The two-dimensional anti-Perron effect of changing all positive characteristic exponents of the linear approximation to four different negative exponents, respectively, of four non-trivial solutions of a differential system with a perturbation of higher order of smallness, has been realized.

We prove the existence of a two-dimensional linear system ẋ=A(t)x , t≥ t_0 , with bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an infinitely differentiable m -perturbation f(t,y) having an order m>1 of smallness in a neighborhood of the origin y=0 and an order of growth not exceeding m outside it, such that the perturbed system ẏ=A( t)y+ f(t,y) , y∈ℝ^2 , t≥ t_0 , has a solution y(t) with a negative Lyapunov exponent.

We prove the existence of a two-dimensional linear system x˙ = A(t)x, t ≥ t0, withbounded infinitely differentiable coefficients and all positive characteristic exponents, as wellas an infinitely differentiable m-perturbation f(t, y) having an order m 1 of smallness ina neighborhood of the origin y = 0 and an order of growth not exceeding m outside it, such thatthe perturbed system y˙ = A(t)y + f(t, y), y ∈ R2, t ≥ t0, has a solution y(t) with a negativeLyapunov exponent.

A linear version of the anti-Perron effect of changing all positive Lyapunov characteristic exponents to negative ones is realized. For arbitrary numbers λ _n≥…≥λ _1>0 and μ _1≤…≤μ _n<0 , the existence of the following n -dimensional linear systems is proved: an original system ẋ= A(t)x , t≥ t_0 , with characteristic exponents λ _i(A)=λ _i , i=1,… ,n , and a perturbed system ẏ= A(t)y+Q(t)y with perturbation matrix Q(t)→ 0 as t→∞ and characteristic exponents λ _i(A+Q)=μ _i , i =1,… ,n . Moreover, the coefficient matrices of the original and perturbed differential systems are bounded and infinitely differentiable on the half-line [t_0,+∞ ) .

We prove the existence of $$n$$ -dimensional lineardifferential systems with the first approximation having all positive characteristic exponents, withexponentially decaying perturbations, and with exactly $$n-1$$ linearly independent solutions with negativeLyapunov exponents. Thus, in the linear case we obtain an anti-Perron version—a versionopposite to the well-known Perron effect of changing the values of negative exponents of the linearapproximation to positive ones for solutions of a differential system with a perturbation ofhigher-order smallness in a neighborhood of the origin and admissible growth outside it.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

For any parameters $$m>1$$ , $$\lambda _1\le \lambda _2<0 $$ , and $$\varepsilon >0 $$ and for two sequences $$\{S_{in}\} $$ of uniformly bounded arbitrary Suslin sets $$S_{1n}\subset [\lambda _1+\varepsilon ,b_1]$$ and $$S_{2n}\subset [\max \{\lambda _2+\varepsilon ,b_1\},b_2]$$ , we prove the existence of a two-dimensional nonlinear differential system with a linear approximation that has characteristic exponents $$\lambda _1$$ and $$\lambda _2 $$ and with a disturbance of the $$m $$ th order of smallness in a neighborhood of the origin and possible growth outside it such that all nontrivial solutions of this system are infinitely extendible and have finite Lyapunov exponents. For any $$n\in \mathbb {N} $$ , these exponents form the following sets: $$S_{1n} $$ for solutions with initial values $$(c_1,0)\ne 0 $$ , where $$|c_1|\in (n-1,n] $$ , and $$S_{2n} $$ for solutions with initial values $$(c_1,c_2) $$ where $$|c_2|\in (n-1,n] $$ . In particular, for any bounded Suslin sets $$S_{-}\subset (-\infty ,0)$$ and $$S_{+}\subset (0,+\infty ) $$ we have also established the existence of a nonlinear system whose Lyapunov exponents for all nontrivial solutions form these two sets (singletons in the Perron case).

For an arbitrary bounded Suslin set S ⊂ (0, +∞) and arbitrary parameters m > 1 and λ1 ≤ λ2 < 0, we construct a two-dimensional differential system ẏ = A(t)y + f (t, y), y ∈ ℝ2, t ≥ t0, with infinitely differentiable matrix A(t) and with vector function f (t,y) infinitely differentiable with respect to its arguments such that all of its nonzero solutions are infinitely extendable to the right and S is their set of characteristic exponents. Further, the characteristic exponents of the linear approximation system ẋ = A(t)x, x ∈ ℝ2, are λ1(A) = λ1 ≤ λ2(A) = λ2, its coefficients are bounded on the half-line [t0, +∞), and the perturbation f (t, y)is of order m > 1 in a neighborhood of the origin y = 0 and of an admissible order of growth outside it: ‖ f (t,y)‖ ≤ const ‖y‖m, y ∈ ℝ2, t ⁥ t0.

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y0) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y0)] of these solutions is a function of the second Baire class of their initial vectors y0 ∈ ℝn {0}.

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.

We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R 2.

In the paper [ Differ. Uravn ., 2007, vol. 43, no. 2, pp. 191–202], we defined the noncoinciding irreducibility sets N 2 ( a, σ ) and N 3 ( a, σ ), σ ∈ (0, 2 a ], of all n -dimensional linear differential systems with piecewise continuous coefficient matrices A(t) bounded on the half-line [0,+∞) with norms || A ( t )|| ≤ a < +∞ for each of which there exists a linear differential system that cannot be reduced to it by Lyapunov transformations and whose coefficient matrix B ( t ) satisfies the condition || B ( t ) - A ( t )|| ≤ const × e − σt , t ≥ 0, or the more general condition that the Lyapunov exponent of the difference B ( t ) - A ( t ) does not exceed - σ , respectively. In the present paper, we study the properties of irreducibility sets treated as functions of the parameters σ and a .