Discrete dynamics and difference equations: proceedings of the twelfth International Conference on Difference Equations and Applications, Lisbon, Portugal, 23 - 27 July 2007

Annotation. This volume holds a collection of articles based on the talks presented at ICDEA 2007 in Lisbon, Portugal. The volume encompasses current topics on stability and bifurcation, chaos, mathematical biology, iteration theory, nonautonomous systems, and stochastic dynamical systems Contents......Page 10 Preface......Page 8 Part 1 Papers by Main Speakers......Page 14 1. Introduction and motivation......Page 16 2. Newton maps for quintics......Page 17 3. Symbolic dynamics......Page 20 References......Page 26 1.1. Economics introduction......Page 28 1.2. Mathematics introduction......Page 30 2. Description of the family H and results......Page 31 3. Essential Nash map......Page 36 4. Making use of the symmetry......Page 42 5. Proof of the Main Theorem......Page 56 References......Page 57 1. Introduction......Page 59 2. Classification......Page 62 3. Cases 3 and 8......Page 64 4. Cases 2 and 7......Page 65 5. Cases 5 and 10......Page 67 6. Cases 4 and 9......Page 73 7. Cases 1 and 6......Page 77 Acknowledgment......Page 86 References......Page 87 1.1. Hyperbolic dynamics......Page 88 1.2. The frontier between order and chaos......Page 90 2.1. Examples: cookie-cutters and tent maps......Page 92 2.3. Teichmuller spaces......Page 93 3. Hyperbolic diffeomorphisms on surfaces......Page 96 3.1. Smoothness of the holonomies......Page 97 3.2. Teichmüller spaces......Page 98 3.3. Extension of A. N. Livšic and Ja. G. Sinai's eigenvalue formula......Page 101 3.4. Explosion of smoothness......Page 102 4.1. p-ordered and p-tent maps......Page 103 4.2. Feigenbaum-Coullet-Tresser universality......Page 104 4.3. Rigidity in the frontier between order and chaos......Page 106 4.4. The renormalization operator......Page 107 4.5. Hyperbolicity of the renormalization operator......Page 108 5.1. Hyperbolic dynamics......Page 110 5.2. The frontier between order and chaos......Page 111 References......Page 112 1. Introduction......Page 117 2. Idea of Investigation, Problems arising in so doing, and Main Results......Page 120 3. Metrics and Extended Phase Spaces......Page 122 4. Attractor in the Space CΔ......Page 123 5. Attractor in the Space C#......Page 125 6. Long-Term Properties of Solutions......Page 127 7. Application to boundary value problems......Page 129 8. Conclusion......Page 130 References......Page 131 1. Introduction......Page 133 2. Lyapunov-Schmidt reduction......Page 136 3. The bifurcation equation......Page 140 4. Additional structures......Page 141 4.1. Equivariance......Page 142 4.2. Reversibility......Page 143 4.3. Symplectic mappings......Page 145 5. Approximation of the reduced mapping......Page 147 6. Stability of bifurcating periodic orbits......Page 151 7. A basic example......Page 153 References......Page 156 Part 2 Invited Papers......Page 158 2. Pinto’s golden tilings......Page 160 2.3. Boundary condition......Page 161 3. Golden diffeomorphisms......Page 162 3.2. Golden arc exchange systems......Page 163 3.3. Golden renormalization......Page 164 4.1. Golden tilings......Page 165 5. Conclusion......Page 166 References......Page 167 Non-Exponential Stability and Decay Rates in Nonlinear Stochastic Homogeneous Difference Equations......Page 168 1. Introduction......Page 169 2. Auxiliary Definitions......Page 170 3. Discretized It- formula......Page 171 4. Stability and instability......Page 172 5.1. A comparison theorem......Page 173 5.2. Rate of decay of ln |xn|......Page 174 References......Page 175 1. Introduction......Page 176 2. Statement of the Problem......Page 177 3.2. Defining the discrete-time model......Page 179 3.4. Prior analysis in the literature......Page 180 4. Main Results......Page 181 5. Conclusions and Further Work......Page 184 References......Page 185 1. Introduction and preliminaries......Page 186 2. Functional relations for g arising from rule 184......Page 188 3. Transition matrices......Page 190 References......Page 193 1. Introduction......Page 194 2.1. The Jordan canonical forms......Page 195 3. The orbit of the critical point 0 in M2 (R)......Page 197 4.1. The existence of non-critical attractive cycles......Page 198 4.2. Non-critical attracting cycles of period 2 when C is of type III......Page 200 References......Page 201 1. Introduction......Page 202 2. Conversion to a Linear Difference Equation......Page 203 3. The Cushing-Henson Conjecture......Page 205 References......Page 206 1. Introduction......Page 207 2. Solution Estimates......Page 208 3. Global Attractivity......Page 212 References......Page 216 1. Introduction......Page 217 2. Periodic Trichotomy of Eq. (1)......Page 218 4. Periodically Forced Pielou's Equation......Page 219 5. The Beverton-Holt Equation......Page 222 References......Page 225 1. Introduction......Page 227 2. Main results......Page 228 2.1. Synchronization of piecewise linear maps......Page 229 2.2. Conjugacy and synchronization......Page 231 3. During oscillators's example and symbolic synchronization......Page 234 References......Page 235 1. Introduction......Page 236 2. Iterated smooth functions......Page 237 2.1. Localization of critical points and critical values......Page 238 2.2. Spectrum......Page 240 References......Page 243 1. Introduction......Page 244 2. The main properties of the comparative index......Page 246 3. Separation results......Page 248 References......Page 251 1. Introduction......Page 252 2. Settings......Page 253 3. Conductance......Page 255 4. Unimodal and bimodal applications......Page 256 References......Page 259 1. Introduction......Page 261 2. Model......Page 262 3. Stability as Strong Ergodicity......Page 264 4. Primitivity......Page 265 5. Traffic Distribution......Page 267 References......Page 269 1. Introduction......Page 271 2. Data analysis......Page 272 3. Intermittent dynamics of Paiva......Page 274 4. Nonlinear prediction......Page 275 5. Conclusions......Page 278 References......Page 279 1. Introduction......Page 280 2. Time scale symplectic systems......Page 283 3. Reid roundabout theorems......Page 285 4. Jacobi systems for control problems......Page 292 4.1. Control problem with shift in the state variable......Page 293 4.2. Control problem without shift in the state variable......Page 295 4.3. Euler-Lagrange system for functional F......Page 297 Acknowledgments......Page 299 References......Page 300 1. Introduction......Page 302 2. General Properties of the Quadratic Map......Page 303 2.1. Critical Curves......Page 305 3. Global Bifurcations......Page 307 References......Page 310 1. Introduction......Page 312 2. Ordinal patterns under iteration......Page 313 3. Distribution of ordinal patterns......Page 314 4. Permutation entropy and topological permutation entropy......Page 317 References......Page 319 1. Introduction......Page 320 2. Preliminaries......Page 321 3. Main result......Page 322 Acknowledgments......Page 324 References......Page 325 1. Introduction......Page 326 Preliminaries......Page 327 2. Linear Functional Equations in S......Page 328 3. Linear Dependence and Fundamental Solutions......Page 329 References......Page 333 1. Introduction......Page 334 2. Study of the profit rate as a function of time......Page 336 3.2. e (t) and k (t) grow with power model......Page 338 4.1. General model......Page 339 4.2. Model with a discrete dynamical system......Page 340 5. Chaos on the profit rate......Page 343 6. Conclusion......Page 347 References......Page 348 1. Introduction......Page 349 2. Numerical Range Theory......Page 350 3. Merging f(x) in W(Mn)......Page 351 4. Partitions of Ω......Page 354 5. Dynamics of f(x)......Page 355 References......Page 356 1. Introduction......Page 358 2. Classiffications of Nonoscillatory Solutions......Page 359 3. Oscillation Results......Page 362 References......Page 366 1. Introduction......Page 368 2. Oscillations......Page 369 3. Nonoscillations......Page 374 References......Page 376 1. Introduction......Page 378 2. Variational formulation of the problem......Page 379 2.1. Finite element approximations of the dynamic contact problems with friction......Page 381 2.2. Solution strategies for spatially discrete system and time steppining procedures......Page 382 2.3. Implicit methods......Page 383 2.5. Nonlinear equation solving with Newton-Raphson iterative method......Page 384 3. Numerical example and concluding remarks......Page 386 References......Page 387 1. Introduction......Page 388 2. Preliminaries......Page 389 3. Main Results......Page 391 References......Page 395 1. Introduction......Page 396 2.1. u ́(t) +Mu(t) = σ(t), t ε I, M > 0.......Page 398 2.2. u ́(t) = - Mu(t) + σ(t), t ε I, M > 0.......Page 400 2.3. u ́(t) = Mu(t) + σ(t), t ε I, M > 0.......Page 401 2.4. u ́(t) - Mu(t) = σ(t), t ε I, M > 0.......Page 402 References......Page 404 1. Introduction......Page 405 2. Main results......Page 406 References......Page 412 1. Introduction......Page 413 2. A Certain Class of Boundary Value Problems......Page 414 3. A Bimodal Boundary Condition......Page 416 4. The Spatial Distribution of Vortices for a Bimodal BVP......Page 419 References......Page 421 1. Introduction......Page 423 (1) The discrete spectral problems for several (2+1)-dimensional differential-difference hierarchies......Page 424 (2) Conservation laws for the (2+1)-dimensional BO lattice hierarchy......Page 429 (3) Conservation laws for the (2+1)-dimensional BS three lattice-field equations......Page 431 (4) Conservation laws for the (2+1)-dimensional BS four lattice-field equations......Page 433 (5) Conservation laws for several (1+1)-dimensional lattice-field equations......Page 435 References......Page 436 Combinatorics of Newton maps on quintic equations / F. Balibrea -- Hyperbolic dynamics in Nash maps / R.A. Becker [und weitere] -- Attractive cycles of an artificial neural network / S.S. Cheng and X.-L. Liu -- Hyperbolic and minimal sets / A.A. Pinto -- Difference equations with continuous time : Theory and applications / E. Yu. Romanenko and A.N. Sharkovsky -- Lyapunov-Schmidt : A discrete revision / A. Vanderbauwhede -- Pinto's golden tilings / J.P. Almeida -- Non-exponential stability and decay rates in nonlinear stochastic homogeneous difference equations / J.A.D. Appleby, G. Berkolaiko and A. Rodkina -- Dynamical consistency of solutions of continuous and discrete stochastic equations with a finite time explosion / J.A.D. Appleby, C. Kelly and A. Rodkina -- Interval maps and cellular automata / L. Bandeira, M.J. Martinho and C.C. Ramos -- Difference equations on matrix algebras / A.N. Baptista, C.C. Ramos and N. Martins -- The Beverton-Holt difference equation / M. Bohner, S. Stevic and H. Warth -- Global attractivity of a multiplicative delay population dynamics model / E. Braverman and S.H. Saker -- Periodically forced rational difference equations / E. Camouzis -- Chaotic synchronization of piecewise linear maps / A. Caneco, C. Gracio and J.L. Rocha -- On the iteration of smooth maps / M.F. Correia, C.C. Ramos and S. Vinagre -- The comparative index and the number of focal points for conjoined bases of symplectic difference systems / J.V. Elyseeva -- Conductance and mixing time in discrete dynamical systems / S. Fernandes and S. Jayachandran -- Pulsating equilibria : Stability through migration / T. Fujimoto, M. Li and P. Mossay -- Nonlinear prediction in complex systems using the Ruelle-Takens embedding / R. Goncalves and A.A. Pinto -- Reid roundabout theorems for time scale symplectic systems / R. Hilscher and V. Zeidan -- The global properties of a two-dimensional competing species model exhibiting mixed competition / P.J. Johnson and M.E. Burke -- Ordinal pattern distributions in dynamical dystems / K. Keller and M. Sinn -- Discretized pantograph equation with a forcing term : Note on asymptotic estimate / P. Kundrat -- General theory of linear functional equations in the space of strictly monotonic functions / J. Laitochova -- A dynamical system for a long term economic model / R. Luis and H. Oliveira -- Numerical range, numerical radii and the dynamics of a rational function / H. Melo and J. Cabral -- On first-order nonlinear difference equations of neutral type / M. Migda and J. Migda -- Oscillatory mixed differential difference equations / S. Pinelas -- Dynamic contact problems in linear viscoelasticity / N. Pop -- Regularly varying decreasing solutions of half-linear dynamic equations / P. Rehak and J. Vitovec -- Some considerations on fuzzy dynamical systems / R. Rodriguez-Lopez -- Asymptotic behavior of solutions of third order difference equations / E. Schmeidel and M. Zbaszyniak -- Difference equations and nonlinear boundary value problems for hyperbolic systems / A.N. Sharkovsky, R. Severino and S. Vinagre -- Conservation laws of several (2 + 1)-dimensional differential-difference hierarchies / Z.-N. Zhu