X.We note that in most cases, the upper bound R will be taken to be finite.The sole exception to this convention would occur when we are dealing with the case of instability for the solutions of the differential equations when the solutions become un- bounded and hence the region must a ccomodate them.
In addition, the differential equation (2.1) will be subject to the initial condition X(to) x o Moreover, we will consider only those equations for which the solution is uniquely determined by the initial point and this unique solution to the differential equation (2.1) satisfying the initial condition will be denoted by F(t, t o, x o) In addition to the differential equation (2.1), we also consider the asso- where g(t,Y) is also a function from Dto R into Em.If the additional term g(t,Y) is small in some sense, it is reasonable to expect that the behavior of the solutions of the perturbed equation will be similar to that for the solutions of the original equation, provided that the initial values for the two equations are sufficiently close.In this regard, the assumed unique solution to the per- turbed differential equation (2.2) satisfying the initial condition Y(t o) Yo will be denoted by G(t,to,Yo).
The present investigation will be carried out by using a certain class of continuous real scalar functions V(t,X) also defined on Dto R and satisdng the requirement V(t,0) 0 for all t t o The following additional pro- perties will all be required.Let M o represent the class of all real-valued monatone increasing functions, a(r), defined and positive for r 0 and such that a(0) O. In terms of this, a real scalar function V(t,X) is said to be positive definite if there exists a function a(r) of class Mo such that V(t,X) >a( X II ) for all t > t o A real scalar function V(t,X) is said to be positive semi- definite if V(t,X)> 0 for all t >/t o Entirely similar defimtions hold for such functions being either negative definite or negative semi-deflnite.
Moreover, @orresponding to a function V(t,X), we define its total deriva- tive with respect to the differential equation (2.1) as VV(t, t V(t,X) + V(t,X).f(t,X),where vv(t,x) B)v(t,x).
Here, x I ,..., x m denote the ccmponents of X. V' (t,X) is obviously a measure of the growth or decay of the function V(t,X) with regard to increasinE t along the trajectories represented by the solutions of the differential equation (2.1).
It should be noted that, in general, this can be calculated without direct knowledge of the actual solutions.
We now introduce the types of possible behavior for the solutions of the perturbed differential equation (2.2) which will be of interest to us in the sequel.
DEFINITION I: The solutions of the perturbed differential equation (2.2) are said to be _q,bl wth r_!t to th n_r,,bd dlffernial tion (2.1) if, for all E > 0 and for all t o I, there exists a ( e,t o ) > 0 such that Yo Xoll < implies that G(t,to,Yo) -F(t,to,Xo)l < e for all t > t o for every solution G(t,to,Yo) of the perturbed equation (2.2).
DEFINITION 2: The solutions of the perturbed differential equation (2.2) are said to be aavmpttical!ystable with respct to the unperturbed dif- ferential euati0n (2.1) if they are stable with respect to the equation (2.1) and if, for all t o I, there exists a o(to) > 0 such that ly Xoll 6 o implies that G(t,to,Yo) F(t,to,Xo) / 0 as t / (R) for every solution G (t,to,Yo) of the perturbed equation (2.2).
The above two definitions are equivalent to the statement that all solutions of the perturbed differential equation which start sufficiently close to the initial value of the unperturbed solution respectively remain close to it or eventually approach it.The latter case is essentially the same concept as Brauer's asymptotic equivalence.
The next definition expresses an intermediate type of behavior whereby the perturbed solutions initially may diverge fr the unperturbed solution, but eventually becne arbitrarily close to the latter.The concept is somewhat similar to that introduced by LaSalle and Rath [4 ] DEFINITION 3. The solutions of the perturbed differential equation (2.2) are said to be e__tuallv stable with respect to the unperturbed differential euuation (2.1) if, for every e > O, there exists a g e ,t o > 0 such that, for any Xo, Yo-Xoll <6 implies that G(t,to,Yo) F(t,to,Xo) < e for all t T, for some T > t o for every solution G(t,to,Yo) of the perturbed equation (2.2).
Finally, we give two further definitions of modes of behavior which will be considered.DEFINITION 4: The soluticms of the perturbed differential equation (2.2) are said to be exoonentiallv stable with respect to the unDerturbed dif- ferential euation (2.1) if there exist positive numbers a and B and a such that t o e I, Yo-Xo| <8o Imply that -a (t-t o I G(t,to,Yo) F(t,to,Xo)l B II Yo XoI e for all t Z t o for every solution G(t,to,Yo) of equation (2.2).
EEFINITION 5: The solutions of the perturbed differential equation (2.2) are said to be un_stable with re _sect to the unperturbed differential eauation (2.1) if, for every e > 0 and every to e I, there exists some Yo with I Yo-Xol < e and such that IG(tl,to,Yo) F(tl,to,Xo)l >.
for some > t o The above definition requires that for each solution of the unperturbed equa- tion (2.1), a solution of the perturbed equation (2.2) can be found which starts arbitrarily close to the unperturbed solutic and which eventually diverges from it.
We note that all of these definitions are independent of the behavior of the solutions of the unperturbed equation.In fact, we specifically indicate that the equilibria of the original differential equations may be stable, asymptotic- ally stable or even unstable.This is illustrated by the following: EXAMPLE I: Consider the unperturbed differential equation X' aX, with a > O, whose asymptotically stable solution is given by F(t,to,Xo x o e 289 Further, consider the associated perturbed equation Y' whose solution is given by (a+ As a consequence, G(t,to,Yo) F(t,to,Xo) e-a(t-to) [Yoe-b(t-to If b > O, this difference approaches 0 as t +(R) and thus the perturbed solu- tions are asymptotically, and in fact emponentially, stable with respect to the unperturbed equation.On the other hand, if b < O, then the perturbed solutions are unstable with respect to the unperturbed equation.where f(t) is any function which is defined and non-integrable on [t o (R) ).
The unstable solution to this equation is given by F(t,to,Xo) Further, consider the associated perturbed equation Y' where II g(t,Y)l 4 a llh(t) for some sufficiently small positive constant a and for some function h(t) which is integrable on [t o (R) ).The solution is given by G(t'to'Yo) Yo + to f(s) ds + to g(s,Y(s)) ds, and hence t a(t,to,yo) F(t,to,Xo)#l .<I#Y o Xol# + a to h(s) ds, which can be made arbitrarily small.Therefore, the perturbed solutions are stable with respect to the unperturbed differential equation.
We now present several theorems which supply sufficient conditions for the above types of behavior to hold in terms of the existence of continuous real scalar, Liapunov-type, functions V(t,X).
THEOREM I.If there exists a function V(t,X) on Dto R such that a) V(t,X) is positive definite b) V(t,X) is continuous for X 0 c) V'(t,Y(t) X(t)) is negative semi-definite, then the solutions of the perturbed differential equation (2.2) are stable with respect to the unperturbed differential equation (2.1), provided that for all t >-to, IiG(t,to,Yo) F(t,to,xo)l# % R.
PROOF: Since V(t,X) is positive definite, there is a function a(r) of class M o such that v(t,x) a(l x i ).
Now, given any a choose Yo sufficiently close to x o so that lyo-Xo < and V(to,y o-x o) < a().
It then follows that for all t t o; G(t,to,Yo) F(t,to,Xo)#/ < e for, if not, there would be some t I >t o such that liG(tl,to,Yo) F(tl,to,Xo)## > e This, however, would imply that > V(to,o Xo) V(tl,G(tl,to,yo) F(tl,to,Xo)) which is a contradiction.
It should be noted that the above theorem, as well as the ones which follow, depends strongly on the condition that, for all t ?. to, I G(t,to,Y o) F(t,to,Xo) II <.R. (3.1) This condition ,aranees that both the function V'(t,Y-X) remains well- defined and that the difference of the two solutions remains on Dto R. The following result gives one fairly simple set of criteria for the functions f(t,X) and g(t,Y) which insures that this holds.
THEOR 2: If f(t,X) satisfies a generalized Lipschitz condition If(t,X I) f(t,X 2)II < L(t) fiX I X 2## where L(t) is integrale on [to, (R) ) and if g(t,Y) satisfies I g(t,Y)ll .< a h(t)II for some sufficiently small positive constant a and for some function h(t) which is integrable on to, (R) ), then if Yo is chosen sufficiently close to xo, condition (3.1) holds for all t Z t o. and PROOF: We have t F(t,to,Xo) x o + to f(s,x(s ds t G(t,t o,yo) Yo * t o f(s,Y(s)) ds + J to g(s,Y(s)) ds.
As a consequence I{ O(t,to,Yo) F(t,to,X o) II JiYo-Xoli + to jlf(s Y) f(s X)li ds + Ii g(s,Y)i# ds tO < II Yo Xoll + L(s) G(S,to,Yo) F(s,to,Xo)ll ds + a lh(s)II ds t o I I Yo Xoll + a to II h(s)ll ds + L(s) ii (S,to,Yo) F(s,to,Xo)llds t o A + t t o L(s) IIG(S,to,Yo) F(S,to,Xo)I{ ds, where, we observe, the quantity A can be made arbitrarily small.We now apply the following form of Gronwall's Inequality to the aove relation" If Z(t) > 0 and P(t) < Q(t) + to Z(s) P(s) ds, We therefore obtain which can be made smaller than any given R by choosing the constant a sufficiently small and by choosing Yo sufficiently close to x We note that in the above, o K(t) represents an indefinite integral of L(t).
We now turn to a result giving sufficient conditions for asymptotic behavior for the two solutions.THEOR 3. If there exists a function V(t,X) on Dto R such that a) V(t,X) is bounded below b) V'(t,Y(t) X(t)) is negative definite, then the solutions of the perturbed differential equation (2.2) are asymptotically stable with respect to the unperturbed differential equation (2.1) provided that condition (3.1) holds for all t >. t o PROOF.Since V'(t,Y-X) is negative definite, there exists a function a(r) of class M o such that V'(t,Y-X) % a(lY-XI).
Noreover, we have that V(t,G(t,to,Yo) F(t,to,Xo)) V(to,Yo x o) + to 4 V(to,Y o x o) to V'(s,G(s,to,Yo) F(s,to,Xo)) ds a( O(S,to,yo) (S,to,Xo) s.
Taking the limit as t /(R) and using the fact that V(t,X) is bounded below by sce B, we find that lira t + (R) t o a(i G(s,to,Yo) F(s,to,Xo)ll ds .<V(to,Y o x o) B, which implies that, as t / (R), a( G(t,to,Yo) F(t,to,Xo)I ) / O.
Therefore, since a(r) is monotonically increasing, it follows that I I G(t,to,Yo) F(t,to,Xo)II / 0 as t / @, thus proving the theorem.
THEOP 4. If there exists a function V(t,X) on Dto R such that a) V(t,X) is positive definite b) V(t,X) is continuous as X 0 e) v,(t,-x)<-b(t), where to b(s) ds O, then the solutions of the perturbed differential equation (2.2) are eventually stable with respect to the unperturbed differential equation (2.1), provided that condition (3.1) holds for all t >. t o PROOF: Since V(t,X) is positive definite, there exists a function a(r) of class o such that V(t,X) > a(IXlI).Now, suppose that the solutions of (2.2) are not eventually stable with respect to (2.1).Then, for any e > 0 and any Xo, there exist sequences { z k )/ x o and {t k} / (R) as k / @ such that II G(tk,to,Zk) F(tk,to,Xo) I >.e.
THEOPd." it" there exists s eunction V(t,X) on Dto R such that a) liX;I p <V(t,X) < a 21x p for some positi constants a I and a 2 and for se p > O, b) V,(t,Y-X) < -a;Y-X p for some positive constant a, then the solutions of the peurd differential equation (2.2) are eonentially stable with respect to the unperturbed differential equation (2.1) provided that condition (3.1) holds for all t > t o PROOF" From the conditions on V(t,X) and V'(t,Y-X), we find V'(t,Y-X) -< -a 3 It Y-XII p <. -(a3/a2) V(t,Y-X) Therefore, upon integrating, we obtain V(t,G(t,to,Yo) F(t,to,Xo)) < V(to,Y o x o) e-a4 (t to) where we have written a 4 a3/a2 Moreover, it follows that V(t,G(t,to,Yo) F(t,to,X o)) > a I Ii G(t,to,Yo) F(t,to,X o) # p and hence As a result, I G(t,to,Yo) F(t,to,Xo)ll p < (I/a I) V(to,Y o x o) e -a4(t to) II G(t,to,Yo) F(t,to,Xo)il < B fly o Xol e-(a4/p)(t to) which completes the proof.
Finally, we conclude this section with a criterion for the solutions of (2.2) to be unstable with respect to equation (2.1).THEORI 6: Suppose there exists a real scalar ftmction V(t,X) such that a) for each e >0 and each t > t o and each solution F(t,to,Xo) of (2.1), there exists a solution G(t,to,Yo) of (2.2) such that G(t,to,Yo) F(t,to,Xo)I < e and S.P. GORDON V(t,G(t,to,Yo F(t,to,Xo)) < O; Corresponding. to each solution F(t,to,Xo), the set off all points flor which V(t,Y-X) < 0 is bounded by the hypersurfaces IY-X II R and V 0 and may consist off several component domains; b) In at least one of the component domains D* in which V(t,Y-X) < 0 corresponding to each solution F(t,to,Xo) V(t,X) is bounded below; c) In the domain D*, v,(t,-x) .<-( v(t,x) ), Cot some function a(r) of class Mo, then the solutions of the perturbed differential equation (2.2) are unstable with espect to the differential equation (2.1).
PROOF: Let F(t,to,Xo) be any solution off (2.1) and choose any point (tl'Yl Xl in D* such that V(tl,Y 1 Xl) -b < O where x I F(tl,to,Xo).Consider the solution G(t,tl,YI) of (2.2).We thus have V(t,G(t,tl,Yl) F(t,tl,Xl) V(tl,Y I x I) + V'(s,G(S,tl,YI) F(S,tl,Xl)) ds o.However, by assumption, V(t,Y-X) is bounded below in D* and hence, the points (t,G(t,,yI) F(t,tl,Xl)) must leave D* as t +.This can only happen across the boundary I Y-X II R, for any arbitrarily large R.Moreover, since Yl can be chosen arbitrarily close to Xl, the solutions of (2.2) are unstable with respect to (2.1).

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
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