On lower-bound estimates of the Lyapunov dimension and topological entropy for the Rossler systems

15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Delay Systems 15th IFAC Workshop on Time Time9-11, Delay Systems Sinaia, Romania, September 2019 15th IFAC Workshop on Time Delay Systems Sinaia, Romania, September 9-11, 2019 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 Available online at www.sciencedirect.com Sinaia, Romania, September 9-11, 2019 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 Sinaia, Romania, September 9-11, 2019

ScienceDirect

IFAC PapersOnLine 52-18 (2019) 97–102

On lower-bound estimates of On lower-bound estimates of On lower-bound estimates of On lower-bound estimates of the Lyapunov dimension and topological On lower-bound estimates of the Lyapunov dimension and topological the Lyapunov dimension and topological the Lyapunov dimension and topological entropy for the Rossler systems entropy for the Rossler the Lyapunov dimension andsystems topological entropy for systems entropy∗,∗∗,∗∗∗ for the the Rossler Rossler systems ∗ entropy for the Rossler systems E.V. ∗∗∗ Kuznetsov ∗,∗∗,∗∗∗ Mokaev Kuznetsov N.V. N.V. ∗,∗∗,∗∗∗ Mokaev T.N. T.N. ∗∗ Kudryashova Kudryashova E.V. Kuznetsov Kuznetsov Kuznetsov Kuznetsov

N.V. T.N. E.V. ∗,∗∗,∗∗∗ Mokaev ∗ Kudryashova ∗ ∗ ∗∗∗∗ N.V. Mokaev T.N. Kudryashova E.V. ∗ ∗∗∗∗ Kuznetsova O.A. ∗ Danca ∗∗∗∗ N.V. ∗,∗∗,∗∗∗ Mokaev T.N. ∗∗M.-F. Kudryashova E.V. ∗∗ Kuznetsova O.A. M.-F. Kuznetsova O.A. Danca M.-F. ∗ Danca ∗∗∗∗ ∗,∗∗,∗∗∗ Kuznetsova O.A. ∗∗∗∗ N.V. Mokaev T.N. M.-F. Kudryashova E.V. Kuznetsova O.A. ∗ Danca Danca M.-F. ∗Kuznetsova O.A. ∗ Danca M.-F. ∗∗∗∗ ∗ of Mathematics and Mechanics, ∗ Faculty Faculty of of Mathematics Mathematics and and Mechanics, Mechanics, ∗ Faculty Faculty of Mathematics and Mechanics, ∗St. Petersburg State University, Russia Faculty of Mathematics and Mechanics, St. Petersburg State University, Russia St. Petersburg State University, Russia ∗∗ ∗St. of Petersburg State University, Russia ∗∗ Dept. Mathematical Information Technology, Faculty of Mathematics and Mechanics, ∗∗ Dept. St. Petersburg State University, of Mathematical Information Technology, Technology, ∗∗ Dept. of Mathematical Information Russia of Mathematical Information Technology, ∗∗ Dept. University of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland St. Petersburg State University, Russia Dept. of Mathematical Information Technology, University of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland University of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland ∗∗∗ ∗∗ University of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland ∗∗∗ Institute for Problems in Mechanical Engineering RAS, Dept. of Mathematical Information Technology, ∗∗∗ University of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland Institute for Problems in Mechanical Engineering RAS, Russia Russia Institute for Problems in Mechanical Engineering RAS, Russia ∗∗∗ ∗∗∗∗ Institute for Problems in Mechanical Engineering RAS, Russia ∗∗∗ ∗∗∗∗University Romanian Institute of Science and Technology, of Jyv¨ a skyl¨ a , Jyv¨ a skyl¨ a , Finland ∗∗∗∗ Institute for Problems in Mechanical Engineering RAS, Russia Romanian Institute of Science and Technology, Romanian Institute of Science and Technology, ∗∗∗∗ ∗∗∗ Romanian Institute of Science and Technology, ∗∗∗∗ Cluj-Napoca, Romania Institute Romanian for Problems in Mechanical Engineering RAS, Russia Institute of Science and Technology, Cluj-Napoca, Romania Cluj-Napoca, Romania ∗∗∗∗ Romania RomanianCluj-Napoca, Institute of Science and Technology, Cluj-Napoca, Romania Cluj-Napoca, Romania Abstract: In In this this paper, paper, on on the the example example of the the R¨ R¨ ssler systems, systems, the the application application of of the the Pyragas Pyragas Abstract: of ooossler Abstract: In this paper, on the example of the R¨ ssler systems, systems, the the application application of of the the Pyragas Pyragas Abstract: In this paper, on the example of the R¨ o ssler time-delay feedback control technique for verification of Eden’s conjecture on the maximum of Abstract: In this paper, on technique the example the R¨osslerof the application the Pyragas time-delay control for verification Eden’s on of time-delay feedback feedback control technique for of verification ofsystems, Eden’s conjecture conjecture on the theofmaximum maximum of time-delay feedback control technique for verification of Eden’s conjecture on the maximum of local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To Abstract: In this paper, on the example of the R¨ o ssler systems, the application of the Pyragas time-delay feedback control technique for verification of Eden’s conjecture on the maximum of local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To local Lyapunov dimension, for estimation of the entropy is To this end, end, numerical experiments onthe computation finite-time local Lyapunov dimensions and time-delay feedback controland technique for verification oftopological Eden’slocal conjecture on the maximumand of local Lyapunov dimension, and foron estimation of topological entropy is demonstrated. demonstrated. To this numerical experiments computation finite-time Lyapunov dimensions this end, end, numerical experiments onthe computation of the finite-time local Lyapunov dimensions and this numerical experiments on computation finite-time local Lyapunov dimensions and finite-time topological entropy on a R¨ o ssler attractor and embedded unstable periodic orbits local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To this end, numerical experiments on computation of finite-time local Lyapunov dimensions and finite-time topological entropy on a R¨ o ssler attractor and embedded unstable periodic orbits finite-time topological topological entropy entropy on on a R¨ R¨ossler ssler attractor and and embedded unstable unstable periodic periodic orbits orbits finite-time are performed. performed. The problem problem of reliable reliable numerical computation of theLyapunov mentioned dimension-like this end, numerical experiments of finite-time local and finite-time topological entropy onona acomputation R¨oonumerical ssler attractor attractor and embedded embedded unstable dimensions periodic orbits are The of computation of the dimension-like are performed. performed. The problem problem of reliable reliable numerical computation of the mentioned mentioned dimension-like are The of numerical computation of the mentioned dimension-like characteristics along the trajectories over large time intervals is discussed. finite-time topological entropy on a R¨ o ssler attractor and embedded unstable periodic orbits are performed. The problem of reliable numerical computation of the mentioned dimension-like characteristics characteristics along along the the trajectories trajectories over over large large time time intervals intervals is is discussed. discussed. characteristics the trajectories over large time intervals is are performed. along The problem of reliable numerical computation ofdiscussed. the mentioned dimension-like characteristics along the trajectories over large time intervals isreserved. discussed. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights characteristics along the trajectories over attractors, large time intervals isdimension, discussed. Lyapunov Keywords: chaos, hidden and self-excited Lyapunov Keywords: chaos, hidden and self-excited attractors, Lyapunov dimension, Lyapunov Keywords: chaos, hidden and self-excited attractors, Lyapunov dimension, Lyapunov Keywords: chaos, hidden and orbit, self-excited attractors, Lyapunov dimension, Lyapunov Lyapunov exponents, chaos, unstable periodic orbit, time-delay feedback control dimension, Keywords: hidden and self-excited attractors, Lyapunov exponents, unstable periodic time-delay feedback control exponents, unstable periodic orbit, time-delay feedback control exponents, unstable periodic orbit, time-delay feedback control Keywords: hidden and orbit, self-excited attractors, Lyapunov exponents, chaos, unstable periodic time-delay feedback control dimension, Lyapunov exponents, unstable periodic orbit, time-delay feedback control 1. INTRODUCTION System R (a 1. System (3) (3) for for arbitrary arbitrary real real parameters parameters a a333 ,,, bbb333 ∈ ∈R R (a (a333 = = 1. INTRODUCTION INTRODUCTION System (3) for arbitrary real parameters a ∈ = 1. INTRODUCTION System (3) for arbitrary real parameters a , b ∈ R (a 0), has the following equilibria: 3 3 1. INTRODUCTION System (3) following for arbitrary real parameters a3 , b3 ∈ R (a33 = = 0), has the the following equilibria: 0), has equilibria: has following 1. INTRODUCTION  arbitrary equilibria:  a +b a3 ,ab3+b∈ R (a3 = System (3) for real Consider the the following following R¨ ssler systems systems [R¨ [R¨ ssler, 1976a,b, 1976a,b, 0), 0), has Othe the equilibria: + following − parameters Consider R¨ oo ssler oo ssler, 3 3 3 3    Consider the following R¨ o ssler systems [R¨ o ssler, 1976a,b, + − a +b a +b (7) =following 0, 0, 0, 0 ,, O O3− = = 0, a33a+b 3 , − a3 3 3 . + = 0, 3 Consider the following following R¨ R¨ ssler systems systems [R¨ [R¨ ssler, 1976a,b, 1976a,b, 0), has O the . (7) ,− 1979]: 3 3 3 3 + = − = 0, O (7) 0, 0, 00equilibria: , O 0, a a+b − a3aa+b +b 3 3 Consider the oossler oossler, 1979]: 3 3, 3 3 . 3 3 1979]: + = 0, 0, 0 , − = 0, a33a O (7) O +b +b 3 3 3 , − a3a 3 . 3 3 a a 1979]: O . (7) = 0, 0, 0 , O = 0, , − 3 3     Consider following R¨ oassler systems [R¨ o ssler, 1976a,b, ± 3 3 a a 1979]: + − a +b a +b 3 3 y, z ˙ = b − c z + xz, (1) ˙x˙ = x = −y −ythe − z, y ˙ = x + 3 3 3 3 ± shows 1 y, 1 − c1 z + xz, Stability analysis of O that if 0 < −a < b , then ± 3 3 O . (7) = 0, 0, 0 , O = 0, , − − z, y ˙ = x + a z ˙ = b (1) 3 1 − c1 1 z + xz, Stability analysis of O shows that if 0 < −a < b , then z ˙ = b (1) ˙x˙ = x = −y −y − − z, z, yy˙˙ = =x x+ +a a11 y, 3 3 3 3 a a 1 ± Stability analysis of O shows that if 0 < −a < b , then 1979]: 3 3 3 3 < b3 , then + − y, zz˙˙ = bb12 x− cc1czz2 z+ xz, (1) 3 1 ± ifshows Stability analysis of O that if 00 < < −a x ˙˙ = −y − z, yy˙˙ = x + a − + xz, (2) 3 then 3 ,Othen + is locally − is stale, and 0 < b < −a 2b , O = − + xz, (1) 3 2 y, + − 3 3 3 1 1 1 Stability analysis of O shows that if < −a < b 3 3 x = −y − z, = x + a y, z ˙ = b x − c z + xz, (2) is locally locally stale, stale, and and3± if if 00 < < bb33 < < −a −a33 < < 2b 2b33 ,,3 then then3 O O33− is is O33+ is x ˙˙ = −y − z, yy˙˙ = x + a22 y, zz˙˙ = bb22 x − cz22 zz++ xz, (2) O x −y − z, x + x − xz, (2) 2xz, + is locally − is stale, 00 < bb3that < −a O analysis ofand O3 if shows if 30 < < 2b −a < b3 ,O then 2 zy+ locally stable. 3 ,,3 then 3 3 x ˙˙ = = −y − z, yy˙˙ = = x +a azz˙˙122 y, y, z˙3 zz= =+ b122a x− −c1cc− xz, (1) (2) Stability 2 ), (3) x, = −b is locally stale, and if < < −a < 2b then O is O locally stable. 2 y+ 3 (y 2 3 3 3 locally stable. 3 3 + a (y − ), (3) x = −y − z, = x, = −b + − y 22 ), x = −y − z, = x, = −b stable. x ˙˙˙ = −y − z, yyy˙˙˙ = x + azz˙˙2 y, z˙333 zz=+ b2a x333 (y − c− xz, (3) (2) locally stale, and if 0 < b3 < −a3 < 2b3 , then O3 is O3 is locally 2 zy+ + a (y − ), (3) x = −y − z, = x, = −b locally stable. values + a,, 3 bb(y1,2,3 − y,, 2cc),1,2 ∈ x˙ =arbitrary −y − z, real y˙ =parameters x, z˙ = −b3a with R. For some some values of of parameters parameters systems systems (1),(2),(3) (1),(2),(3) exhibit exhibit with arbitrary real parameters az1,2,3 ∈ (3) R. For For some values of parameters systems (1),(2),(3) exhibit locally stable. 1,2,3 with a R. 1,2,3 1,2 ∈ + a,, 3 bb(y1,2,3 − y,, cc),1,2 (3) x˙ =arbitrary −y −ifz,c22 real y˙ 4a =parameters x, z˙ =the −bfollowing For some values of parameters systems (1),(2),(3) exhibit 3az1,2,3 chaotic behavior. To get a visualization of chaotic attractor with arbitrary real parameters ∈ R. 1,2,3 1,2,3 1,2 System (1) ≥ b has equilibria: For some values of parameters systems (1),(2),(3) exhibit chaotic behavior. To get a visualization of chaotic attractor 1 1 with arbitrary a , b , c ∈ R. System (1) if if cc21121 real ≥ 4a 4aparameters b has the following equilibria: chaotic behavior. To get a visualization of chaotic attractor 1,2,3 1,2,3 1,2 1 1 System (1) ≥ b has the following equilibria: 1 1 chaotic behavior. To get a visualization of chaotic attractor √ one needs to choose an initial point in the basin of For some values of parameters systems (1),(2),(3) exhibit System (1) if c ≥ 4a b has the following equilibria: 2 with arbitrary real parameters a , b , c ∈ R. 1 1 √ chaotic behavior. To get a visualization of chaotic attractor 1,2,3 1,2,3 1,2 1 one needs to choose an initial point in the basin of 2 √cc122 −4a System (1) if± c12 ≥±4a1±b1 has the following one needs to choose an initial point in the basin of b1 c1 ±equilibria: ± ± −4a111 b b11 , (4) c11 ± ± √c one needs to choose an initial point in the basin of attraction of the attractor and observe how the trajectory, ± ± ± ± ± chaotic behavior. To get a visualization of chaotic attractor −4a c 1 = (a p , −p , p ), where p = O ± = (a 1 pif 2 −4a b , (4) √ ± ± ± ± System (1) c ≥ 4a b has the following equilibria: 1 one needs to choose an initial point in the basin of attraction of the attractor and observe how the trajectory, 1 1 1 2a , −p , p ), where p = O c ± c 1 1 attraction of the attractor and observe how the trajectory, 1 1 1 1 p± , −p± , p± ), where p± = c ± 2a ± = (a1 , (4) O 2 1 1 −4a 1 c b 1 attraction of the attractor and observe how the trajectory, 1 1 , (4) starting from this initial point, after a transient process ± = (a1 p± , −p± , p± ), where p± = 1 √2a O 11 one needs to choose an initial point in the basin of 1 attraction of the attractor and observe how the trajectory, starting from this initial point, after a transient process 2 ), where p = c ± 2a , (4) = (a p , −p , p O 2 −4a 1 starting from this initial point, after a transient process 1 c b 1 2 ±= 4a1 b1 . It is [Barrio which 2a 1 1et al., 1 ± coincide ± for ±c ± known 11 2 starting from this initial point, after aahow transient process 1 visualizes the attractor. An attractor is called a self= 4a b . It is known [Barrio et al., which coincide for c attraction of the attractor and observe the trajectory, 1 1 , (4) = (a p , −p , p ), where p = O = 4a b . It is known [Barrio et al., which coincide for c 1 2 starting from this initial point, after transient process 1 visualizes the attractor. An attractor is called a self1 1 1 +. It is known [Barrio − 1 2a1 visualizes the attractor. An attractor is called a selfet al., which coincide for + − 1b 1 . is 2011] the unstable and visualizes the attractor. An attractor is called a self+ − excited attractor if its basin attraction = 4a 4aO is known [Barrio et O al., which coincide for cc2121 = 1 starting from initial point, after a transient 2011] that that the equilibrium equilibrium O isIt always always unstable and O 1 b1 1 is visualizes the this attractor. Anof attractor is intersects called process a with selfexcited attractor if its basin of attraction intersects with + − 2011] that the equilibrium O always unstable and O 1 1 excited attractor if its basin of attraction intersects with 1 1 + − 2011] that the equilibrium O is always unstable and O = 4a b . It is known [Barrio et al., which coincide for c is linearly stable iff the parameters a , c belongs to the excited attractor if its basin of attraction intersects with 1 11 is always 1 , cunstable 1 belongsand 1 any open neighborhood of an equilibrium, otherwise, it 1the parameters   visualizes the attractor. An attractor is called a self2011] that the equilibrium O O is linearly stable iff a to the 1 , c1 1 belongs to the excited attractor if its basin of attraction intersects with any open neighborhood of an equilibrium, otherwise, it is is is linearly stable iff the parameters a 1 1  1 + − any open neighborhood of an equilibrium, otherwise, it is  is linearly iff parameters a ,, ccunstable belongs to the region {(a ,, cc1 ))equilibrium a ≤ 1, ccO11 > 2a a ∈ 1or 1{(a 2011] that the is always and any open neighborhood of an equilibrium, otherwise, it is   1stable 1the 1} 1 ,, c 1) 1O called a hidden attractor [Leonov et al., 2011, Leonov and excited attractor if its basin of attraction intersects with   is linearly stable iff the parameters a belongs to the 1 region {(a a ≤ 1, > 2a } or {(a c ) a ∈ 1 1 1 1 1 1 1 1 1 1 √ any open neighborhood of an equilibrium, otherwise, it is called a hidden attractor [Leonov et al., 2011, Leonov and region {(a , c ) a ≤ 1, c > 2a } or {(a , c ) a ∈   1 1 1 1 1 1 1 1 called a hidden attractor [Leonov et al., 2011, Leonov and √ 2a region {(a , c ) a ≤ 1, c > 2a } or {(a , c ) a ∈ 1 √ 1 1 1 1 1 1 1 1   is linearly stable iff the parameters a , c belongs to the called a hidden attractor [Leonov et al., 2011, Leonov and 2a Kuznetsov, 2013, Leonov et al., 2015, Kuznetsov, 2016a]. 1 1 (1, 2), cc1 1∈ ∈, c(2a (2a )}, and the parameter b satisfies any open neighborhood of an equilibrium, otherwise, it is 1 region {(a ) ≤ 1, c > 2a } or {(a , c ) a ∈   2 −1 1 ,,aa12a √ 1 1 1 1 1 1 1 called a hidden attractor [Leonov et al., 2011, Leonov and Kuznetsov, 2013, Leonov et al., 2015, Kuznetsov, 2016a]. (1, 2), )}, and the parameter b satisfies 2 −1 Kuznetsov, 2013, Leonov et al., 2015, Kuznetsov, 2016a]. (1, cc11 ∈ the √2), 1 11 ,,aaa12a  a1 ∈ called 1 2 region {(a , c(2a ≤)}, 1, cand 2a parameter } or {(a1 , cbb111) satisfies Kuznetsov, 2013, Leonov et al., 2015, Kuznetsov, 2016a]. 1 −1 (1, 2), (2a )}, and satisfies 2a It was discovered numerically by R¨ oossler that in the phase 1) 1 1 >the 1 a hidden attractor [Leonov et al., 2011, Leonov and 1 1∈ c2122 1parameter a1221 −1 Kuznetsov, 2013, Leonov et al., 2015, Kuznetsov, 2016a]. It was discovered numerically by R¨ ssler that in the phase (1, 2), c ∈ (2a , )}, and the parameter b satisfies √ ∗ 1 1 1 It was discovered numerically by R¨ ossler that in the phase c12 , where a2a the c) ≤ bband ≤ the c 1 1 −1 It was discovered numerically by R¨ in space system with parameters a = bb1 2016a]. = 0.2, 4a the inequalities: (a, c) ≤ where 2013,(1) Leonov al., (1, inequalities: 2), c1 ∈ (2abbb1∗∗11∗1, (a, )}, b1 satisfies Kuznetsov, 1 that c12111 ,, parameter the inequalities: (a, c) ≤ bb111 ≤ ≤ It was of discovered numerically by 2015, R¨oossler ssler that in the the phase space of system (1) with et parameters aKuznetsov, = 0.2, 0.2, =phase 0.2, 4a a21 −1 1 space of system (1) with parameters a 0.2, bb11 = 0.2, c11 , where 4a the inequalities: b (a, c) ≤ ≤ where 1 = ∗ 1  1 space of system (1) with parameters a = 0.2, = 0.2, 4a c = 5.7 and system (3) with parameters a = 0.386, the ∗inequalities: ba1∗1(a, c) ≤ b1 4≤ 4a 1 1 21 , 3where2 It was discovered numerically by R¨ o ssler that in the phase 1 3 2 space of system (1) with parameters a = 0.2, b = 0.2, c = 5.7 and system (3) with parameters a = 0.386, cc11a3 + 2a2 − c a + c2 1 1 1 3 ∗ 4 a c = 5.7 and system (3) with parameters a = 0.386, (a , c ) = 2 − a + b 1 1 3 2 1 , c1 ) = 2(ab2a+1) 1a 1 a1 + c 2 4 3 2 2c)  1 1 1 1 1 thebb∗1∗1inequalities: (a, ≤ b ≤ , where (a 2 − a + c + 2a − c c = 5.7 and system (3) with parameters a = 0.386, 1 b = 0.2 there exist chaotic attractors of different shapes, 1 1 1 1 1 1 2 1 3 space of system (1) with parameters a = 0.2, b = 0.2, (a , c ) = 2 − a + c a + 2a − c a + c 4 3 2 2 a+1) 1 1 1 1 4a  1 3 1 2 1 1 1 1 1 1 1 1 2(a 2 c = 5.7 and system (3) with parameters a = 0.386, 1 1 1 1 1 b = 0.2 there exist chaotic attractors of different shapes, 3 1 2 − a41 + c1 a31 + 2a21 − c1 a1  bb∗1 (a a+1) b331 = 0.2 there exist chaotic attractors of different shapes, 1  ± 2 + 2(a1221 +1) (a1 ,, cc1 )) = = 2(a + cc2112 = 0.2 there exist chaotic attractors of different shapes,  ± , are self-excited with respect to both equilibria O 2 2 − a1 + c1 a1 + 2a1 − c1 a1  3 cbbwhich = 5.7 and system (3) with parameters a = 0.386, 1  ± 2(a1a+1) 1 3 1,3 ∗ 1 1 44 3 3 2 2 = 0.2 there exist chaotic attractors of different shapes, which are self-excited with respect to both equilibria O  1  6 2 3 ± ,, which are self-excited with respect to both equilibria O 1,3 ) = 2 a + c a + 2a − c a + c b1 (a1 , c1+(c . (5) − 4a + c − a ) a − 4a + 2c a 4 3 2 2 6  2 1,3 1 1 1 2 1 1 1 1 1 1 a ± , which are self-excited with respect to both equilibria O 2 6 4 3 2 1+ 1 .. (5) 1− 1+ 1− respectively. (5) + 2c a − 4a + c +(c11 2(a −a a1 +1) ) − 4a b = 0.2 there exist chaotic attractors of different shapes, 1,3 1 1 3 c +(c − ) a 4a 2c a 4a 1 1 1 1 1 which are self-excited with respect to both equilibria O , 4 3 2 2 6 respectively. 1 1 1 1 1 1 1 respectively. 1,3 +(c ± (5) respectively. +(c11 − −a a11 ))a a161 − − 4a 4a141 + + 2c 2c11 a a131 − − 4a 4a121 + + cc121 .. (5) which are self-excited with respect to both equilibria O , respectively. 1,3 2 6 4 3 2 One of the building blocks of chaotic attractor are embedAs it it was was +(c mentioned ina[Algaba [Algaba et2c al., 2015], in+the the region . (5) One − 4a1in c1 region of the building blocks of chaotic attractor are embed1 a12015], 1 − a1 ) in 1 − 4a1 + As mentioned et al., One of the building blocks of chaotic attractor are embedrespectively. As it was mentioned in [Algaba et al., 2015], in the region ± One of blocks of chaotic are periodic orbits (see As it [Algaba et the ± of parameters parameters where in the equilibria O12015], existin (1) ded One of the the building building blocks of (UPOs) chaotic attractor attractor are embedembedded unstable unstable periodic orbits (UPOs) (see e.g. e.g. [Afraımovic [Afraımovic ± exist As it was was mentioned mentioned inthe [Algaba et al., al.,O 2015], insystem the region region ded unstable periodic orbits (UPOs) (see e.g. [Afraımovic of where equilibria (1) ± exist system of parameters where the equilibria O system (1) 1 ded unstable periodic orbits (UPOs) (see e.g. [Afraımovic et al., 1977, Auerbach et al., 1987, Cvitanovi´ c ,, 1991]). A 1 One of the building blocks of chaotic attractor are embed± of parameters where the equilibria O exist system (1) As it was mentioned in [Algaba et al., 2015], in the region is equivalent to system (2) with respect to the following ded unstable periodic orbits (UPOs) (see e.g. [Afraımovic et al., 1977, Auerbach et al., 1987, Cvitanovi´ c 1991]). A 1 exist et al., 1977, Auerbach et al., 1987, Cvitanovi´ c , 1991]). A of parameters where the equilibria O system (1) is equivalent to system (2) with respect to the following is equivalent to system (2) with respect the following 1 et al., 1977, Auerbach et al., 1987, Cvitanovi´ cc[Afraımovic ,, 1991]). A ± to proof of the existence of UPOs for discrete and continuousded unstable periodic orbits (UPOs) (see e.g. is equivalent to system (2) with respect to the following linear change of variables: et al., 1977, Auerbach et al., 1987, Cvitanovi´ 1991]). A proof of the existence of UPOs for discrete and continuousof parameters where the equilibria O exist system (1) proof of the existence of UPOs for discrete and continuousis equivalent (2) with respect 1 to the following linear change to of system variables: linear change of variables: proof of the existence of UPOs for discrete and continuoustime dynamical systems can be done, for example, uset al., 1977, Auerbach et al., 1987, Cvitanovi´ c , 1991]). A linear change of variables: + + + proof of the existence of UPOs for discrete and continuoustime dynamical systems can be done, for example, usis equivalent to system (2) with respect to the following time dynamical systems can be done, for example, uslinear change ofa variables: +, +, +, x → x + yy → yy − p zz → zz + p (6) 1p + + + time dynamical systems can be done, for example, usx → x + a p , → − p , → + p , (6) ing a special analytical-numerical technique [Galias, 1999, proof of the existence of UPOs for discrete and continuous1 x → x + a p , y → y − p , z → z + p , (6) + + + 1 p+ , dynamical systems can be technique done, for [Galias, example,1999, using aa special special analytical-numerical technique [Galias, 1999, linear change of variables: ing analytical-numerical x y → yy − z→ z+ p++,, (6) time +, 1p , ing aa special analytical-numerical [Galias, 1999, x→ →x x+ +a aa −p p+ → +a + , , c2 z +. 2006b, Galias and Tucker, Barrio al., 2015]. and parameters a1 ,,ybb→ −p = cc1 z+ 12 = time dynamical systems can 2008, be technique done, for et example, us2 = 1pp + ing special analytical-numerical technique [Galias, 1999, 2006b, Galias and Tucker, 2008, Barrio et al., 2015]. and parameters a −p = + a .. (6) + a +,, c ++ 2p= 1 ,y b→ 2 = 2 z 1z 1pp 2006b, Galias and Tucker, 2008, Barrio et al., 2015]. and parameters a = a = −p c = c + a p + + 2 1 2 2 1 1 x → x + a , y − p , → + , (6) 1 2006b, Galias and Tucker, 2008, Barrio et al., 2015]. and ,, cc2 = cc1 + a +. 1p ing a special analytical-numerical technique [Galias, 1999, 2006b, Galias and Tucker, 2008, Barrio et al., 2015]. and parameters parameters a a22 = =a a11 ,, bb22 = = −p −p+ = + a p . 2 1 1 + 2006b, Galias and Tucker, 2008, Barrio et al., 2015]. and parameters a©2 2019. = a1The , b2 Authors. = −p+ ,Published c2 = c1 by + aElsevier 1 p . Ltd. All Copyright © 2019 IFAC 219rights 2405-8963 Copyright reserved. Copyright © © 2019 2019 IFAC IFAC 219 Copyright 219 Peer review responsibility of International Federation of Automatic Copyright © under 2019 IFAC 219 Control. Copyright © 2019 IFAC 219 10.1016/j.ifacol.2019.12.213 Copyright © 2019 IFAC 219

2019 IFAC TDS 98 Sinaia, Romania, September 9-11, 2019

N.V. Kuznetsov et al. / IFAC PapersOnLine 52-18 (2019) 97–102

15

25

25 10

20

O−

10

uupo1 (t)

10

5

0 10 5 0

y

-5 -10

-10

-5

0

5

x

10

15

0 10

5

0

-5

-10

y

(a)

(b)

-15

-10

-5

5

0

x

2 uupo 0

15

z

5

20

1 uupo 0

15

z

z

10

15

uupo2 (t)

5 0 10

5

0

-5

y

-10

-15

-10

-5

5

0

10

15

x

(c)

Fig. 1. The UPO uupo1 with period τ1 = 5.8811 (red) and the UPO uupo2 with period τ2 = 11.7586 (green) in system (1) with parameters a1 = 0.2, b1 = 0.2, c1 = 5.7, stabilized using TDFC method. Since for system (1) (and, also, (2)) corresponding results about the existence of low-periodic UPOs were obtained in [Galias, 2006a], further in this paper we will restrict ourselves to the case of system (1). One of the effective methods 1 for the numerical visualization of the UPOs is the time-delay feedback control (TDFC) approach, suggested by K. Pyragas [Pyragas, 1992]. The main idea behind the Pyragas’ approach is to stabilize UPOs by constructing a control force proportional to the difference between the current state of the system and an earlier state of the system (delayed by some time interval). From the mathematical perspective, the Pyragas method is as follows. Let uupo (t) be an UPO with period τ > 0, uupo (t − τ ) = uupo (t), satisfying a differential equation u˙ = f (u). To compute the UPO, we add the TDFC:   u˙ = f (u) + KB C ∗ u(t − T ) − u(t) , (8) where B, C are column vectors, K > 0 is a feedback gain and operator ∗ denotes transposition. If T = τ ,  then KB C ∗ u(t − T ) − u(t) = 0 along the UPO, and the periodic solution of system (8) coincides with the periodic solution of system the u˙ = f (u). There exist various modifications of the TDFC that allow to calculate the values of feedback gain and period adaptively, within the stabilization procedure [Chen and Yu, 1999, CruzVillar, 2007, Lin et al., 2010, Lehnert et al., 2011, Pyragas and Pyragas, 2011, 2013]. Remark that TDFC has some limitations (see e.g. [Hooton and Amann, 2012, Kuznetsov et al., 2015]) which nevertheless do not occur in the R¨ossler system. For system (1) with parameters a1 = 0.2, b1 = 0.2, c1 = 5.7, we solved time-delayed differential equation (8) with B ∗ = C ∗ = (0, 1, 0) and for different values of feedback gain K stabilized two UPOs, uupo1,2 , by the trajectories on the time interval t ∈ [0, 1000] with initial data u0 = (1, 1, 1) on [0, −τ1,2 ]. Using the feedback gain K = 0.3 the trajectory tends to the UPO uupo1 with period τ1 = 5.8811 2 , using the feedback gain K = 0.2 1

The first known approach to control chaos and detect the UPOs was the so-called OGY method suggested by Ott, Grebogi, Yorke [Ott et al., 1990]. The idea behind the method was to use a small time-dependent perturbation in the form of feedback to an accessible system parameter. 2 Periodic orbits in a continuous-time system often regarded as period-k orbits in accordance with the smallest positive number k

220

the trajectory tends to the UPO uupo2 with period τ2 = 11.7586 (see Fig. 1), wherein uupo1,2 are also the solutions of the initial system (1) (i.e. a system without the control). Here the values of period τ1,2 can be found e.g. using the algorithm by Lin et al. [2010]. upo the UPO Then for the initial points  upo  u0 , chosen onupo upo upo 1 2 u (t), t ∈ [0, τ ] , either u = u (t), t ∈ u1 = 1 2  ˜(t, uupo [0, τ2 ] , we numerically compute the trajectory u 0 ) of system (8) without the stabilization (i.e. with K = 0) on sufficiently large time interval [0, T = 500] (see Fig. 1b,1c). One can see that on the initial small time interval [0, T1 ≈ 60], even without the control, the obtained trajectory u ˜(t, uupo 0 ) traces approximately the ”true” periodic orbit uupo (t, uupo 0 ). But for t > T1 without control the trajectory upo u ˜(t, uupo and wind on the attractor A 3 . 0 ) diverge from u

For systems (1),(2),(3) it was numerically verified [Kuznetsov et al., 2014] the conjecture stating that the maximum of the Lyapunov dimension on attractor is reached at an equilibrium (a special case of the Eden’s conjecture [Eden, 1989] which claims that for an arbitrary attractor its maximum of the Lyapunov dimension is reached at an equilibrium or on a unstable periodic orbit. In general, a conjecture on the Lyapunov dimension of self-excited attractor [Kuznetsov, 2016b, Kuznetsov et al., 2018] is that for a typical system the Lyapunov dimension of a self-excited attractor does not exceed the Lyapunov dimension of one of unstable equilibria, the unstable manifold of which intersects with the basin of attraction and visualize the attractor. In this article we are going to verify numerically the Eden’s conjecture for system (1)

of distinct points {uj = Πj (u0 ) | j = 0, . . . , k − 1} with u0 = Πk (u0 ) on a certain Poincar´ e map Π. 3 Rigorous analysis of the time interval choices for reliable numerical computation of trajectories for the Lorenz system leads to the following results [Kehlet and Logg, 2013, Liao and Wang, 2014, Kehlet and Logg, 2017]. The time interval for reliable computation with 16 significant digits and error 10−4 is estimated as [0, 36], with error 10−8 is estimated as [0, 26], and reliable computation for a longer time interval, e.g. [0, 10000] in [Liao and Wang, 2014], is a challenging task that requires significant increase of the precision of the floating-point representation and the use of supercomputers. Analytical aspects of this problem are concerned with the so-called shadowing theory (see e.g. [Pilyugin, 2011]) which for some classes of systems can guarantee the existence of a ”true” trajectory in the vicinity of its approximation.

2019 IFAC TDS Sinaia, Romania, September 9-11, 2019

N.V. Kuznetsov et al. / IFAC PapersOnLine 52-18 (2019) 97–102

99

(and (2)) using the procedures for estimation of finite-time Lyapunov dimension. 2. LYAPUNOV DIMENSION, EDEN CONJECTURE AND TOPOLOGICAL ENTROPY Below we follow the concept of the finite-time Lyapunov dimension, which is convenient for carrying out numerical experiments with finite time. The finite-time local Lyapunov dimension [Kuznetsov, 2016b, Kuznetsov et al., 2018] can be defined via an analog of the Kaplan-Yorke formula with respect to the set of finite-time Lyapunov exponents: 3 dimL (t, u) = dKY L ({LEi (t, u)}i=1 ) = LE (t,u)+··+ LE

(t,u)

j(t,u) , (9) j(t, u) + 1 | LEj(t,u)+1 (t,u)| m where j(t, u) = max{m : i=1 LEi (t, u) ≥ 0}. Then the finite-time Lyapunov dimension (of dynamical system generated by (1) on compact invariant set A) is defined as dimL (t, A) = sup dimL (t, u). (10)

u∈A

The Douady–Oesterl´e theorem [Douady and Oesterle, 1980] implies that for any fixed t > 0 the finite-time Lyapunov dimension, defined by (10), is an upper estimate of the Hausdorff dimension: dimH A ≤ dimL (t, A). The best estimation is called the Lyapunov dimension [Kuznetsov, 2016b]: dimL A = inf sup dimL (t, u) = lim inf sup dimL (t, u). t>0 u∈A

t→+∞ u∈A

The notion of Lyapunov dimension is closely connected (see e.g. [Boichenko et al., 2005]) with another important characteristic of dynamical and control systems which is called topological entropy. Topological entropy was proposed in [Adler et al., 1965] as an analog of KolmogorovSinai entropy that can be introduced without consideration of invariant measures. It plays an important role in the development of large-scale control systems in which the control tasks are distributed among numerous processors via a communication network [Matveev and Pogromsky, 2017]. As the size of such systems grows, limitations caused by the network finite capacity becomes irremovable via the design of control algorithms. Here, topological entropy comes to the fore, since according to the fundamental data rate theorem [Nair et al., 2007] the rate at which the channel is capable of reliable data communication should exceed the topological entropy of the open-loop system. It is known that R¨ ossler system (1) is non-dissipative in the sense that it does not possess a bounded convex absorbing set [Leonov and Reitmann, 1986] containing a global attractor. Nevertheless, for parameters a1 = 0.2, b1 = 0.2, c1 = 5.7 in numerical experiments it is possible to observe an invariant attracting set A which could be localized numerically within a cuboid C (Fig. 2). On the set A we define a dynamical system {ϕt }t≥0 , generated  t by equations (1). Here ϕ (x0 , y0 , z0 ) is a solution of (1) with the initial data (x0 , y0 , z0 ). In Fig. 2 is shown the grid of points Cgrid filling the attractor: the grid of points fills cuboid C = [−9.5, 12] × [−11, 8] × [0, 23] with the distance between points equals to 0.5. The time interval is [0, T = 500], k = 5000, τ =

221

z

y

x

Fig. 2. Localization of the chaotic attractor of system (1) with parameters a1 = 0.2, b1 = 0.2, c1 = 5.7 by the cuboid C = [−9.5, 12] × [−11, 8] × [0, 23] (red) and the corresponding grid of points Cgrid (black). 0.1, and the integration method is MATLAB ode45 with predefined parameters. The infimum on the time interval is computed at the points {tk }N 1 with time step τ = ti+1 − ti = 0.5. Note that if for a certain time t = tk the computed trajectory is out of the cuboid, the corresponding value of finite-time local Lyapunov dimension is not taken into account in the computation of maximum of the finite-time local Lyapunov dimension (e.g. there are trajectories with initial data in cuboid, which tend to infinity). For the considered set of parameters we use MATLAB realization 4 of the adaptive algorithm of finite-time Lyapunov dimension and Lyapunov exponents computation [Kuznetsov et al., 2018] and obtain the following values: (i) maximum of the finite-time local Lyapunov dimensions at the points of grid, maxu∈Cgrid dimL (t, u), at the time points t = tk = 0.1 k (k = 1, .., 5000); (ii) finite-time Lyapunov dimensions dimL (500, ·) for the stabilized UPOs with periods τ1 = 5.8811 and τ2 = 11.7586; (iii) approximate value of topological entropy which is defined by the maximum of the finite-time local topological entropies   around the points of grid: maxu∈Cgrid Hloc 500, u ; (iv) the values of finite-time local local topological entropy  on the two UPOs: Hloc 500, uupo1,2 .

For numerical computation dimen of finite-time Lyapunov  sion for the UPO uupo = uupo (t), t ∈ [0, τ ] we choose an ∈ uupo and apply adaptive algorithm toinitial point uupo 0 gehter with Pyragas control to keep the computation along the uupo (t). Starting from the same point uupo on UPO we 0 also compute finite-time Lyapunov dimension along the trajectory but without stabilization (i.e. when K = 0 in (8)) which stops to trace the UPO and starts to wind on the chaotic attractor. The corresponding  upo results obtained  upo1 for two stabilized UPOs, u = u 1 (t), t ∈ [0, τ1 ] and   uupo2 = uupo2 (t), t ∈ [0, τ2 ] , with periods τ1 = 5.8811 and τ2 = 11.7586, respectively, are presented in Fig. 3 and Fig. 4. The results are given in Table 1. 4

Various realizations of algorithms that compute finite-time Lyapunov exponents and Lyapunov dimension for discontinuous, piecewise continuous and fractional order systems can be found e.g. in [Danca, 2015, 2018, Danca and Kuznetsov, 2018, Danca et al., 2018].

2019 IFAC TDS 100 Sinaia, Romania, September 9-11, 2019

N.V. Kuznetsov et al. / IFAC PapersOnLine 52-18 (2019) 97–102

0.3

2.06 2.055

0.25

2.05 2.045

25

0.2

20

z 10

2.035

0.15

uupo1 (t)

5 0 10

2.04

1 uupo 0

15

5

0

-5

y

-10

-15

-10

-5

5

0

10

2.03 2.025

15

0.1

2.02 2.015

x

0.05

0

100

200

(a)

300

400

500

2.01

0

100

200

(b)

300

400

500

(c)

upo1 3 1 1 )}1 ) for the time interval ) = dKY Fig. 3. Numerical computation of LE1 (t, uupo ) and dimL (t, uupo 0 0 L ({LEi (t, u0 upo1 t ∈ [0, 500] along the period-1 UPOs u (t) (red) and along the trajectory integrated without stabilization 1 (blue). Each trajectory starts from the point uupo = (6.491, −7.0078, 0.1155) (dark red). 0 0.3

2.06 2.055

0.25

2.05 2.045

25

0.2

20

z 10

2.035

uupo2 (t)

0.15

5 0 10

2.04

2 uupo 0

15

5

0

-5

y

-10

-15

-10

-5

5

0

10

2.03 2.025

15

0.1

2.02 2.015

x

0.05

0

100

200

(a)

300

400

(b) 2 LE1 (t, uupo ) 0 upo2

500

2.01

0

100

200

300

400

500

(c)

2 dimL (t, uupo ) 0

upo2 3 dKY )}1 ) L ({LEi (t, u0

Fig. 4. Numerical computation of and = for the time interval t ∈ [0, 500] along the period-2 UPOs u (t) (green) and along the trajectory integrated without stabilization 2 (blue). Each trajectory starts from the point uupo = (5.3914, −3.2889, 0.1099) (dark green). 0 Table 1. Approximation of the finite-time Lyapunov dimensions and entropy. t = 60

t = 500

maxu∈Cgrid dimL (t, u) 2 dimL (t, uupo ) 0 1 dimL (t, uupo ) 0

2.0209 2.0227 2.0406

2.0160 2.0200 2.0283

maxu∈Cgrid Hloc (t, u) Hloc (t, uupo2 ) Hloc (t, uupo1 )

0.1754 0.1756 0.3126

0.1250 0.1571 0.2219

corresponding to the unstabilized trajectory is lower than the part of the graph corresponding to the UPO (see Fig. 3b). The same situation is observed for the period2 UPO, but the obtained time intervals are smaller: the corresponding values of the largest Lyapunov exponents 2 LE1 (t, uupo ) coincide up to the 5th decimal place inclusive 0 1 on the interval [0, Tmatch ≈ 1.1τ2 ], up to the 4th decimal 2 place inclusive on the interval [0, Tmatch ≈ 3.6τ2 ], up to 3 the 3th decimal place inclusive on the interval [0, Tmatch ≈ 5.6τ2 ].

The comparison of the obtained values of LE1 (t, uupo 0 ) and upo 3 KY ) = d ({LE (t, u )} ) computed along the dimL (t, uupo i 1 0 0 L stabilized UPO and the trajectory without stabilization gives us the following results 5 . On the initial part of the time interval, one can indicate the coincidence of these values with a sufficiently high accuracy. For the period1 UPO and for the unstabilized trajectory the largest 1 Lyapunov exponents LE1 (t, uupo ) coincide up to the 5th 0 1 decimal place inclusive on the interval [0, Tmatch ≈ 5τ1 ], up to the 4th decimal place inclusive on the interval 2 [0, Tmatch ≈ 9τ1 ], up to the 3th decimal place inclusive on 3 3 the interval [0, Tmatch ≈ 12τ1 ]. After t > Tmatch the difference in values becomes significant and the corresponding graphics diverge in such a way that the part of the graph 5

The idea to compare finite-time Lyapunov dimensions of trajectories computed along UPOs with and without Pyragas control was discussed in [Kuznetsov and Mokaev, 2019].

222

For the considered values of parameters a1 = 0.2, b1 = 0.2, c1 = 5.7 the Jacobian at the equilibria O1± has the following simple eigenvalues λ2,3 (O1+ ) = −4.5 · 10−6 ± 5.428i, λ1 (O1+ ) = 0.1930, − λ1,2 (O1 ) = 0.0970 ± 0.9952i, λ3 (O1− ) = −5.6870. Therefore, we get + 3 dimL O1+ = dKY L ({Reλi (O1 )}i=1 ) = 3, − 3 dimL O1− = dKY L ({Reλi (O1 )}i=1 ) = 2.0341.

(11)

Let us mention here previous results on dimension of R¨ossler attractor. In [Peitgen et al., 2004, p. 644] it is stated that the fractal dimension is between 2.01 and 2.02, in [Fuchs, 2013, p. 80] it is stated that the correlation dimension is equal to 2.01. In literature we found the following values for the Lyapunov dimension: 2.014 [Froehling et al., 1981], 2.01 [Sano and Sawada, 1985], 2.0132 [Sprott,

2019 IFAC TDS Sinaia, Romania, September 9-11, 2019

N.V. Kuznetsov et al. / IFAC PapersOnLine 52-18 (2019) 97–102

2003, 2007, Fuchs, 2013], and 2.09635 [Awrejcewicz et al., 2018]. Using the formula of the local topological entropy of a system around an equilibrium point: Hloc (ueq ) = 3 1 j=1 max{Re[λj (ueq )], 0}, we get the the following ln 2 values of the local topological entropy of system (1) around O1± : Hloc (O1+ ) = 0.2784, Hloc (O1− ) = 0.2799. 3. CONCLUSION In this note we have confirmed the Eden conjecture for the R¨ ossler system (1) 6 and obtained the following relations between the Lyapunov dimensions: 3 = dimL O1+ > 2.0341 = dimL O1− > 2.0283 = dimL (500, uupo1 ) > 2.02 = dimL (500, uupo2 ) > 2.0160 = maxu∈Cgrid dimL (500, u) and various values of topological entropy: 0.2799 = Hloc (O1− ) > 0.2784 = Hloc (O1+ ) > 0.2219 = Hloc (500, uupo1 ) > 0.1571 = Hloc (500, uupo2 ) > 0.1250 = maxu∈Cgrid Hloc (500, u). Then for any invariant set or attractor contaning period1 UPO: A ⊃ uupo1 , we have the following lower-bound estimates for the Lyapunov dimension dimL A ≥ 2.0283 ≈ dimL (uupo1 ), and the topological entropy: H(A) ≥ 0.2219 ≈ Hloc (uupo1 ). The above numerical values are close to the exact values computed via multipliers of uupo1,2 . For the upper-bound estimates one can use special analytical methods (see, e.g. [Leonov, 1991, Boichenko and Leonov, 1998, Kuznetsov, 2016b]). ACKNOWLEDGEMENTS This work was supported by the Russian Science Foundation 19-41-02002. REFERENCES Adler, R., Konheim, A., and McAndrew, M. (1965). Topological entropy. Transactions of the American Mathematical Society, 114(2), 309–319. Afraımovic, V., Bykov, V., and Silnikov, L. (1977). On the origin and structure of the Lorenz attractor. 234(2), 336–339. Algaba, A., Freire, E., Gamero, E., and Rodr´ıguez-Luis, A. (2015). An exact homoclinic orbit and its connection with the R¨ ossler system. Physics Letters A, 379(16-17), 1114–1121. Auerbach, D., Cvitanovi´c, P., Eckmann, J.P., Gunaratne, G., and Procaccia, I. (1987). Exploring chaotic motion through periodic orbits. Physical Review Letters, 58(23), 2387. Awrejcewicz, J., Krysko, A., Erofeev, N., Dobriyan, V., Barulina, M., and Krysko, V. (2018). Quantifying chaos by various computational methods. Part 1: Simple systems. Entropy, 20(3), 175. 6

For system (3) with a3 = 0.386, b3 = 0.2 for period-1 and period-2 UPOs uupo1,2 corresponding local Lyapunov dimensions are: dimL uupo1 = 2.316, dimL uupo2 = 2.241, and local topological entropies are: Hloc (uupo1 ) = 0.134, Hloc (uupo2 ) = 0.092. Similar results can be obtained for the Lorenz system.

223

101

Barrio, R., Blesa, F., and Serrano, S. (2011). Qualitative and numerical analysis of the Rossler model: Bifurcations of equilibria. Computers and Mathematics with Applications, 62(11), 4140–4150. doi:10.1016/j.camwa. 2011.09.064. Barrio, R., Dena, A., and Tucker, W. (2015). A database of rigorous and high-precision periodic orbits of the Lorenz model. Computer Physics Communications, 194, 76–83. Boichenko, V. and Leonov, G. (1998). Lyapunov’s direct method in estimates of topological entropy. Journal of Mathematical Sciences, 91(6), 3370–3379. Boichenko, V., Leonov, G., and Reitmann, V. (2005). Dimension theory for ordinary differential equations. Teubner, Stuttgart. Chen, G. and Yu, X. (1999). On time-delayed feedback control of chaotic systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46(6), 767–772. Cruz-Villar, C.A. (2007). Optimal stabilization of unstable periodic orbits embedded in chaotic systems. Revista mexicana de f´ısica, 53(5), 415–420. Cvitanovi´c, P. (1991). Periodic orbits as the skeleton of classical and quantum chaos. Physica D: Nonlinear Phenomena, 51(1-3), 138–151. Danca, M.F. (2015). Lyapunov exponents of a class of piecewise continuous systems of fractional order. Nonlinear Dynamics, 81(1-2), 227–237. Danca, M.F. (2018). Lyapunov exponents of a discontinuous 4D hyperchaotic system of integer or fractional order. Entropy, 20(5), 337. Danca, M.F., Feˇckan, M., Kuznetsov, N., and Chen, G. (2018). Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system. Nonlinear Dynamics, 91(4), 2523– 2540. Danca, M.F. and Kuznetsov, N. (2018). Matlab code for Lyapunov exponents of fractional-order systems. International Journal of Bifurcation and Chaos, 28(05), 1850067. Douady, A. and Oesterle, J. (1980). Dimension de Hausdorff des attracteurs. C.R. Acad. Sci. Paris, Ser. A. (in French), 290(24), 1135–1138. Eden, A. (1989). An abstract theory of L-exponents with applications to dimension analysis (PhD thesis). Indiana University. Froehling, H., Crutchfield, J., Farmer, D., Packard, N., and Shaw, R. (1981). On determining the dimension of chaotic flows. Physica D: Nonlinear Phenomena, 3(3), 605–617. Fuchs, A. (2013). Nonlinear dynamics in complex systems. Springer. Galias, Z. (1999). All periodic orbits with period n ≤ 26 for the h´enon map. In Proceedings European Conference on Circuit Theory and Design, ECCTD’99, volume 1, 361–364. Citeseer. Galias, Z. (2006a). Counting low-period cycles for flows. International Journal of Bifurcation and Chaos, 16(10), 2873–2886. Galias, Z. (2006b). Short periodic orbits and topological entropy for the Chua’s circuit. In 2006 IEEE International Symposium on Circuits and Systems, 1667–1670. IEEE.

2019 IFAC TDS 102 Sinaia, Romania, September 9-11, 2019

N.V. Kuznetsov et al. / IFAC PapersOnLine 52-18 (2019) 97–102

Galias, Z. and Tucker, W. (2008). Rigorous study of short periodic orbits for the lorenz system. In 2008 IEEE International Symposium on Circuits and Systems, 764– 767. IEEE. Hooton, E. and Amann, A. (2012). Analytical limitation for time-delayed feedback control in autonomous systems. Phys. Rev. Lett., 109, 154101. Kehlet, B. and Logg, A. (2017). A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations. Numerical Algorithms, 76(1), 191–210. Kehlet, B. and Logg, A. (2013). Quantifying the computability of the Lorenz system using a posteriori analysis. In Proceedings of the VI Int. conf. on Adaptive Modeling and Simulation (ADMOS 2013). Kuznetsov, N. (2016a). Hidden attractors in fundamental problems and engineering models. A short survey. Lecture Notes in Electrical Engineering, 371, 13–25. doi:10. 1007/978-3-319-27247-4\ 2. (Plenary lecture at International Conference on Advanced Engineering Theory and Applications 2015). Kuznetsov, N. (2016b). The Lyapunov dimension and its estimation via the Leonov method. Physics Letters A, 380(25-26), 2142–2149. doi:10.1016/j.physleta.2016.04. 036. Kuznetsov, N., Leonov, G., Mokaev, T., Prasad, A., and Shrimali, M. (2018). Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dynamics, 92(2), 267–285. doi:10.1007/ s11071-018-4054-z. Kuznetsov, N., Leonov, G., and Shumafov, M. (2015). A short survey on Pyragas time-delay feedback stabilization and odd number limitation. IFAC-PapersOnLine, 48(11), 706–709. doi:10.1016/j.ifacol.2015.09.271. Kuznetsov, N. and Mokaev, T. (2019). Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension. Journal of Physics: Conference Series, 1205(1). doi:10.1088/1742-6596/1205/1/012034. art. num. 012034. Kuznetsov, N., Mokaev, T., and Vasilyev, P. (2014). Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Commun Nonlinear Sci Numer Simulat, 19, 1027–1034. Lehnert, J., H¨ ovel, P., Flunkert, V., Guzenko, P., Fradkov, A., and Sch¨ oll, E. (2011). Adaptive tuning of feedback gain in time-delayed feedback control. Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(4), 043111. Leonov, G. (1991). On estimations of Hausdorff dimension of attractors. Vestnik St. Petersburg University: Mathematics, 24(3), 38–41. [Transl. from Russian: Vestnik Leningradskogo Universiteta. Mathematika, 24(3), 1991, pp. 41-44]. Leonov, G. and Kuznetsov, N. (2013). Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 23(1). doi:10.1142/ S0218127413300024. art. no. 1330002. Leonov, G., Kuznetsov, N., and Mokaev, T. (2015). Homoclinic orbits, and self-excited and hidden attractors in a 224

Lorenz-like system describing convective fluid motion. The European Physical Journal Special Topics, 224(8), 1421–1458. doi:10.1140/epjst/e2015-02470-3. Leonov, G., Kuznetsov, N., and Vagaitsev, V. (2011). Localization of hidden Chua’s attractors. Physics Letters A, 375(23), 2230–2233. doi:10.1016/j.physleta.2011.04. 037. Leonov, G. and Reitmann, V. (1986). Das R¨ossler-System ist nicht dissipativ im Sinne von Levinson. Mathematische Nachrichten, 129(1), 31–43. Liao, S. and Wang, P. (2014). On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0,10000]. Science China Physics, Mechanics and Astronomy, 57(2), 330–335. Lin, W., Ma, H., Feng, J., and Chen, G. (2010). Locating unstable periodic orbits: When adaptation integrates into delayed feedback control. Physical Review E, 82(4), 046214. Matveev, A. and Pogromsky, A. (2017). Two Lyapunov methods in nonlinear state estimation via finite capacity communication channels. IFAC-PapersOnLine, 50(1), 4132–4137. Nair, G., Fagnani, F., Zampieri, S., and Evans, R. (2007). Feedback control under data rate constraints: An overview. Proceedings of the IEEE, 95(1), 108–137. Ott, E., Grebogi, C., and Yorke, J. (1990). Controlling chaos. Physical review letters, 64(11), 1196. Peitgen, H.O., J¨ urgens, H., and Saupe, D. (2004). Chaos and fractals: New frontiers of science. Springer-Verlag New York. Pilyugin, S. (2011). Theory of shadowing of pseudotrajectories in dynamical systems. Differencialnie Uravnenia i Protsesy Upravlenia (Differential Equations and Control Processes), 4, 96–112. diffjournal.spbu.ru/EN/numbers/2011.4/article.1.2.html. Pyragas, K. (1992). Continuous control of chaos by selfcontrolling feedback. Phys. Lett. A., 170, 421–428. Pyragas, V. and Pyragas, K. (2011). Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay. Physics Letters A, 375(44), 3866–3871. Pyragas, V. and Pyragas, K. (2013). Adaptive search for the optimal feedback gain of time-delayed feedback controlled systems in the presence of noise. The European Physical Journal B, 86(7), 306. R¨ossler, O. (1976a). Different types of chaos in two simple differential equations. Zeitschrift Naturforschung Teil A, 31, 1664–1670. R¨ossler, O. (1976b). An equation for continuous chaos. Physics Letters A, 57(5), 397–398. R¨ossler, O. (1979). Continuous chaos - four prototype equations. Annals of the New York Academy of Sciences, 316(1), 376–392. Sano, M. and Sawada, Y. (1985). Measurement of the Lyapunov spectrum from a chaotic time series. Physical Review Letters, 55(10), 1082. Sprott, J. (2003). Chaos and time-series analysis. Oxford University Press, Oxford. Sprott, J. (2007). Maximally complex simple attractors. Chaos: An Interdisciplinary Journal of Nonlinear Science, 17(3), 033124.