Research Proposal Lyapunov method in stability analysis of invariant tori Petro Feketa, University of Applied Sciences Erfurt, Germany

Summary Many systems in science and engineering can be modelled as coupled or forced oscillators which behavior may have quasiperiodic character. The central object of investigation in the quasiperiodic oscillations theory is invariant torus which is a carrier of quasiperiodic motions. The properties of stability, attractivity and robustness of the invariant torus are vital to understand the behavior of solutions to dynamical system in the vicinity of invariant manifold. One of the powerful tool to study stability properties of invariant torus of nonlinear system is Lyapunov method. It links the existence of a sign-definite function that satisfies specific dissipation conditions (a so-called Lyapunov function) with stability properties of invariant manifold. The constraints imposed on Lyapunov function has to be satisfied in the vicinity of a torus in order to guarantee the desired stability properties. Recently for a particular class of linear extensions of dynamical systems it has been proven1 that the stability of invariant toroidal manifold heavily depends on the properties of the systems’ ingredients not on the whole torus, but in some smaller domains. It was reported that in some particular cases it is sufficient to set constraints only in non-wandering set of dynamical system to derive the desired stability and robustness properties of invariant toridal manifold. However the methods that were used there heavily rely on the linearity of some components of dynamical system and do not fit to analyze truly nonlinear systems. The purpose of this project is to develop new relaxed Lyapunov-like sufficient conditions for stability of invariant tori of nonlinear systems. We expect that new theorems will set constraints on Lyapunov function not on the whole surface of the torus, but only in non-wandering set of dynamical system. In the second part of the research we plan to extend the developed results onto an interesting class of dynamical system that possesses an invariant torus but undergoes impulsive perturbations at nonfixed moments. Hence the trajectories of such system issued from the outside of the invariant manifold are piecewise continuous. This requires a careful investigation of a trade-off between continuous and discontinuous dynamics since they can play different roles in a view of contribution towards stability. A dwell-time condition that restricts the frequency of impulses will be developed in order to guarantee the desired stability properties of the invariant manifold. A successful realization of the proposed research will benefit into the qualitative theory of ordinary differential equations and deepen the understanding of multi-frequency oscillations. It will be useful in the investigations of systems that have a simple structure of limit sets and recurrent trajectories and will provide a simplified sufficient conditions for stability of invariant tori in these cases.

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the author of the proposal is a co-author of the mentioned results

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Project Description 1 State of the art and preliminary work Consider an autonomous system of differential equations dz = F (z), z ∈ Rk (1) dt that possesses m-dimensional invariant toroidal manifold Tm . For a comprehensive description of the dynamics in the vicinity of the invariant toroidal manifold it is convenient to introduce so-called local coordinates (ϕ1 , . . . , ϕm , x1 , . . . , xn ), n = k−m, where ϕ = (ϕ1 , . . . , ϕm ) is a point on the torus Tm and x = (x1 , . . . , xn ) is from Euclidean space in transversal direction to the torus. The change of variables is performed in such a way that the invariant toroidal manifold gets a representation x = 0, ϕ ∈ Tm in new coordinates. System (1) can be transformed into dϕ dx = a(ϕ, h), = f (ϕ, x), (2) dt dt with f (ϕ, 0) ≡ 0. The last condition guarantees the existence of invariant toroidal set x = 0, ϕ ∈ Tm that is called trivial. The problems of existence, stability and an approximate construction of non-trivial invariant toroidal manifolds for system (2) are treated carefully in [17]. The central object of investigation there is the so-called linear extension of dynamical system on torus dϕ dx = a(ϕ), = A(ϕ)x + f (ϕ), (3) dt dt where a ∈ CLip (Tm ) is an m-dimensional vector function, A, f ∈ C(Tm ) are n × n square matrix and n-dimensional vector function respectively; C(Tm ) stands for a space of continuous 2π-periodic with respect to each of the variables ϕj , j = 1, . . . , m functions defined on the surface of the torus Tm . The main ingredient in the investigation of the existence and stability analysis of non-trivial invariant tori of system (3) is Green function introduced by Samoilenko in [15]. The existence of such function is sufficient for the existence of non-trivial invariant torus for system (3). Later numerous of works by different authors have developed and extended this approach to broad classes of equations including impulsive [8, 12], stochastic [19], infinite-dimensional [11], and equations with delay [16]. This method of investigation got a Green-Samoilenko function method name [11]. A deep connection of the existence of invariant tori and quadratic functions was explored in [6]. A Lyapunov-like approach was proposed for stability analysis of invariant tori and their robustness properties characterization. A question of the preservation of invariant tori under perturbations of the right-hand side was also considered. In [6, 14, 15] using different methods it has been proven that sufficiently small perturbations do not ruin the invariant torus, which has made it a convenient object for investigation of quasi-periodic motions of dynamical system. As it is widely known, quasi-periodic solution may be easily transformed into a periodic one by a small perturbation of right-hand side. The existence of invariant tori that is a carrier of quasi-periodic trajectories ensures the existence of multi-frequency oscillations in the system. It makes this theory well-adapted for the applications in electrotechnics and radiophysics with complex oscillatory processes of several frequencies.

Own results in connection with the proposed project The results of the applicant and his collaborators that are connected with the proposed project can be divided into three topics.

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Multi-frequency systems The results from this section mostly concern the problems of the existence, stability and preservation under perturbations of non-trivial invariant tori of multi-frequency systems of the type (3). Thus in [3], a linear extension of dynamical system on torus in the Lappo-Danilevskii case has been studied. The sufficient conditions for invariant tori existence when the matrix A commutes with its integral were obtained. In [9] a novel perturbation theory of linear extensions of dynamical system on torus has been developed. The peculiarities of the approach is that the perturbation term required to be sufficiently small only in non-wandering set of dynamical system rather than on the whole surface of torus in order to preserve the invariant manifold. Finally in [10], it was shown that the perturbation of the right-hand side that vanishes in infinity does not ruin the exponentially stable invariant torus. Discontinuous dynamical systems In [8] and [12] some particular classes of multi-frequency systems with impulses were considered and the sufficient conditions for the existence of discontinuous invariant toroidal sets were obtained. In [2] the linear extension of dynamical system with discontinuous right-hand side was considered. The iterative method for the invariant tori construction was justified. Finally, in [4] the perturbation theory for multi-frequency systems with impulses were developed using the ideas from the paper [9]. Exponential dichotomy An exponential dichotomy property of linear systems of ODEs is tightly connected with the existence of non-trivial invariant tori of multi-frequency systems of the type (3). Moreover the roughness property of exponential dichotomy enables to prove the preservation of invariant tori under sufficiently small perturbations. In [5] a novel theorem on roughness of exponential dichotomy has been proven. It relaxes the constraints on perturbation term in the case when it commutes with matrix A and requires it to be small only for large arguments. This approach can potentially be applied to the problem of preservation of invariant tori under the appropriate perturbations of the right-hand side.

2 Objectives and work program 2.1 Objectives The objectives of the project are as follows: (i) To develop relaxed Lyapunov-like conditions for stability (asymptotic stability) of trivial invariant torus which will set constraints on the properties of Lyapunov function in a smaller domain than in the classical theory. (ii) To develop the analogues sufficient conditions for multi-frequency systems that undergo impulsive perturbations at non-fixed moments. (iii) To investigate a trade-off between continuous and discontinuous dynamics and to develop an easy-to-be-verified dwell-time conditions that guarantee stability (asymptotic stability) of invariant torus.

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2.2 Work programme incl. proposed research methods I. Relaxed Lyapunov-like conditions for invariant tori stability. We plan to consider the nonlinear system of differential equations in local coordinates dϕ = a(ϕ), dt

dx = f (ϕ, x), dt

(4)

where a ∈ CLip (Tm ), f : Z → Rn is continuous 2π-periodic with respect to ϕv , v = 1, . . . , m function, Z = {(ϕ, x) ∈ Tm × Rn : ϕ ∈ Tm , x ∈ J¯h },

J¯h = {x ∈ Rn : kxk ≤ h, h > 0}.

Let f (ϕ, 0) ≡ 0 ∀ϕ ∈ Tm so system (4) has the trivial invariant torus x = 0, ϕ ∈ Tm . Denote by ϕt (ϕ) a solution to the first equation from (4) that satisfies the initial condition ϕ0 (ϕ) = ϕ. Utilizing a positive definite function V (ϕ, x) that satisfies a certain dissipation condition for all (ϕ, x) ∈ Z one may conclude on stability (asymptotic stability) of the trivial invariant torus. In our project we aim to relax this condition and require the analogues properties from the derivative of function V hgradϕ V (ϕ, x), a(ϕ)i + hgradx V (ϕ, x), f (ϕ, x)i ≤ 0 to hold not for all (ϕ, x) ∈ Z, but in some smaller domain ZΩ = {(ϕ, x) ∈ Tm × Rn : ϕ ∈ Ω, x ∈ J¯h }, where Ω is a non-wandering set of dynamical system ϕ˙ = a(ϕ). Note that ZΩ ⊂ Z. Definition 2.1 ([7]). A point ϕ is called wandering if there exist its neighbourhood U (ϕ) and a positive number T > 0 such that U (ϕ) ∩ ϕt (U (ϕ)) = ∅ for t ≥ T. Let W be the set of all wandering points of dynamical system and Ω = Tm \ W be the set of nonwandering points. From the compactness of torus it follows that the set Ω is nonempty and compact. The key ingredient to prove the relaxed Lyapunov proposition is the fact that for any ϕ0 ∈ Tm the corresponding trajectory ϕt (ϕ0 ) spends only a bounded time outside the non-wandering set Ω. Moreover there exists a uniform estimate for this time that does not depend on initial condition ϕ0 . It provides an intuition that the restrictions and constraints in the set Ω are much more important than outside this set. This hypothesis was used successfully in the problem of preservation of invariant torus of linear extension of dynamical system under the perturbations of the right-hand side. It has been proven in [9] that for the preservation of an exponentially stable invariant torus the perturbation term should be sufficiently small only in non-wandering set Ω rather on the whole surface of the torus as it was known before from [6, 14, 17]. We would like to mention the potential benefits of this new result. In the case of a simple structure of limit sets and recurrent trajectories of the dynamical system, the domain of constraints on Lyapunov function can be dramatically downsized. For example, consider the system ϕ˙ = sin2 ϕ2 defined on the circle T1 . Its non-wandering set consists of a single point ϕ0 = 0. All trajectories tend to it as |t| → ∞. So we need to check the conditions on Lyapunov function only in a single point instead of the whole circle T1 . II. Stability of invariant tori of discontinuous dynamical systems. Consider the nonlinear system of differential equations in local coordinates that undergoes impulsive perturbations dϕ = a(ϕ), dt

dx = f (ϕ, x), dt ∆x|ϕ∈Γ = I(ϕ, x),

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ϕ ∈ Tm \ Γ, ϕ ∈ Γ,

(5)

Figure 1: A simple example of phase portrait of the system of the type (5) defined in T1 × R with Γ = {ϕ ∈ T1 : ϕ = 0}. Red arrows denote the vector field of the system without impulses. Impulsive jumps are highlighted with green. In a multidimensional case the set Γ can be of a more complex form that leads to a more complex behavior of trajectories. where a ∈ CLip (Tm ), f, I : Z → Rn are continuous 2π-periodic functions with respect to ϕj , j = 1, . . . , m, Γ is a predefined smooth compact submanifold of the torus Tm of codimension 1. The dynamics of the system (5) can be described as follows: for any initial condition (ϕ0 , x0 ) ∈ {Tm \Γ}×Rn the components ϕ ∈ Tm and x ∈ Rn move along the trajectory of differential equations from (5). At a time τ when the trajectory ϕt (ϕ0 ) meets the predefined set Γ ⊂ Tm the point x(τ ) ∈ Rn is instantly transferred to a new position x(τ ) + I (ϕτ (ϕ0 ), x(τ )). Then point x ∈ Rn moves along the trajectory of differential equation from (5) until the subsequent intersection of ϕt (ϕ0 ) with the set Γ. Let f (ϕ, 0) ≡ 0 and I(ϕ, 0) ≡ 0 ∀ϕ ∈ Tm so system (4) has the trivial invariant torus x = 0, ϕ ∈ Tm . Despite the fact that system (5) possesses a smooth invariant manifold, its trajectories issued from the outside of this manifold are piecewise continuous. Hence the study of stability and attractivity of the manifold requires techniques from impulsive differential equations theory [18]. System of the type (5) was firstly considered by Perestyuk in [13] for the case of a(ϕ) ≡ ω = const and linear functions f and I. The papers [8, 12, 20] prolong this research and study the existence and stability of non-trivial discontinuous invariant toroidal sets. To the best to our knowledge there is the only paper [1] where the sufficient conditions for stability and asymptotic stability of trivial invariant manifold have been developed for the case of fixed moments of impulsive jumps. However these results are quite conservative and require both continuous and discontinuous dynamics to contribute towards stability. In this project we aim to investigate a trade-off between continuous and discontinuous parts of the system (5) in order to guarantee stability and attractivity of the invariant manifold. The most interesting cases are when the continuous dynamics contribute towards stability and the discontinuous one plays

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against it and vice versa. Then the restrictions on the frequency of impulsive occurrences should be imposed in order to ensure the desired stability properties. From this viewpoint (5) can be considered as the problem of impulsive stabilization: non-impulsive system itself has an unstable invariant manifold, but it may become stable (asymptotically stable) with an appropriate choice of impulsive perturbations and their frequency.

3 Bibliography [1] S. Dudzyanyi and N. Perestyuk. On the stability of a trivial invariant torus of one class of impulsive systems. Ukrainian Math. J., 50(3):387–399, 1998. [2] P. Feketa. Invariant manifolds of a certain class of discontinuous dynamical system. Bulletin of Taras Shevchenko National Univ. of Kyiv. Mathematics, Mechanics Series, 23:24–28, 2010. [3] P. Feketa and Y. Perestyuk. On invariant tori of multi-frequency systems in the Lappo-Danilevskii case. Bulletin of Taras Shevchenko National Univ. of Kyiv. Ser.: Physics&Mathematics, 2012(3):105–110, 2012. [4] P. Feketa and Y. Perestyuk. Perturbation theorems for a multi-frequency system with impulses. Nonlinear Oscil., 18(2):280–289, 2015. [5] O. Leontiev and P. Feketa. A new criterion for the roughness of exponential dichotomy on R. Miskolc Math. Notes, 16(2):987–994, 2015. [6] Y. A. Mitropolsky, A. M. Samoilenko, and V. L. Kulik. Dichotomies and Stability in Nonautonomous Linear Systems. Taylor & Francis, 2003. [7] V. V. Nemytskii and V. V. Stepanov. Qualitative Theory of Differential Equations. Courier Corporation, 1989. [8] M. Perestyuk and P. Feketa. Invariant sets of impulsive differential equations with particularities in ω-limit set. Abstr. Appl. Anal., 2011:14, 2011. [9] M. Perestyuk and P. Feketa. On preservation of the invariant torus for multifrequency systems. Ukrainian Math. J., 65(11):1661–1669, 2014. [10] M. Perestyuk and P. Feketa. On preservation of an exponentially stable invariant torus. Tatra Mt. Math. Publ., 63(1):215–222, 2015. [11] M. Perestyuk and V. Y. Slyusarchuk. Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations. Ukrainian Math. J., 61(7):1123–1136, 2009. [12] M. O. Perestyuk and P. V. Feketa. Invariant manifolds of one class of systems of impulsive differential equations. Nonlinear Oscil., 13(2):260–273, 2010. [13] N. Perestyuk. Invariant sets of a class of discontinuous dynamical systems. Ukrainian Math. J., 36(1):58–62, 1984. [14] R. J. Sacker and G. R. Sell. A spectral theory for linear differential systems. J. Differ. Equations, 27:320–358, 1978. [15] A. Samoılenko. Preservation of an invariant torus under perturbation. Izvestiya: Mathematics, 4(6):1225–1249, 1970. ´ [16] A. Samoilenko and A. El’nazarov. On invariant tori of countable systems of differential equations with delay. Ukrainian Math. J., 51(9):1454–1458, 1999. [17] A. M. Samoilenko. Elements of the Mathematical Theory of Multi-Frequency Oscillations. Kluwer Academic Publishers, Dordrecht et al., 1991. [18] A. M. Samoilenko and N. Perestyuk. Impulsive Differential Equations. World Scientific, 1995. [19] A. M. Samoilenko and O. Stanzhytskyi. Qualitative and asymptotic analysis of differential equations with random perturbations. World Scientific, 2011. [20] V. Tkachenko. The Green function and conditions for the existence of invariant sets of impulse systems. Ukrainian Math. J., 41(10):1187–1190, 1989.

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