A Hartman-Grobman theorem for algebraic dichotomies

Algebraic dichotomy is a generalization of an exponential dichotomy (Lin, JDE2009). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem. Besides, we prove that the homeomorphism in the linearization theorem (and has a H\"{o}lder continuous inverse). Comparing with exponential dichotomy, algebraic dichotomy is more complicate. The exponential dichotomy leads to the estimates $\int_{-\infty}^{t}e^{-\alpha(t-s)}ds$ and $\int_{t}^{+\infty}e^{-\alpha(s-t)}ds$ which are convergent. However, the algebraic dichotomy will leads us to $\int_{-\infty}^{t}\left(\frac{\mu(t)}{\mu(s)}\right)^{-\alpha}ds$ or $\int_{t}^{+\infty}\left(\frac{\mu(s)}{\mu(t)}\right)^{-\alpha}ds$, whose the convergence is unknown in the sense of Riemann.


Brief history on dichotomy and C 0 linearization
In 1930, Perron [2] introduced the concept of the (classical or uniform) exponential dichotomy. Exponential dichotomy theory plays an important role in the differential equations.
On the other hand, linear equations are mathematically well-understood but nonlinear systems are relatively difficult to investigate. For this reason, linearization of differential equations is very important. A basic contribution to the linearization problem for autonomous differential equations is the Hartman-Grobman theorem (see [24,25]). Some improvements of the Hartman-Grobman theorem to infinite dimensional space can be found in Bates and Lu [26], Hein and Prüss [27], Lu [28], Pugh [29] and Reinfelds [30,31]. Palmer successfully generalized the Hartman-Grobman theorem to nonautonomous differential equations (see [32]) y ′ = A(t)y + f (t, y).

Motivation and novelty
In this paper, we pay particular attention to the effect of the algebraic dichotomy imposing on the linearization of the differential equations. Palmer's linearization theorem requires two essential conditions: (i) the nonlinear term f is uniformly bounded and Lipschitzian; (ii) the linear system possesses an exponential dichotomy. In the present paper, we try to reduce the second condition. Motivated by Lin's algebraic dichotomy (see Lin [1]) and the works of Palmer [32] and Zhang et al [12], we study the C 0 linearization with the algebraic dichotomy. Further more, we prove that the homeomorphism and its inverse are Hölder continuous under the assumption of the algebraic dichotomy. When the algebraic dichotomy reduces to the exponential dichotomy, our results generalize and improve the previous ones. Comparing with exponential dichotomy, algebraic dichotomy is more general. The exponential dichotomy leads to the estimates t −∞ e −α(t−s) ds and +∞ t e −α(s−t) ds which are convergent. However, the algebraic dichotomy will leads us to −α ds, whose the convergence is unknown in the sense of Riemann. This brings more difficulties to our research.
The structure of our paper as follows. In Section 2, we give our main results. In Section 3, we give some preliminary results. In Section 4, we give rigorous proofs to show the regularity of the equivalent function H(t, x) and G(t, y). Finally, we give an example to illustrate our linearization theorem.
Consider the following two non-autonomous systems is a n × n continuous and bounded matrix defined on for any solution x(t) of system (2.2). Clearly, T (t, t) = Id and In the following, we always assume that µ(t) is growth rate.
To show its generality, we give an example of algebraic dichotomy.
From system (2.7), we get where V 1 , V 2 are constants.
Definition 2.5. [9] We say that h and k are fulfill a compensation law on R if there exists a positive constant C h,k such that The system is said to be a h-system, if it has a (h, h) dichotomy.
Clearly, a system having an (h, k)-dichotomy with compensation law belongs to the class of h-systems.
Definition 2.6. Suppose that there exists a function H : R × R n → R n such that (i) for each fixed t, H(t, ·) is a homeomorphism of R n into R n ; (ii) ||H(t, x) − x|| is uniformly bounded with respect to t; (iii) assume that G(t, ·) = H −1 (t, ·) also has property (ii); (iv) if x(t) is a solution of system ( 2.1), then H(t, x(t)) is a solution of system ( 2.2); and if y(t) is a solution of system ( 2.2), then G(t, y(t)) is a solution of system ( 2.1).
If such a map H t (H t := H(t, ·)) exists, then system ( 2.1) is topologically conjugated to system ( 2.2) and the transformation H(t, x) is called an equivalent function.
Now we are in a position to state our main results.
Theorem 2.7. Suppose that system ( 2.2) admits an algebraic dichotomy, and f (t, x) satisfies Then nonlinear system ( 2.1) is topologically conjugated to their linear partẋ = A(t)x, and the equivalent function Denote H −1 (t, ·) = G(t, ·), then G(t, y) also satisfies Theorem 2.8. Suppose that the conditions in Theorem 2.7 are satisfied. Moreover, assuming that α > γ. Then there exist constants p, q > 0, 0 < p ′ , q ′ < 1 such that Remark 2.9. Assuming that µ(t) = e t in Theorem 2.7, Theorem 2.7 reduces to the classical Palmer linearization theorem, (see [32]). However, Palmer did not study the regularity of equivalent function H(t, x). We remark that there are good results for the linearization of (h, k, µ, ν)-dichotomy ( [12,13]). However, they did not study the regularity of homeomorphisms mapping the nonlinear systems onto its linearization.

Preliminary results
We split the proof of Theorem 2.7 into several lemmas. Suppose that X(t, t 0 , x 0 ) is the solution of system (2.1) with the initial value condition X(t 0 ) = x 0 , and Y (t, t 0 , y 0 ) is the solution of system (2.2) with the initial value condition Y (t 0 ) = y 0 .
We start with a fundamental lemma, which shows that linear system (2.2) has no other bounded solutions except for the zero solution under our hypothesis. Proof. Let x(t) be the bounded solution of system (2.2). There exists n order real vector x(s), such that If P (s)x(s) = 0, consider the case t ≤ s. We have From (2.3), (2.5) and (2.6), we get Thus, we obtain Hence, This contradicts with the boundedness of x(t), thus P (s)x(s) = 0. Similarly, we get Q(s)x(s) = 0, when t > s. Thus, This conclusion is key to the following lemmas.
Proof. For any given (τ, ξ), taking Differentiating Z 0 (t), we get Z 0 (t) is a solution of system (3.2). Now we prove that Z 0 (t) is the unique bounded solution of system (3.2).

Proof.
Let Defining the following mapping: Furthermore, Thus, F is a self-mapping in Ω. Moreover, Now we prove the uniqueness. Let Z 2 (t) be another bounded solution. By the variation formula, we get Since )ds is convergent, and denoting it as x 1 . Thus, we get Therefore, Thus, Z 1 (t) = Z 2 (t). The bounded solution of system (3.3) is unique. This solution is related to (τ, ξ), denoting it as g(t, (τ, ξ)) . From the above proof, we get Lemma 3.4. Let x(t) be any solution of system ( 2.1), system has a unique bounded solution Z = 0.
Proof. Obviously, Z = 0 is a bounded solution of system (3.5). Next, we prove the uniqueness of the bounded solution. Let Z 3 (t) be another bounded solution. By the variation formula, we get Similar to the proof of Lemma 3.3, we get

This shows that M(t) is a bounded solution of
Thus, y(t) = y 0 (t), H(t, G(t, y)) = y.

Proofs of main results
Proof of Theorem 2.7. We are going to show that H(t, ·) satisfies the four conditions of Definition 2.1.
From Lemmas 3.5 and 3.6, we know that Condition (iv) is true. Hence, the system (2.1) and its linear system (2.2) are topologically conjugated.
Proof of Theorem 2.8. We prove this theorem in two steps.
This completes the proof of Theorem 2.8.

Data Availability Statement
My manuscript has no associated data. It is pure mathematics.

Conflict of Interest
The authors declare that they have no conflict of interest.