Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

Abstract

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 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] for 𝑖=0,1,…,𝐽 of the solution. Two examples are provided.

1. Introduction

This paper investigates the errors of the solutions of nonlinear differential equations 𝑦(̇𝑡)=𝑓(𝑦(𝑡),𝑡), where 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛), provided they exist and are unique for each given admissible initial condition 𝑦(𝑡₀)=𝑦₀, with respect to the solutions of its approximate differential equations 𝑥̇(𝑡)=∑_𝑘=0(𝑓^(𝑘)(𝑥(𝑡_𝑖),𝑡_𝑖)/𝑘!)(𝑥(𝑡)−𝑥(𝑡_𝑖))^𝑘; 𝑥(𝑡₀)=𝑥₀, for any given nonnegative integer ≤𝑛, obtained from truncated Taylor expansions of the solutions about certain sampling points 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] for 𝑖=0,1,…,𝐽. It is assumed that if a unique solution exists on some interval [𝑡₀,𝑡_𝐽] and that the choice of the sampling points is such that the intersample intervals [1–4] are subject to a certain maximum allowable upper-bound then the error of the solution in the whole interval [𝑡₀,𝑡_𝐽] satisfies a prescribed norm bound. Using the obtained results, the shadowing property [5–10] 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 [12–18]. 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 [19–28]. 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 𝑓∈𝐶^(𝑛+1)((𝑎,𝑏);𝐑^𝑛) and divide the real interval (𝑎,𝑏) into 𝐽 subintervals with points 𝑦_𝑛∈[𝑎,𝑏] such that 𝑎≡𝑦₀<𝑦₁<𝑦₂<⋯<𝑦_𝐽−1<𝑦_𝐽≡𝑏. (1) Then 𝑓𝑦_𝑖+1=𝑓𝑦₀+^ℎ𝑖₀𝑓𝑦+𝑦₀𝑦𝑑𝑦=𝑓₀+^𝑖_𝑗=−1^ℎ𝑗+1_ℎ𝑗𝑓𝑦+𝑦₀𝑦𝑑𝑦=𝑓₀+^𝑖_𝑗=−1^ℎ𝑗₀𝑓𝑦+𝑦₀+ℎ_𝑗𝑦𝑑𝑦=𝑓₀+^𝑖𝑛_{𝑗=−1𝑘=0}𝑓^(𝑘)𝑐_𝑗𝑘!^ℎ𝑗₀𝑦+𝑦₀+ℎ_𝑗−𝑐_𝑗+1^𝑘+1𝑑𝑦𝑛!^𝑖_𝑗=−1^ℎ𝑗₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑗−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡𝑑𝑦, (2) where ℎ_𝑛=𝑦_𝑛+1−𝑦_𝑛, ℎ_𝑛=𝑦_𝑛+1−𝑦₀=∑^𝑛_𝑖=0ℎ_𝑖 for 𝑛=0,1,…,𝐽−1 with ℎ₋₁=0, and 𝑓𝑦_𝑖+1ℎ_𝑖+1=𝑓𝑦₀+^𝑖𝑛_{𝑗=−1𝑘=0}𝑓^(𝑘)𝑐_𝑗𝑘!^ℎ𝑗₀𝑦+𝑦₀+ℎ_𝑗−𝑐_𝑗+1^𝑘+1𝑑𝑦𝑛!^𝑖_𝑗=−1^ℎ𝑗₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑗−𝑡^𝑛𝑓^(𝑛+1)(𝑡)+𝑑𝑡𝑑𝑦_𝑛𝑘=0𝑓^(𝑘)𝑐_𝑖+1𝑘!^ℎ𝑖+1₀𝑦+𝑦₀+ℎ_𝑖+ℎ_𝑖+1−𝑐_𝑖+2^𝑘+1𝑑𝑦𝑛!^ℎ𝑖+1₀^𝑦𝑗+1_𝑐𝑗+1𝑦+𝑦₀+ℎ_𝑖+ℎ_𝑖+1−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡𝑑𝑦; (3)∀𝑦∈[𝑦_𝑖,𝑦_𝑖+1] and any real 𝑐_𝑖∈[𝑦_𝑖,𝑦_𝑖+1] for 𝑖=0,1,…,𝐽−1; ∀ℎ_𝑖∈[0,ℎ_𝑖] for 𝑖=0,1,…,𝐽−2.

Proof. It follows from a well-known preparatory theorem to Bernstein’s theorem [5] that 𝑓𝑦=^𝑛_𝑘=0𝑓^(𝑘)𝑐_𝑖𝑘!𝑦−𝑐_𝑖^𝑘+1𝑛!^𝑦_𝑐𝑖𝑦−𝑡^𝑛𝑓^(𝑛+1)(𝑡)𝑑𝑡. (4)

Now, consider the nonlinear ordinary differential equation 𝑦̇(𝑡)=𝑓𝑦(𝑡),𝑡;𝑦𝑡₀=𝑦₀ (5) in the real interval 𝐑^𝑛×[𝑡₀,𝑡_𝐽] such that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽]. The following result follows from Lemma 1.

Theorem 2. The unique solution of (5) in [𝑡₀,𝑡_𝐽] is given by 𝑦(𝑡)=𝑦𝑡₀+^𝑖−2_𝑗=0^ℎ𝑗_0𝑘=0𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑡𝑡𝑖−1𝑘=0}𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (6)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽} and ∀(∈𝐙₀₊)≤𝑛, where 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] are any arbitrary strictly ordered points such that 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1.

Proof. Note that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽] so that the solution 𝑦(𝑡) on [𝑡₀,𝑡_𝐽] is unique, provided that 𝑡_𝐽=𝑡_𝐽(𝜃₀,𝑡₀) for the given 𝑡₀∈𝐑 and some 𝜃₀∈𝐑^𝑛 is such that 𝑦(𝑡)∈[𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]; ∀𝑡∈[𝑡₀,𝑡_𝐽] and 𝑡_𝐽∈(𝑡₀,𝑡_𝐽] since 𝑓∶[𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽]→𝐑^𝑛 is local Lipschitz-continuous as a result. Such a unique solution is given by 𝑦(𝑡)=𝑦_𝑎+^𝑡_𝑎𝑓𝑦(𝜏),𝜏𝑑𝜏;∀𝑡∈𝑡₀,𝑡_𝐽. (7) Take any set of 𝐽 strictly ordered points 𝑡_𝑛∈[𝑡₀,𝑡_𝐽] satisfying 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1, so that 𝑦(𝑡)=𝑦𝑡_𝑖−1+^𝑡_𝑡𝑖−1𝑓𝑦(𝜏)𝑡,𝜏𝑑𝜏=𝑦₀+^𝑖−2_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑓𝑦(𝜏),𝜏𝑑𝜏+^𝑡_𝑡𝑖−1𝑓𝑦(𝜏),𝜏𝑑𝜏; (8)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}, with 𝑦(𝑡₀), so that, by choosing the real 𝑐_𝑖∈[𝑡_𝑖−1,𝑡_𝑖] as 𝑐_𝑖=𝑡_𝑖 for 𝑖=0,1,…,𝐽−1, one gets from (3) in the proof of Lemma 1 into (4): 𝑦(𝑡)=𝑦𝑡₀+^𝑖−2_𝑗=0^ℎ𝑗₀^𝑛_𝑘=0𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘+1𝑛!^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗^𝑛×𝑓^(𝑛+1)𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑡−𝑡𝑖−10𝑛𝑘=0}𝑓^(𝑘)𝑦𝑡_𝑖−1,𝑡_𝑖−1×𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘+1𝑛!^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑛×𝑓^(𝑛+1)𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (9)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}. Note that, since 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛), then 𝑓∈𝐶⁽⁺¹⁾(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) for any nonnegative integer ≤𝑛. Thus, we obtain the result from a similar expression of (9) by replacing 𝑛 by (≤𝑛) while truncating the Taylor series expansion by its (+1)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 𝑦(𝑡₀) on the real interval 𝐑^𝑛×𝐑₀₊, with initial conditions 𝑦^(𝑗)(𝑡₀) for 𝑗=0,1,…,𝑛−1, such that 𝑓∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃₀,𝑦(𝑡₀)+𝜃₀]×[𝑡₀,𝑡_𝐽] for some 𝜃₀∈𝐑^𝑛, and consider also its th order truncation 𝑥̇(𝑡)=_𝑘=0𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥(𝑡)𝑡𝑘!−𝑥_𝑖^𝑘;𝑥𝑡₀=𝑥₀ (10) such that 𝑓^(𝑘)(𝑦(𝑡),𝑡) are bounded in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽] for 𝑘=0,1,…,+1 for some nonnegative integer ≤𝑛 and some 𝜃∈𝐵(𝜃₀,𝑅), where 𝐵(𝜃,𝑅)={𝑧∈𝐑^𝑛∶‖𝑧−𝜃₀‖≤𝑅} for some positive real 𝑅 with 𝑥^(𝑗)(𝑡₀)=𝑦^(𝑗)(𝑡₀) for 𝑗=0,1,…,+1.

Since 𝑓^(𝑘)(𝑦(𝑡),𝑡) are bounded in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽] for 𝑘=0,1,…,−1, then the right-hand-side of (10) is Lipschitz-continuous in [𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽]⊆[𝑦(𝑡₀)−𝜃,𝑦(𝑡₀)+𝜃]×[𝑡₀,𝑡_𝐽]. Therefore, the unique solution of the truncated differential equation (10) in [𝑎,𝑏] is 𝑥(𝑡)=𝑥𝑡₀+_𝑘=0^𝑖−2_𝑗=0^ℎ𝑗₀𝑓^(𝑘)𝑥𝑡_𝑗,𝑡_𝑗𝑥𝑘!𝜏+𝑡_𝑗−𝑥𝑡_𝑗^𝑘+𝑑𝜏_{𝑡−𝑡𝑖−10}𝑓^(𝑘)𝑥𝑡_𝑖−1,𝑡_𝑖−1×𝑥𝑘!𝜎+𝑡_𝑖−1−𝑥𝑡_𝑖−1^𝑘𝑑𝜏; (11)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽}, ∀(∈𝐙₀₊)≤𝑛, where 𝑡_𝑖∈[𝑡₀,𝑡_𝐽] are arbitrary strictly ordered points such that 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝐽−1<𝑡_𝐽 with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1. The error in between the exact solution of (10) and that of its truncated form (5) is 𝑒(𝑡)(𝑡)(𝑡)==𝑦−𝑥_{𝑖−2𝑗=0𝑘=0ℎ𝑗0}𝑓^(𝑘)𝑦𝑡_𝑗,𝑡_𝑗𝑦𝑘!𝜏+𝑡_𝑗−𝑦𝑡_𝑗^𝑘−𝑓^(𝑘)𝑥𝑡_𝑗,𝑡_𝑗×𝑥𝑘!𝜏+𝑡_𝑗−𝑥𝑡_𝑗^𝑘+1𝑑𝜏!^𝑖−2_𝑗=0^ℎ𝑗₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗+𝑑𝜎𝑑𝜏_{𝑘=0𝑡−𝑡𝑖−10}𝑓^(𝑘)𝑦𝑡_𝑖−1,𝑡_𝑖−1×𝑦𝑘!𝜏+𝑡_𝑖−1−𝑦𝑡_𝑖−1^𝑘−𝑓^(𝑘)𝑥𝑡_𝑖−1,𝑡_𝑖−1×𝑥𝑘!𝜏+𝑡_𝑖−1−𝑥𝑡_𝑖−1^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖−1}₀^𝜏₀𝑦𝜎+𝑡_𝑖−1−𝑦𝑡_𝑖−1×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑖−1,𝜎+𝑡_𝑖−1𝑑𝜎𝑑𝜏; (12)∀𝑡∈[𝑡_𝑖−1,𝑡_𝑖]; ∀𝑖∈𝐽={1,2,…,𝐽} and ∀(∈𝐙₀₊)≤𝑛.

Proof. Property (i) follows directly Theorem 2 applied to the truncated differential equation (10) leading to the solution (11) in [𝑡₀,𝑡_𝐽]. 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 [𝑎,𝑏].

Lemma 4. Assume that the following hypothesis holds.
(A1)  𝑓(𝑦(𝑡),𝑡) and its first (+1) derivatives are uniformly bounded from above on a bounded subset of their existence domain with the specific boundedness constraint:sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑦(𝑡)+𝐾₁,sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁, (13a)(13b)for 𝑗=0,1,…,+1 and some 𝐾,𝐾₁∈𝐑₀₊ with 𝐾<1 if 𝐾₁∈𝐑₊. Then, the following properties hold.
(i) Assume that the intersample intervals ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 fulfill the constraint ℎ_𝑖𝑎≤ℎ≤min_𝑖,1−𝜌_𝑥/2Λ_𝑥1−𝜌_𝑥/2,𝜌_𝑥2Λ_𝑥1−𝜌_𝑥/2⁺¹𝐽−1+𝜌_𝑥/2 (14) for 𝑖=0,1,…,𝐽−1 and any given real constant 𝜌_𝑥∈(0,2), where Λ_𝑥=_𝑘=0𝐾^𝑘𝑘!sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡)+𝐾,𝑡₁1−𝐾⁺¹1−𝐾𝑎_𝑖∶=minarg𝑡∈𝐑₊>𝑡_𝑖∶𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌_𝑥2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1. (15)(16) Then, the approximated solution fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2 provided that 𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶𝑥(𝑡)−𝑥𝑡₀≤𝜌_𝑥2. (17)
(ii) Assume that the intersample intervals ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 fulfill the constraint ℎ_𝑖𝑏≤ℎ≤min_𝑖,1−𝜌/2Λ1−𝜌/2,𝜌2Λ1−𝜌/2⁺¹𝐽−1+𝜌/2 (18) for 𝑖=0,1,…,𝐽−1 and any given real constant 𝜌∈(0,1), where Λ=⁺¹_𝑘=0𝐾^𝑘𝑘!sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡)+𝐾,𝑡₁1−𝐾^𝑘1−𝐾𝑏_𝑖∶=minarg𝑡>𝑡_𝑖∶𝑦(𝑡)−𝑦𝑡_𝑖≤𝜌2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1. (19)(20) Then, the true solution fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑦(𝑡)−𝑦(𝑡₀)‖≤𝜌/2 provided that 𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶𝑦(𝑡)−𝑦𝑡₀≤𝜌2. (21)
(iii) If 𝜌_𝑥=𝜌∈(0,1) and, furthermore, ℎ_𝑖𝑐≤ℎ≤min_𝑖,1−𝜌/2Λ1−𝜌/2,𝜌2Λ1−𝜌/2⁺¹;𝐽−1+𝜌/2𝑖=0,1,…,𝐽−1,𝑐_𝑖∶=minarg𝑡>𝑡_𝑖∶max𝑦(𝑡)−𝑦𝑡_𝑖,𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌2;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1,𝑡₁=minarg𝑡∈𝐑₊>𝑡₀∶max𝑦(𝑡)−𝑦𝑡₀,𝑥(𝑡)−𝑥𝑡₀≤𝜌2 (22)(23)(24) then the true, the approximated and the error solution fulfill sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑦(𝑡)−𝑦𝑡₀≤𝜌2,sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑥(𝑡)−𝑥𝑡₀≤𝜌2, (25) and the error in between them, 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡), fulfills sup_{𝑡∈[𝑡0,𝑡𝐽]}𝑒(𝑡)−𝑒𝑡₀≤𝜌. (26)

Proof. Proceeding recursively, one gets from Assumption A1 that sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁≤𝐾²sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑗−1)𝑦(𝑡),𝑡+𝐾₁(1+𝐾)≤𝐾^𝑘𝐹₀+𝐾₁^𝑘−1_𝑖=0𝐾^𝑖≤𝐾^𝑘𝐹₀+𝐾₁1−𝐾^𝑘1−𝐾≤𝐹₀+𝐾₁1−𝐾 (27) if 𝐾<1 and 𝐾₁≠0, and sup_{𝑦(𝑡)𝑡∈[𝑦0𝑡−𝜃,𝑦𝐽+𝜃],𝑡∈[𝑡0,𝑡𝐽]}𝑓^(𝑘)𝑦(𝑡),𝑡≤𝐾^𝑘𝐹₀, (28) if 𝐾₁=0, where 𝐹₀=sup_{𝑦(𝑡)∈𝑦𝑡0𝑡−𝜃,𝑦𝐽+𝜃,𝑡∈[𝑡0,𝑡𝐽]}𝑓⁽⁰⁾𝑦(𝑡),𝑡=sup_{𝑦(𝑡)∈𝑦𝑡0𝑡−𝜃,𝑦𝐽+𝜃,𝑡∈[𝑡0,𝑡𝐽]}𝑓𝑦(𝑡),𝑡<+∞. (29)
Case (a). If 𝐾<1 and 𝐾₁≠0 proceed by complete induction by assuming that sup_{𝑡∈[𝑡0,𝑡𝑖]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2 since the condition (𝑡(∈𝐑₊)>𝑡_𝑖∶‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2) guarantees that sup_{𝑡∈[𝑡0,𝑡1]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2. Thus, one gets from (11) that 𝑥(𝑡)−𝑥𝑡_𝑖≤^𝑖_𝑘=0𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘=_𝑘=0^𝑖−1_𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘+_𝑘=0ℎ_𝑖𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝑥(𝑡)𝑡1−𝐾−𝑥_𝑖^𝑘≤_𝑘=0^𝑖−1_𝑗=0ℎ_𝑗𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑥𝑘+_𝑘=0ℎ_𝑖𝐾𝑘!^𝑘𝐹₀+𝐾₁1−𝐾^𝑘𝑥(𝑡)𝑡1−𝐾−𝑥_𝑖^𝑘=Λ_𝑥𝑘=0𝜌_𝑥2^𝑘𝑖−1_𝑗=0ℎ_𝑗+ℎ_𝑖Λ_𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘≤𝑖ℎΛ_𝑥𝑘=0𝜌_𝑥2^𝑘+ℎ_𝑖Λ_𝑥𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1𝑥(𝑡)−𝑥𝑡_𝑖;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1, (30)(31) where Λ_𝑥=∑_𝑘=0(1/𝑘!)(𝐾^𝑘𝐹₀+(𝐾₁(1−𝐾^𝑘)/(1−𝐾))) and ℎ≥max_{0≤𝑖≤𝐽−1}ℎ_𝑖, with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖, for 𝑖=0,1,…,𝐽−1, so that 1−ℎ_𝑖Λ_𝑥𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1×𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥𝑘=0𝜌_𝑥2^𝑘=𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−𝜌_𝑥,/2 (32) or 𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−ℎ_𝑖Λ_𝑥∑_𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−11−𝜌_𝑥≤/2𝑖ℎΛ_𝑥1−𝜌_𝑥/2⁺¹1−ℎ_𝑖Λ_𝑥1−𝜌_𝑥/2/1−𝜌_𝑥/21−𝜌_𝑥≤𝜌/2_𝑥2 (33) provided that 0<𝜌_𝑥<2, and 1>ℎ_𝑖Λ_𝑥(∑_𝑘=0(‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖^𝑘−1)) which is guaranteed if ℎ_𝑖<min(𝑎_𝑖,(1−𝜌_𝑥/2)/(Λ_𝑥(1−𝜌_𝑥/2))) holds with 𝑎_𝑖 for 𝑖=0,1,…,𝐽−1, defined in (16), provided that ‖𝑥(𝑡)−𝑥(𝑡_𝑗)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡_𝑗,𝑡_𝑗+1) for 𝑗(≤𝑖)=0,1,…,𝑖−1, and then (33) and ℎ_𝑗<((1−𝜌_𝑥/2)/(Λ_𝑥(1−𝜌_𝑥/2))) for 𝑗=0,1,…,𝑖−1 are jointly guaranteed for the given 𝑖=0,1,…,𝐽−1 if ℎ_𝑖<min1−𝜌_𝑥/2Λ_𝑥1−𝜌_𝑥/2,𝜌_𝑥2Λ_𝑥1−𝜌_𝑥/2⁺¹𝐽−1+𝜌_𝑥/2 (34) provided that ‖𝑥(𝑡)−𝑥(𝑡_𝑗)‖≤𝜌_𝑥/2 for 𝑡∈[𝑡_𝑗,𝑡_𝑗+1) for 𝑗=0,1,…,𝑖−1, the last condition being identical to 𝑡_𝑖+1≤𝑎_𝑖∶=minarg𝑡>𝑡_𝑖∶𝑥(𝑡)−𝑥𝑡_𝑖≤𝜌_𝑥2. (35) The above two conditions (34)-(35) reduce to (14). Then, one gets from complete induction from (31), if (14) holds, the following.
sup_{𝑡∈[𝑡0,𝑡𝑖]}‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2⇒sup_{𝑡∈[𝑡0,𝑡𝑖+1)}‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤𝜌_𝑥/2 and one gets also by continuity extension,
sup_{𝑡∈[𝑡0,𝑡]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡₀,𝑡_𝐽]. Hence, we got the result for Case (a).
Case (b). If 𝐾₁=0 then 𝑥(𝑡)−𝑥𝑡_𝑖≤𝑖ℎΛ_𝑥0𝑘=0𝜌_𝑥2^𝑘+ℎ_𝑖Λ_𝑥0𝑘=0𝑥(𝑡)−𝑥𝑡_𝑖^𝑘−1𝑥(𝑡)−𝑥𝑡_𝑖; (36)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1), where Λ_𝑥0=∑_𝑘=0((𝐾^𝑘𝐹₀)/(𝑘!))≤Λ_𝑥, so that (1−ℎΛ_𝑥0)‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖ ≤𝑖ℎΛ_𝑥0𝐸 for 𝑖=0,1,…,𝐽 and, thus, one gets the following.
‖𝑥(𝑡)−𝑥(𝑡_𝑖)‖≤((𝑖ℎΛ_𝑥0𝐸)/(1−ℎΛ_𝑥0))≤𝜌_𝑥/2; ∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1) for 𝑖=0,1,…,𝐽 and one gets from complete induction the same conclusion sup_{𝑡∈[𝑡0,𝑡]}‖𝑥(𝑡)−𝑥(𝑡₀)‖≤𝜌_𝑥/2; ∀𝑡∈[𝑡₀,𝑡_𝐽) 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 Λ≥Λ_𝑥 from (15) and (19) and noting also that 𝑎_𝑖 in (16) is replaced with 𝑏_𝑖 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 𝜌_𝑥 and 𝜌 to take a maximum value less than 1/2.

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 𝑦(𝑡)≤sup_{𝑡0≤𝜏≤𝑡𝐽}𝑦(𝜏≤𝑦𝑡₀)+𝑡_𝐽−𝑡₀𝐾sup_{𝑡0≤𝜏≤𝑡𝐽}𝑦(𝜏) (37) leading to sup_{𝑡0≤𝑡≤𝑡𝐽}𝑦(𝑡≤𝐾𝑡)^𝐽−𝑡₀1−𝐾𝑡_𝐽−𝑡₀𝑦𝑡₀if^𝐽−1_𝑖=0ℎ_𝑖<1𝐾. (38) Thus, (25)-(26) might be guaranteed with sup_{𝑡0≤𝑡≤𝑡𝐽}‖𝑦(𝑡)−𝑦(𝑡₀)‖ ≤(1/(1−𝐾(𝑡_𝐽−𝑡₀)))‖𝑦(𝑡₀)‖≤(𝜌/2) if ‖𝑦(𝑡₀)‖≤(𝜌/2)(1−𝐾(𝑡_𝐽−𝑡₀))<(𝜌/2). Thus, there is no need to compute the solution of the true differential equation (5) and sup_{𝑡𝑖≤𝑡<𝑡𝑖+1}‖𝑦(𝑡)−𝑦(𝑡₀)‖≤(𝜌/2) for 𝑖=0,1,…,𝐽−1 in (20) and (23) if ‖𝑦(𝑡₀)‖≤(𝜌/2)(1−𝐾(𝑡_𝐽−𝑡₀)).

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 maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,𝑒(𝑡)−𝑒𝑡₀≤𝜌<1;max_{𝑡𝑡∈0,𝑡𝐽}‖𝑒(𝑡)‖≤𝑒𝑡₀+𝜌; (39)∀𝑡∈[𝑡₀,𝑡_𝐽], and max_{0≤𝑖≤𝐽−1}𝑒(𝑡)−𝑒𝑡_𝑖𝑡≤𝜌;∀𝑡∈_𝑖,𝑡_𝑖+1𝑓𝑜𝑟𝑖=0,1,…,𝐽−1 (40) for 𝑖=0,1,…,𝐽−1.

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 𝑥(𝑡_𝑖)=𝑥(𝑡⁻_𝑖)+𝑔(𝑡_𝑖) with {𝑔(𝑡_𝑖)}⊂𝐑^𝑛 at 𝑡=𝑡_𝑖 fulfilling ‖𝑔(𝑡_𝑖)‖≤𝑔_𝑖≤𝑔 for some given 𝑔∈𝐑, ∀𝑖∈𝐽. Note that a perturbation at the initial 𝑡=𝑡₀ is considered by choosing 𝑥(𝑡₀)=𝑦(𝑡₀)+𝑔₀ for some nonzero 𝑔₀=𝑔(𝑡₀)∈𝐑. The perturbed solution can be generated, in particular, from impulsive controls of amplitudes 𝑔(𝑡_𝑖) at the sequence {𝑡_𝑖∶𝑖∈𝐽}. The exact and approximate solutions (6) and (11) in [𝑡₀,𝑡_𝐽], provided that they exist, are 𝑦(𝑡)=𝑦𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖}₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑡𝑑𝜎𝑑𝜏;∀𝑡∈_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1,𝑥(𝑡)=𝑥𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘𝑑𝜏+𝑈𝑡−𝑡_𝑖+1𝑔𝑡_𝑖+1;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑖=0,1,…,𝐽−1, (41)(42) where 𝑈(𝑡) is the Heaviside function. The error between the exact and perturbed approximated solutions becomes 𝑒(𝑡)=𝑒𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘−𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖×𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘+1𝑑𝜏!^{𝑡−𝑡𝑖}₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑑𝜎𝑑𝜏−𝑈𝑡−𝑡_𝑖+1𝑔𝑡_𝑖+1; (43)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1]; 𝑖=0,1,…,𝐽−1. Now, one gets from (25)-(26) of Lemma 4: 𝑒(𝑡)−𝑒𝑡_𝑖≤_𝑘=02^𝑘+1𝑘!𝑡−𝑡_𝑖𝑀_𝑖𝑘𝜌2^𝑘+1!𝑡−𝑡_𝑖²2𝑀_𝑖,+1𝜌2+𝑔_𝑖;∀𝑡∈𝑡_𝑖,𝑡_𝑖+1, (44) where 𝑀_𝑖𝑘=sup_{𝑦(𝑡)𝑡∈[𝑦𝑖𝑡−𝜃,𝑦𝑖+𝜃],𝑡∈[𝑡𝑖,𝑡𝑖+1]}𝑓^(𝑗)𝑦(𝑡),𝑡≤𝐾^𝑘𝐹₀+𝐾₁1−𝐾^𝑘≤𝐾1−𝐾_𝑘+1𝜌2+𝐾₁1−𝐾^𝑘≤𝜌1−𝐾2+𝐾₁1−𝐾 (45) from (27) and one gets after using the triangle inequality, 𝑒(𝑡)−𝑒𝑡_𝑖≤^𝑚_{𝑗=𝑖𝑘=0}2^𝑘+1𝑘!𝑡−𝑡_𝑗𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗²2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑗; (46)∀𝑡∈[𝑡_𝑖+𝑚,𝑡_𝑖+𝑚+1] for 𝑚=0,1,…,𝐽−𝑖−1; 𝑖=0,1,…,𝐽−1. One obtains easily from (46), either by complete induction or via recursive calculation, that𝑒(𝑡)−𝑒𝑡_𝑖≤^𝐽−1_𝑗=0𝑘=02^𝑘+1𝑘!𝑡−𝑡_𝑗𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗²2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑗≤𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+𝐽ℎ2!×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔,‖𝑒(𝑡)‖≤𝑒𝑡₀+^𝐽−1_𝑗=0𝑡−𝑡_𝑗𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑗2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹×𝜌1−𝐾2+𝑔_𝑗≤𝑒𝑡₀+𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+𝐽ℎ2!𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔; (47a)(47b)(47c)(47d)∀𝑡∈[𝑡₀,𝑡₀+∑^𝐽−1_𝑖=0ℎ_𝑖](⊆[𝑡₀,𝑡₀+𝐽ℎ]) with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖 for 𝑖=0,1,…,𝐽−1 and any given nonnegative integer ≤𝑛. 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 {𝑔(𝑡_𝑖)}⊂𝐑^𝑛 at 𝑡=𝑡_𝑖 fulfilling ‖𝑔(𝑡_𝑖)‖≤𝑔_𝑖≤𝑔≥‖𝑒(𝑡₀)‖ for 𝑖=1,2,… and some 𝑔∈𝐑₊. Then, there are numbers ℎ∈𝐑₊, 𝐽=𝐽(ℎ)∈𝐙₊, 𝜀₁=𝜀₁(ℎ,𝑔)∈𝐑₊, and 𝜀=𝜀(𝜀₁,‖𝑒(𝑡₀)‖) such that maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,max_{𝑡∈𝐑0+}𝑒(𝑡)−𝑒𝑡₀≤𝜀₁;max_{𝑡∈𝐑0+}‖𝑒(𝑡)‖≤𝜀 (48) on [𝑡₀,𝑡_𝐽] for any strictly ordered sequence of (𝐽+1) nonnegative real numbers {𝑡_𝑖∶𝑖=0,1,…,𝐽}, subject to the constraints 𝑡_𝐽=𝑡₀+^𝐽−1_𝑖=0ℎ_𝑖,ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ,𝑖=0,1,…,𝐽−1 (49) satisfying the constraints (22)–(24) of Lemma 4 subject to (18).

Proof. Note that fixing ∑^𝐽−1_𝑖=0ℎ_𝑖=𝑡_𝐽−𝑡₀≤𝐽ℎ, with ℎ=max_{0≤𝑖≤𝐽−1}(𝑡_𝑖+1−𝑡_𝑖), and the use of (46), (47a), and (47b) leads to𝑒(𝑡)−𝑒𝑡_𝑖≤^𝐽−1_𝑖=0ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖≤𝜀₁=𝜌+^𝐽−1_𝑖=0𝑔_𝑖,𝑒(𝑡)−𝑒𝑡_𝑖≤𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔≤𝜀₁=𝜌+𝐽𝑔; (50a)(50b)∀𝑡∈[𝑡₀,𝑡_𝐽]; 𝑖=0,1,…,𝐽−1 by using Lemma 4 and Theorem 6 for any given prefixed 𝜌∈𝐑₊. The result then follows since 𝑔₀=‖𝑒(𝑡₀)‖≤𝑔 and either 𝜀=𝜀₁+𝑒𝑡₀,𝜀₁=𝜌+^𝐽−1_𝑖=0𝑔_𝑖,_{𝐽−1𝑖=0}ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+ℎ2!𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2≤𝜌 (51) or 𝜀=𝜀₁+𝑒𝑡₀,𝜀₁=𝜌+𝐽ℎ𝑔,_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2≤𝜌; (52) 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 {𝑡_𝑖∶𝑖=0,1,…,𝐽}.

Theorem 8. Consider the forced versions of the differential equations (5) and (10): 𝑦̇(𝑡)=𝑓𝑦(𝑡),𝑡+𝑔𝜏,𝑦(𝜏),𝑦𝑡₀=𝑦₀,𝑥̇(𝑡)=_𝑘=0𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥(𝑡)𝑡𝑘!−𝑥_𝑖^𝑘((𝜏,𝑥𝑡+𝑔𝜏,𝑥))₀=𝑥₀ (53)(54) under the additive forcing perturbation 𝑔∈𝐶^(𝑛+1)(𝐑^𝑛×(𝑡₀,𝑡_𝐽);𝐑^𝑛) satisfying Assumption (A2) of Lemma 4 fulfilling 𝑔(𝑦(𝑡),𝑡)=𝜆(𝑡)𝑦(𝑡) and 𝑔(𝑥(𝑡),𝑡)=𝜆(𝑡)𝑥(𝑡)+𝑈(𝑡−𝑡_𝑖+1)𝑔_𝑖+1; ∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1] with ‖𝑔_𝑖+1‖≤𝑔 for 𝑖=0,1,…,𝐽−1 and some 𝑔∈𝐑₊ and 𝜆∶[𝑡₀,𝑡_𝐽]→𝐑^𝑛 being a bounded piecewise-continuous function. Then, there are numbers ℎ∈𝐑₊, 𝐽=𝐽(ℎ)∈𝐙₊, 𝜀₁=𝜀₁(ℎ,𝑔)∈𝐑₊ and 𝜀=𝜀(𝜀₁,‖𝑒(𝑡₀)‖) such that maxmax_{0≤𝑖≤𝐽−1}𝑒𝑡_𝑖+1−𝑒𝑡_𝑖,max_{𝑡∈𝐑0+}𝑒(𝑡)−𝑒𝑡₀≤𝜀₁,max_{𝑡∈𝐑0+}‖𝑒(𝑡)‖≤𝜌 (55) on [𝑡₀,𝑡_𝐽] for a strictly ordered finite set of (𝐽+1) nonnegative real numbers {𝑡_𝑖∶𝑖=0,1,…,𝐽}, subject to the constraints 𝑡_𝐽=𝑡₀+∑^𝐽−1_𝑖=0ℎ_𝑖, ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ; 𝑖=0,1,…,𝐽−1, the constraints (22)–(24) subject to (18), and either^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖<1,_{𝐽−1𝑖=0}𝑔_𝑖<∞,∑𝑔≥^{𝐽−1𝑖=0}ℎ_𝑖𝜆_𝑖1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖𝑒𝑡₀, (56a)(56b)or𝐽ℎ𝜆<1,𝐽𝑔<∞,𝑔≥𝐽ℎ𝜆𝑒𝑡1−𝐽ℎ𝜆⁰. (57a)(57b)

Proof. Fix ∑^𝐽−1_𝑖=0ℎ_𝑖=𝑡_𝐽−𝑡₀≤𝐽ℎ, with ℎ=max_{0≤𝑖≤𝐽−1}(𝑡_𝑖+1−𝑡_𝑖). Equations (53)-(54) have the following solutions: 𝑦(𝑡)=𝑦𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘+1!^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑦(𝜏)𝑑𝜎+𝑔,𝜏𝑑𝜏,𝑥(𝑡)=𝑥𝑡_𝑖+^{𝑡−𝑡𝑖}_0𝑘=0𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘+𝑔(𝑥(𝜏),𝜏)𝑑𝜏+𝑈𝑡−𝑡_𝑖+1𝑔_𝑖+1; (58)(59)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝐽−1. Note that 𝑔𝑦(𝑡),𝑡−𝑔(𝑥(𝑡),𝑡)=𝜆(𝑡)𝑦(𝑡)−𝑥(𝑡)=𝜆(𝑡)𝑒(𝑡). (60) Thus, the error between both of them becomes 𝑒(𝑡)=𝑒𝑡_𝑖+_𝑘=0^{𝑡−𝑡𝑖}₀𝑓^(𝑘)𝑦𝑡_𝑖,𝑡_𝑖𝑦𝑘!𝜏+𝑡_𝑖−𝑦𝑡_𝑖^𝑘−𝑓^(𝑘)𝑥𝑡_𝑖,𝑡_𝑖𝑥𝑘!𝜏+𝑡_𝑖−𝑥𝑡_𝑖^𝑘(𝑒(𝜏))+1+𝑔,𝜏𝑑𝜏!^{𝑡−𝑡𝑖}₀^𝜏₀𝑦𝜎+𝑡_𝑗−𝑦𝑡_𝑗×𝑓⁽⁺¹⁾𝑦𝜎+𝑡_𝑗,𝜎+𝑡_𝑗𝑑𝜎𝑑𝜏−𝑈𝑡−𝑡_𝑖+1𝑔𝑥𝑡_𝑖+1,𝑡_𝑖+1; (61)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1]. Then, (45) leads to ‖𝑒(𝑡≤𝑒𝑡)‖_𝑖+𝑡−𝑡_𝑖×_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖+1; (62)∀𝑡∈[𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝐽−1. Then sup_{𝑡𝑖≤𝑡≤𝑡𝑖+1}𝑒(𝑡≤𝑒𝑡(‖)‖)_𝑖+𝑡−𝑡_𝑖×_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖+1+ℎ_𝑖𝜆_𝑖sup_{𝑡𝑖≤𝜏≤𝑡𝑖+1}(‖𝑒(𝜏)‖)+𝑔 (63) so that, since 1>ℎ_𝑖𝜆_𝑖, where 𝜆_𝑖=max_{𝑡𝑖≤𝜏≤𝑡𝑖+1}|𝜆(𝜏)| for 𝑖=0,1,…,𝐽−1, one gets sup_{𝑡𝑖≤𝑡≤𝑡𝑖+1}𝑒(𝑡≤1(‖)‖)1−ℎ_𝑖𝜆_𝑖×𝑒𝑡_𝑖+_𝑘=02^𝑘+1ℎ𝑘!^𝑖𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!𝑡−𝑡_𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖+1 (64) which implies sup_{𝑡𝑖≤𝑡≤𝑡𝑖+1}‖𝑒(𝑡)‖−𝑒𝑡_𝑖≤ℎ_𝑖𝜆_𝑖1−ℎ_𝑖𝜆_𝑖𝑒𝑡_𝑖+11−ℎ_𝑖𝜆_𝑖×_𝑘=02^𝑘+1ℎ𝑘!^𝑖𝐾^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1ℎ!^2𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝑔_𝑖+1. (65) If ∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖<1, we also get (66)-(67) below from (65) as well as (68)-(69) if, in addition, 𝐽ℎ𝜆<1: sup_{𝑡0≤𝑡≤𝑡𝐽}𝑒(𝑡≤1(‖)‖)1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖×𝑒𝑡₀+^𝐽−1_𝑖=0ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1ℎ!^𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+^𝐽−1_𝑖=0𝑔_𝑖,sup_{𝑡0≤𝑡≤𝑡𝐽}‖𝑒(𝑡)‖−𝑒𝑡₀≤∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖×𝑒𝑡₀+^𝐽−1_𝑖=0ℎ_𝑖𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1ℎ!^𝑖2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+^𝐽−1_𝑖=0𝑔_𝑖sup_{𝑡0≤𝑡≤𝑡𝐽}𝑒(𝑡≤1(‖)‖)×𝑒𝑡1−𝐽ℎ𝜆₀+𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔,sup_{𝑡0≤𝑡≤𝑡𝐽}‖𝑒(𝑡)‖−𝑒𝑡₀≤𝐽ℎ𝜆1−𝐽ℎ𝜆×𝐽ℎ_𝑘=02^𝑘+1𝐾𝑘!^𝑘+1𝜌2+𝐾₁1−𝐾^𝑘𝜌1−𝐾2^𝑘+1!ℎ2×𝐾⁺²𝜌2+𝐾₁1−𝐾⁺¹𝜌1−𝐾2+𝐽𝑔, (66)(67)(68)(69) where 𝜆=max_{0≤𝑡≤𝑡𝐽}|𝜆(𝑡)|=max_{0≤𝑖≤𝐽−1}max_{𝑡𝑖≤𝜏≤𝑡𝑖+1}|𝜆(𝜏)|=max_{0≤𝑖≤𝐽−1}𝜆_𝑖. Property (i) follows from (65)-(66) by defining 𝜀₀, 𝜀₁, and 𝜀 as in (51) since 𝑔≥max(max_{0≤𝑖≤𝐽−1}‖𝑔_𝑖+1‖,((∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖)/(1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖))‖𝑒(𝑡₀)‖) and Property (ii) follows from (67)-(68) by defining 𝜌, 𝜀₁, and 𝜀 as in (52) since 𝑔≥max(max_{0≤𝑖≤𝐽−1}‖𝑔_𝑖+1‖,((𝐽ℎ𝜆)/(1−𝐽ℎ𝜆))‖𝑒(𝑡₀)‖). Thus, the result has been proven.

Now, three definitions are given concerning the so-called pseudo-orbits, as a counterpart to the true sampled trajectory solution, or orbit, of finite size 𝐽 of the approximate solutions and their perturbed version within the given classes of perturbations. The related concepts are relevant for then quantifying the maximum errors among the real and approximated solutions and parallel issues concerning their counterparts under perturbations of the studied types. More specifically refer to the following.

Definition 9. A sampling sequence 𝑡_𝐽={𝑡_𝑖∶𝑖=0,1,…,𝐽} of strictly ordered sampling points with ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ; 𝑖=0,1,…,𝐽−1 is said to be in the class 𝐶_𝐽ℎ={𝑡_𝑖∈𝑡_𝐽∶𝑡_𝑖+1−𝑡_𝑖≤ℎ;𝑖=0,1,…,𝐽−1}.

Note from Definition 9 that ℎ≤ℎ^′⇒𝐶_𝐽ℎ⊆𝐶_𝐽ℎ′ and that 𝑡_𝐽≡{𝑡_𝑖∶𝑖=0,1,…,𝐽−1}⊂𝐶_𝐽ℎ⇒𝑡_𝐽−𝑡₀≤𝐽ℎ.

Definition 10. A sequence 𝑥_𝐽={𝑥(𝑡_𝑖)∶𝑖=0,1,…,𝐽−1} of 𝐽 samples of the solution of an approximate differential equation (10) is a 𝛿-pseudo 𝐽-orbit of sampling sequence 𝑡_𝐽 for some 𝛿∈𝐑₊ and is denoted by 𝑂(𝑥_𝐽,Γ,𝛿) if max_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑒(𝑡)‖≤𝛿.

If the integer 𝐽 and the real 𝑡_𝐽 are infinite, the corresponding trajectory solutions are referred to as complete pseudo-orbits and orbits. The solution of the true differential equation (5) is a 𝐽-orbit of sampling sequence 𝑡_𝐽. The continuous approximate (resp., true) solution for [𝑡₀,𝑡_𝐽] is the 𝛿-pseudo 𝐽-orbit (resp., 𝐽-orbit) of sampling sequence 𝑡_𝐽. The perturbed solutions under the forcing perturbations of Theorems 7 and 8 are denoted in a similar way leading to the corresponding perturbed pseudo-orbits.

Definition 11. The set of all the 𝛿-pseudo 𝐽-orbits 𝑂(𝑥_𝐽,Γ,𝛿) with max_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑒(𝑡)‖≤𝛿, for some 𝛿∈𝐑₊, obtained for any sampling sequence 𝑡_𝐽∈𝐶_𝐽ℎ, is said to be the class 𝐶𝑂(𝐶_𝐽ℎ,𝛿) of 𝛿-pseudo 𝐽-orbits of sampling sequence 𝑡_𝐽.

The mapping which generates the true solution sequences, for given initial conditions and sampling sequence, has the shadowing property if there is an arbitrarily close orbit for any given 𝛿-pseudo-orbit 𝑂(𝑥_𝐽,Γ,𝛿) in the following precise sense.

Definition 12. The set 𝑌_𝐽 of true solution sequences 𝑦_𝐽={𝑦(𝑡_𝑖)∶𝑡_𝑖∈𝑡_𝐽,𝑖=0,1,…,𝐽−1} of sampling sequence 𝑡_𝐽 possesses the shadowing property on the corresponding set of approximate solutions if, for each given 𝛿∈𝐑₊, there is some 𝑦₀=𝑦₀(𝛿) for which a 𝑂(𝑥_𝐽,Γ,𝛿) exists. It is said that 𝑦₀=𝑦₀(𝛿) shadows 𝑂(𝑥_𝐽,Γ,𝛿).

The subsequent result establishes that if the set of true solution sequences has the shadowing property then the class 𝐶𝑂(𝐶_𝐽ℎ,𝛿) of 𝛿-pseudo 𝐽-orbits of sampling sequence 𝑡_𝐽 is nonempty for any 𝛿∈𝐑₊.

Proposition 13. If the set 𝑌_𝐽 of true solution sequences of sampling sequence 𝑡_𝐽 possesses the shadowing property then 𝐶𝑂(𝐶_𝐽ℎ,𝛿) is nonempty for any 𝛿∈𝐑₊.

Note that 𝐶𝑂(𝐶_𝐽ℎ,𝜌)=⋃_{Γ∈𝐶𝐽ℎ}𝑂(𝑥_𝐽,Γ,𝜌) and note also that 𝐶𝑂(𝐶_𝐽ℎ,𝜌)⊆𝐶𝑂(𝐶_𝐽ℎ′,𝜌) for any ℎ^′≥ℎ. The subsequent result relies on Theorem 7 and Definition 11 for a class of pseudo-orbits 𝐶𝑂(𝐶_𝐽ℎ,𝜌) defined by a sampling sequence class 𝐶_𝐽ℎ. In fact, the characterization becomes global for all approximated solutions on a finite interval [𝑡₀,𝑡_𝐽] for sampling intervals ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖≤ℎ; 𝑖=0,1,…,𝐽−1 and initial conditions subject to a maximum allowable deviation with respect to the initial condition of the true solution provided that the approximate solution exists in a global (rather than local) definition domain.

The so-called shadowing properties, [5–9], of the true solutions with respect to the approximated ones rely on the physical meaning that for sets of appropriate initial conditions, the true solution is arbitrarily close to its approximate version on a certain interval [𝑡₀,𝑡_𝐽] where both solutions exist and are unique. Based on Theorems 6, 7, and 8, the shadowing properties of the true solution to the approximated solution, those ones being the nominal one or the perturbed ones under the class of perturbations of Theorems 7 and 8, are now discussed. It is seen that the shadowing properties at sampling points under Theorems 6, 7, and 8 guarantee the corresponding properties in [𝑡₀,𝑡_𝐽].

The shadowing properties of true solutions of pseudo-orbits for constrained sampling sequences according to the constraints of Theorem 6 are addressed in the subsequent result.

Proposition 14. Consider the true and approximated solutions associated with the differential equations (5) and (10) satisfying the hypotheses and conditions of Theorem 6. Then, such a set of solutions lies in the class 𝐶𝑂(𝐶_𝐽ℎ,𝜀) of 𝜀-pseudo 𝐽-orbits of sampling sequence 𝑡_𝐽=𝑡_𝐽(𝜌) for 𝜌≤𝜀, subject to one of the constraints (13a), (13b), (22)–(24) (Lemma 4, Theorem 6), belonging to a sampling sequence class 𝐶_𝐽ℎ for any 𝜌,𝜀∈𝐑₊ with arbitrary 𝜌≤𝜀 and any given 𝜀. Also, there is a 𝑦₀=𝑦₀(𝜀) which shadows each 𝑂(𝑥_𝐽,Γ,𝜀)∈𝐶𝑂(𝐶_𝐽ℎ,𝜀) for each given 𝜀∈𝐑₊ and 𝜌≤𝜀.

Proof. One gets from Theorem 6 that max_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑒(𝑡)‖≤𝜀=𝑒𝑡₀+𝜌 (70) for an initial condition 𝑦(𝑡₀) of the true differential equation fulfilling |‖𝑦(𝑡₀)‖−‖𝑥(𝑡₀)‖|≤‖𝑒(𝑡₀)‖ and any given real constants 𝜀≥𝜌>0. This defines families of initial conditions 𝑦₀=𝑦₀(𝜀) of the true differential equation which shadow each 𝑂(𝑥_𝐽,Γ,𝜀)∈𝐶𝑂(𝐶_𝐽ℎ,𝜀) for each given 𝜀∈𝐑₊ and 𝜌≤𝜀. For any given 𝜀∈𝐑₊, it suffices to take 0<𝜌≤𝜀 to zero in (22)–(24) of Lemma 4 and (39)-(40) in Theorem 6 to fix an admissible sampling sequence 𝑡_𝐽=𝑡_𝐽(𝜀) and then to get the result.

The perturbed approximated differential equations referred to in Theorems 8 and 7, which is a particular case of Theorem 8 for 𝑔(𝑥(𝑡),𝑡) being zero for 𝑡∉𝑡_𝐽, that is for nonsampling points, are analyzed in the subsequent result which generalizes Proposition 14.

Theorem 15. Consider the true and approximated solutions (58) and (59) associated with the differential equations satisfying the hypotheses and conditions of Theorem 8. Then, such a set of solutions lies in the class 𝐶𝑂(𝐶_𝐽ℎ,𝜀) of 𝜀-pseudo 𝐽-orbits of sampling sequence 𝑡_𝐽=𝑡_𝐽(𝜌), for some 𝜌>0, subject to one of the constraints (13a), (13b), (22)–(24), and (39)-(40) (Lemma 4, Theorem 6) and to either (56a) or (56b) (Theorem 8) with 𝜌≤𝜀−∑^𝐽−1_𝑖=0𝑔_𝑖 and any arbitrary 𝜀∈𝐑₊, belonging to a sampling sequence class 𝐶_𝐽ℎ. Also, there is an initial condition 𝑦₀=𝑦₀(𝜀) which shadows each 𝑂(𝑥_𝐽,Γ,𝜀)∈𝐶𝑂(𝐶_𝐽ℎ,𝜀) for each given 𝜌,𝜀∈𝐑₊ with the perturbation fulfilling ∑^𝐽−1_𝑖=0𝑔_𝑖<𝜀.

Proof. One gets from (55) in Theorem 8 together with either (51) or (52) that max_{𝑡∈[𝑡0,𝑡𝐽]}‖𝑒(𝑡)‖≤𝜀=𝑒𝑡₀+𝜌+^𝐽−1_𝑖=0𝑔_𝑖 (71) with any arbitrary real constant 0<𝜌≤𝜀−∑^𝐽−1_𝑖=0𝑔_𝑖, provided that ∑^𝐽−1_𝑖=0𝑔_𝑖<𝜀, and any given real constant 𝜀 for an (shadowing) initial condition 𝑦(𝑡₀) of the true differential equation fulfilling 𝑦𝑡₀−𝑥𝑡₀≤𝑒𝑡₀≤min𝜀−𝜌−^𝐽−1_𝑖=0𝑔_𝑖,1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖𝑔. (72) Note that a sufficient condition guaranteeing (69) is 𝑒𝑡₀≤min𝜀−𝜀₀−𝐽𝑔,1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖𝑔 (73) since 𝑔≥((∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖)/(1−∑^𝐽−1_𝑖=0ℎ_𝑖𝜆_𝑖))‖𝑒(𝑡₀)‖ from (56b) in Theorem 8. Thus, it suffices to take 0<𝜌≤𝜀 to zero in either (39) or (40) in Theorem 6 to fix an admissible sampling sequence 𝑡_𝐽=𝑡_𝐽(𝜀) so as to get the result.

Remark 16. A particular case of Theorem 15 for the perturbations (42) which are defined only at sampling instants, which has been discussed in Theorem 7, is obtained by the particular constraint below obtained from (72) and (73): 𝑒𝑡₀≤𝜀−𝜌−𝐽𝑔≤𝜀−𝜌−^𝐽−1_𝑖=0𝑔_𝑖. (74)

Remark 17. Note that the condition ∑^∞_𝑖=0ℎ_𝑖𝜆_𝑖≤𝜒<1 of applicability in Theorems 8 and 15 when 𝐽 is infinity can be considered in certain cases when the perturbation vanishes asymptotically as, for instance, when it vanishes as an exponential rate.

4. Simulation Examples

This section contains two numerical examples regarding the theoretical results obtained in Sections 2 and 3.

Example 1. The first example is concerned with the nonlinear model describing the human heart rate during treadmill exercise [29], whose equations are given by 𝑦̇₁=−𝑎₁𝑦₁+𝑎₂𝑦₂,𝑦̇₂=−𝑎₃𝑦₂+𝑎₄𝑦₁1+𝑒^{−(𝑐𝑦1−𝑎5)} (75) with 𝑐=1, 𝑎₁=2.2, 𝑎₂=19.96, 𝑎₃=0.0831, 𝑎₄=0.002526, and 𝑎₅=8.32. This model, with an external control input, has been used to design training protocols for patients with cardiovascular problems [29]. Figure 1 shows the evolution of this system with initial conditions 𝑦₁(0)=𝑦₂(0)=1 on the time interval [0,50] seconds.

Figure 1 

State evolution for the system (75).

For the system (75), the nonlinear function 𝑓(𝑦) is given by 𝑓𝑦=𝑓𝑦₁,𝑦₂=−𝑎₁𝑦₁+𝑎₂𝑦₂−𝑎₃𝑦₂+𝑎₄𝑦₁1+𝑒^{−(𝑐𝑦1−𝑎5)}. (76) The first step to apply the results stated in Section 2 and obtain a truncated approximate model for (75) is to verify that conditions (13a) and (13b) hold. One way to check this fact is to depict the norms of the state vector of the function 𝑓(𝑦) and of its derivative 𝑓^′(𝑦) and observe their behavior. Thus, the following Figures 2 and 3 show the time evolution of these norms. In particular, Figure 2 shows the values of the 2-norm of the state, ‖𝑦‖₂, and the function, ‖𝑓(𝑦)‖₂. The supremum of these norms on this interval are sup_{𝑡∈[0,50]}‖𝑦‖₂=8.94 and sup_{𝑡∈[0,50]}‖𝑓(𝑦)‖₂=17.84. On the other hand, Figure 3 shows the difference between the norm of the function, ‖𝑓(𝑦)‖₂, and the norm of its derivative ‖𝑓^′(𝑦)‖₂. As it can be appreciated, this difference is positive implying that the linear approximation of the function is always bounded by the function itself. The supremum of these two norms on this interval is sup_{𝑡∈[0,50]}‖𝑓(𝑦)‖₂=17.84 and sup_{𝑡∈[0,50]}‖𝑓^′(𝑦)‖₂=17.76. In this way, if we choose 𝐾=0.997 and 𝐾₁=9, we have 17.84=sup_{[]𝑡∈0,50}𝑓(𝑦)₂<0.997sup_{[]𝑡∈0,50}𝑦₂+9=17.91,17.76=sup_{[]𝑡∈0,50}𝑓^′(𝑦)₂<0.997sup_{[]𝑡∈0,50}𝑓(𝑦)₂=17.78. (77) Hence, it is corroborated that both (13a) and (13b) hold. Notice that from a practical point of view, the analytical determination of the constants 𝐾 and 𝐾₁ used in (13a) and (13b) is not necessary since a simple numerical experiment allows us to verify these upper-bounds. In consequence, the results stated in Section 2 can be applied in practical situations easily.

Figure 2 

Relation between the 2-norms of the state, 𝑦, and the function, 𝑓(𝑦).

Figure 3 

Relation between the 2-norms of the function, 𝑓(𝑦), and its derivative, 𝑓′(𝑦).

Once the basic conditions have been checked, a truncated approximate model (10) is generated for this problem by considering =1<2=𝑛. Thus, we have 𝑥̇=𝑓𝑥_𝑖+𝐽𝑥_𝑖𝑥(𝑡)−𝑥_𝑖, (78) where 𝑓^′𝑥_𝑖=𝐽𝑥_𝑖=−𝑎₁𝑎₂𝐽₂₁−𝑎₃,𝐽₂₁=𝑎₄1+𝑒^{−(𝑐𝑥𝑖1−𝑎5)}+𝑎₄𝑐𝑥_𝑖1𝑒^{−(𝑐𝑥𝑖1−𝑎5)}1+𝑒^{−(𝑐𝑥𝑖1−𝑎5)2} (79) with initial conditions 𝑦₁(0)=𝑦₂(0)=𝑥₁(0)=𝑥₂(0)=1. The sampling instants {𝑡_𝑖} have been chosen uniformly in time as 𝑡_𝑖−𝑡_𝑖−1=ℎ=1.5s. The error between the actual and the approximate model with this sampling time is depicted in Figure 4.

Figure 4 

Error between the actual system and the approximate model.

As it can be deduced from Figure 4, the error is very low and, therefore, the exact solution is shadowed by the solution of the approximate model. An important feature appears at this point which is how we should select the sampling time. Lemma 4 and Theorem 6 contain the analytical results providing the formal background on how to select it. However, from a practical point of view a trial-error procedure can be employed to obtain an appropriate sampling time. Thus, Figures 5 and 6 show how a variation in the sampling time impacts the error between the complete system and the approximate model. Figure 5 displays the impact on the first state variable while Figure 6 depicts the influence in the second one. As it is displayed in Figures 5 and 6, the larger the sampling time is, the larger the error between both systems is, as well. Hence, if we fix an upper-bound for the desired error, we may start with a tentative value for the sampling time and increase it if the maximum of the error is below that threshold or decrease it if the error exceeds the desired bound. This procedure allows us to tune an appropriate sampling time by just conducting a series of numerical experiments. Therefore, the mathematical results presented in Section 2 can be applied in a practical way with little effort since the computation of the bounds is not explicitly necessary to construct the approximate truncated model. Afterwards, this approximate model could be used for simulation or control design purposes. For instance, the obtained affine model could simplify the design of the controller with respect to the case when the original nonlinear model is used.

Figure 5 

Effect of the variation of the sampling time, ℎ, in the error of the first state variable, 𝑥1(𝑡).

Figure 6 

Effect of the variation of the sampling time, ℎ, in the error of the second state variable, 𝑥2(𝑡).

Example 2. The second example is related to the Van der Pol equation which exhibits a limit cycle as it is widely known. The equations are given by 𝑦̇₁=𝑦₂,𝑦̇₂=𝜇1−𝑦²₁𝑦₂−𝑦₁ (80) with 𝜇=1, output 𝑧(𝑡)=𝑦₁(𝑡), initial conditions 𝑦₁(0)=4 and 𝑦₂(0)=−0.5, and 𝑓𝑦=𝑓𝑦₁,𝑦₂=𝑦₂𝜇1−𝑦²₁𝑦₂−𝑦₁. (81) The phase portrait of the Van der Pol equation is depicted in Figure 7.

Figure 7 

Phase portrait of the Van der Pol equation.

In this case, the results introduced in Section 3 regarding the error between the actual and the approximate model in the presence of bounded perturbations will be used as a tool to analyze the stability of the limit cycle. Thus, the approximate truncated model is given with =1<2=𝑛 by 𝑥̇=𝑓𝑥_𝑖+𝐽𝑥_𝑖𝑥(𝑡)−𝑥_𝑖, (82) where 𝑓^′𝑥_𝑖=𝐽𝑥_𝑖=𝐽01₂₁𝐽₂₂,𝐽₂₁=−2𝜇𝑥_𝑖1𝑥_𝑖2𝐽−1,₂₁=𝜇−𝜇𝑥²_𝑖1. (83) In this example, the sampling points 𝑥_𝑖 will be generated by using the constant amplitude difference sampling criterion (CADSC) introduced in [4] as a method to generate sampling points in discretization procedures. This method is proposed as a way to generate the sequence of sampling points in a practical way, which shows that the application of the presented theories to real problems is feasible. Thus, the CADSC method proposes to generate a new sampling point when the continuous-time output differs from the previous sampled one a certain threshold. Mathematically, 𝑡_𝑖+1=argmin𝑅₀⁺϶𝑡>𝑡_𝑖∶𝑥₁(𝑡)−𝑥₁𝑡_𝑖=𝛿_𝑖∈𝑅₊, (84) where 𝛿_𝑖 denotes the variation threshold. For this example, consider a constant threshold with a value of 𝛿_𝑖=𝛿=0.15. Figures 8 and 9 display the solution of the actual and the approximate systems.

Figure 8 

Phase portrait of the actual and approximate systems.

Figure 9 

Zoom on the phase portrait of the actual and approximate systems.

It can be appreciated in Figures 8 and 9 that the solution of the actual system is shadowed by one of the approximate models, confirming the results stated in Proposition 14. As the threshold 𝛿 on the sampling criterion enlarges, the sampling takes place in a more separate way, a fact that degrades the quality of the approximate solution as Figure 10 reveals.

Figure 10 

Effect of the variation of the sampling threshold 𝛿 in the approximate model.

Figure 10 shows that the larger the threshold is, the smaller the approximation capabilities of the truncated model is. This feature is due to the fact that a larger threshold implies a greater separation between the sampling points. In Figure 10, 𝛿 is modified from 0.15 to 0.4. Thus, as Lemma 4 states, a large intersampling period might lead to higher errors in the approximated model. At this point we can introduce a bounded perturbation 𝑔(𝑡_𝑖) at sampling points to analyze the stability of the limit cycle. For this, we can firstly select a value for the threshold 𝛿 in such a way that the solution of the approximate model shadows the one of the actual system. Afterwards, we can apply different perturbations to the system in an increasing way. If the limit cycle preserves its shape under this scheme, this would indicate that it is stable. This procedure has been applied in Figure 11.

Figure 11 

Stability of the limit cycle under increasing perturbations.

Since the shape of the limit cycle is maintained, the stability of the original system is deduced from one of the approximate truncated models. Moreover, the shadowing property can be interpreted in terms of stability of the limit cycle in the following way. If the error between the solutions of the actual system and the approximate one under the same perturbation is less in a system 𝐴 than in another system 𝐵, this means that 𝐴 is more stable than 𝐵. Thus, the shadowing property can be viewed as a concept to measure the relative stability of systems by using its Taylor series expansion and construction of approximate models. For instance, consider the van del Pol equation with three different values of 𝜇∈{0.1,1,2}. The behavior of the van der Pol equation depends on the value of 𝜇 as it is widely recognized. Thus, the approximate perturbed model can be used to compute the error between the actual and reduced models in each case and determine for which value of 𝜇 the Van der Pol equation is “more stable,” that is, has a greater relative stability. Thus, we fix the perturbation amplitude to 𝑔(𝑡_𝑖)=0.35 and carry out some numerical experiments with the different values for 𝜇. In this way, Figure 12 displays the error in the first state variable between the actual and approximate models while Figure 13 shows the error in the second state variable for each value of 𝜇.

Figure 12 

Error between 𝑦1(𝑡) and 𝑥1(𝑡) for different values for 𝜇.

Figure 13 

Error between 𝑦2(𝑡) and 𝑥2(𝑡) for different values for 𝜇.

Figures 12 and 13 show that the larger 𝜇 is, the larger the peak error is. Therefore, in this case, systems with smaller 𝜇 have a greater relative stability. Finally, this approximate affine model could be used, as in the previous example, to design a control system based on a reduced model, rather than using the complete nonlinear one. In conclusion, the results presented in the previous sections have been applied in some case studies with little effort, a fact that backs up its potential practical applications.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07.