This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points for of the solution. Two examples are provided.

1. Introduction

This paper investigates the errors of the solutions of nonlinear differential equations , where , provided they exist and are unique for each given admissible initial condition , with respect to the solutions of its approximate differential equations ; , for any given nonnegative integer , obtained from truncated Taylor expansions of the solutions about certain sampling points for . It is assumed that if a unique solution exists on some interval and that the choice of the sampling points is such that the intersample intervals [14] are subject to a certain maximum allowable upper-bound then the error of the solution in the whole interval satisfies a prescribed norm bound. Using the obtained results, the shadowing property [510] of the true solution with respect to the approximate one is investigated in the sense that “shadowing” initial conditions of the true solution exist, for each initial condition of the approximate differential equation, such that any approximated solution trajectory on the interval of interest is arbitrarily close to the true one under prescribed allowable maximum norms of the error between both the true solution and the approximate solutions. The problem is extended to the case when the approximated solution is perturbed either by a sequence of a certain allowable size at the sampling points or with perturbation functions of a certain size in norm about the whole considered interval. The main tool involved in the analysis is an “ad hoc” use of a known preparatory theorem to the celebrated Bernstein’s theorem, [11], which gives an upper-bound for the maximum norm of the error in between both the true and the approximate solutions. The results are potentially extendable to functional equations involving nonlinearities and the presence of delays subject to mixed types of uncertainties [1218]. On the other hand, different characterizations of oscillatory solutions and limit oscillatory solutions (limit cycles) have received important interest in the literature concerning different types of nonlinear dynamic continuous-time, discrete and hybrid systems, and differential equations [1928]. The shadowing property is naturally relevant for the characterization of limit oscillations. Therefore, the formulation is applied in the second example to the characterization of limit cycles generated as solutions to Van der Pol’s equation.

2. Calculation of the Exact Solution from Taylor Series Expansion

Lemma 1. Assume that and divide the real interval into subintervals with points such that Then where , for with , and and any real for ; for .

Proof. It follows from a well-known preparatory theorem to Bernstein’s theorem [5] that

Now, consider the nonlinear ordinary differential equation in the real interval such that is Lipschitz-continuous in . The following result follows from Lemma 1.

Theorem 2. The unique solution of (5) in is given by ; and , where are any arbitrary strictly ordered points such that with for .

Proof. Note that is Lipschitz-continuous in so that the solution on is unique, provided that for the given and some is such that ; and since is local Lipschitz-continuous as a result. Such a unique solution is given by Take any set of strictly ordered points satisfying with for , so that ; , with , so that, by choosing the real as for , one gets from (3) in the proof of Lemma 1 into (4): ; . Note that, since , then for any nonnegative integer . Thus, we obtain the result from a similar expression of (9) by replacing by while truncating the Taylor series expansion by its th term.

A consequence of Theorem 2 by using the same technique of the solution construction is as follows.

Corollary 3. Consider the nonlinear ordinary differential equation (5) with initial condition on the real interval , with initial conditions for , such that is Lipschitz-continuous in for some , and consider also its th order truncation such that are bounded in for for some nonnegative integer and some , where for some positive real with for .

Since are bounded in for , then the right-hand-side of (10) is Lipschitz-continuous in . Therefore, the unique solution of the truncated differential equation (10) in is ; , , where are arbitrary strictly ordered points such that with for . The error in between the exact solution of (10) and that of its truncated form (5) is ; and .

Proof. Property (i) follows directly Theorem 2 applied to the truncated differential equation (10) leading to the solution (11) in . Property (ii) follows from (6) and (11).

Now, a preparatory result follows to be then used to guarantee sufficiency-type errors results in between the true and the approximate solutions in the interval .

Lemma 4. Assume that the following hypothesis holds.
   and its first derivatives are uniformly bounded from above on a bounded subset of their existence domain with the specific boundedness constraint: for and some with if . Then, the following properties hold.
Assume that the intersample intervals for fulfill the constraint for and any given real constant , where Then, the approximated solution fulfills provided that
Assume that the intersample intervals for fulfill the constraint for and any given real constant , where Then, the true solution fulfills provided that
If and, furthermore, then the true, the approximated and the error solution fulfill and the error in between them, , fulfills

Proof. Proceeding recursively, one gets from Assumption A1 that if and , and if , where
Case (a). If and proceed by complete induction by assuming that since the condition guarantees that . Thus, one gets from (11) that where and , with , for , so that or provided that , and which is guaranteed if holds with for , defined in (16), provided that ; for , and then (33) and for are jointly guaranteed for the given if provided that for for , the last condition being identical to The above two conditions (34)-(35) reduce to (14). Then, one gets from complete induction from (31), if (14) holds, the following.
and one gets also by continuity extension,
; . Hence, we got the result for Case (a).
Case (b). If then , where , so that for and, thus, one gets the following.
; for and one gets from complete induction the same conclusion ; as in Case (a) provided that (14) holds. Then, (14) guarantees Property (i) for both Cases (a) and (b). Then, Property (i) has been proven.
Property (ii) is proven “mutatis-mutandis” by noting that from (15) and (19) and noting also that in (16) is replaced with in (20) so that the admissible intersample interval satisfying the constraint (14) is replaced by such an interval satisfying the constraint (18). Finally, Property (iii) follows from Properties (i)-(ii) by equalizing and to take a maximum value less than .

The following comments address the fact that it is not necessary to deal with the solution of the true differential equation (5) to calculate upper-bounds of the solution and error in Lemma 4.

Remark 5. Note that one gets by direct integration from (5) that leading to Thus, (25)-(26) might be guaranteed with if . Thus, there is no need to compute the solution of the true differential equation (5) and for in (20) and (23) if .

One gets directly from Lemma 4 the subsequent result.

Theorem 6. Assume the conditions (13a) and (13b) and (22)–(24) of Lemma 4(iii). Then , and for .

3. Orbits of the Exact Solution, Pseudo-Orbits of the Approximated Solution, and the Shadowing Property

Now, consider a perturbed solution (11) of the approximated differential equation (10) associated with a perturbation with at fulfilling for some given , . Note that a perturbation at the initial is considered by choosing for some nonzero . The perturbed solution can be generated, in particular, from impulsive controls of amplitudes at the sequence . The exact and approximate solutions (6) and (11) in , provided that they exist, are where is the Heaviside function. The error between the exact and perturbed approximated solutions becomes ; . Now, one gets from (25)-(26) of Lemma 4: where from (27) and one gets after using the triangle inequality, for ; . One obtains easily from (46), either by complete induction or via recursive calculation, that with for and any given nonnegative integer . The following result follows directly from the above equations and Theorem 6.

Theorem 7. Consider an approximated perturbed solution (42) under a forcing perturbation sequence at fulfilling for and some . Then, there are numbers , , , and such that on for any strictly ordered sequence of nonnegative real numbers , subject to the constraints satisfying the constraints (22)–(24) of Lemma 4 subject to (18).

Proof. Note that fixing , with , and the use of (46), (47a), and (47b) leads to ; by using Lemma 4 and Theorem 6 for any given prefixed . The result then follows since and either or and the result has been proven.

The following result extends Theorem 7 with results of Theorem 6 for the case when both the exact and approximated differential equations are subject to a piecewise-continuous bounded disturbance which might be dependent on the solution and also can have finite step discontinuities in the sequence .

Theorem 8. Consider the forced versions of the differential equations (5) and (10): under the additive forcing perturbation satisfying Assumption (A2) of Lemma 4 fulfilling and ; with for and some and being a bounded piecewise-continuous function. Then, there are numbers , , and such that on for a strictly ordered finite set of nonnegative real numbers , subject to the constraints , ; , the constraints (22)–(24) subject to (18), and either or

Proof. Fix , with . Equations (53)-(54) have the following solutions: