), 83–104. [37] Mrozek M., Index pairs algorithms , Found. Comput. Math. 6 (2006), 457–493. [38] Mrozek M., Srzednicki R., Topological approach to rigorous numerics of chaotic dynamical systems with strong expansion , Found. Comput. Math. 10 (2010), 191–220. [39] Mrozek M., Srzednicki R., Weilandt F., A topological approach to the algorithmic computation of the Conley index for Poincaré maps , SIAM J. Appl. Dyn. Syst. 14 (2015), 1348–1386. [40] Mrozek M., Żelawski M., Heteroclinic connections in the Kuramoto-Sivashinsky equations , Reliab. Comput. 3 (1997

observed among the multiple time scales in the brain. In this paper we derive a Poincaré map that allows to predict how many revolutions the trajectory stays within an SHC before it switches to a heteroclinic connection of the LHC, provided that we start already in the vicinity of a saddle. The subsequent evolution of the trajectory is not covered by this map: after the switch towards a heteroclinic connection of the LHC it may either turn to a new, second SHC, or stay within the LHC. In section 2 we present the model with the choice of predation rates, in section 3

Introduction .................................................................................................. 89 4.1. System with Impacts............................................................................... 90 4.2. Lyapunov Stability of the Periodic Motion with Impacts ......................... 93 4.3. Ball Bouncing in the Gravitational Field ................................................. 95 4.3.1. Subject of the Study ......................................................................... 95 4.4. The Poincaré Map

in one period, 123, 144 mapping fixed points, 55 nonlinear, 65 method comparison, 2, 6 dynamic integral inequalities, 2, 6 motion limiter, 90 periodic, 118, 147 stability, 161 neural network on a time scale, 21 nonlinear mapping, 65 semigroup quasicontractive, 65 semigroup, 65 one impact in n periods, 121, 142 operator property C, 71 G, 79 resolvent set, 64 ordinary differential equations, 1, 5, 11, 15, 30, 67, 90 period doubling phenomenon, 156 periodic point asymptotically stable, 103 of transformation, 98 Poincaré map, 95, 109

different states of a dynamical system as parameter is varied. At Fig. 5 the value of control parameter (a forcing frequency) is plotted on the horizontal axis and the values of phase coordinate x 2 ( t ) at Poincaré points are plotted on the vertical axis. There is only one value of one point coordinate in Poincaré map for (1, 1)-regime, we see one point along vertical line at bifurcation diagram for ω < 6.07 rad⋅s -1 and ω > 6.38 rad⋅s -1 . There are 2 separate points along vertical line for (2, 2) and (2, 3)-periodic regimes. They are the regimes with 2 T

– eleven stitched up impact occurrences. Each cycle is analogous to a single impact. Figure 5.4 Eleven-cycle solution The same case of bouncing ball dynamics but written in non-dimensional terms (see Eqs. 4.13) is for the reader convenience reiterated here as a set of difference equations (5.5) that represent the Poincaré map for the case under consideration: 1 1 ( ) 1 1 ( ) 1 1 1 2 2( ) ( ) ( )(1 ) ( ), 2 2[( ) ] (1 ) ( ). i i i i X X i i i i i X X i i i Y X Y X V e X X V R V e R Y X (5.5) 5

, if ρ is close enough to O and we follow the orbit for positive values of the time t , it will cut Σ again at some point. We define the Poincaré map 𝒫 : Σ → Σ being 𝒫 ( r ) the point in Σ corresponding to the first cut with Σ of the orbit through ρ in positive time. It is clear that the origin of the analytic differential system 1 is a center if and only if the Poincaré map is the identity. For systems with the linear type form 2 and for systems with the nilpotent form 3 , it is possible to find a parametrization of Σ such that the Poincaré map is

1 : v ′ dr = −0.002788 m ; ξ = 0.412; κ = 1.5. A waveband arises centred on the value of 0.143433 Hz, a value that has no relation with belt velocity. Fig. 14 shows the Poincaré map using a period of 1/ 0.143433 for a time span from 0 to 3,200. Figure 14 Poincaré map: v ′ dr = −0.002788 m ; ξ = 0.412; κ = 1.5. Fig. 14 shows the chaotic behaviour of the system. To quantify the chaos, an approximation to the maximum Lyapunov exponent is calculated from a time series of the solution by the software OpenTSTool [ 80 ], developed by Göttingen University

gravitational field and restrained by an oscillating limiter which represents an oscillating plate. There are moments that the material point collides with the limiter but in the time intervals between impacts, it behaves like a free projectile subjected in its flight to some viscous (linear) damping. Its motion is described by the following equation: 0,x hx g (4.8) 4.4. The Poincaré Map 96 where x is a coordinate of the material point and h is a measure of the energy dissipation due to the damping. Note that, in most similar considerations, it was typical