The book of abstracts - CAMTP

Workshop
Dynamical Systems and Applications
in the framework of the project FP7-PEOPLE-2012-IRSES-316338
within the 7th European Community Framework Programme
Hotel PIRAMIDA, Maribor, Slovenia
23 August 2013 - 24 August 2013
Book of Abstracts
Workshop
Dynamical Systems and Applications
in the framework of the project FP7-PEOPLE-2012-IRSES-316338
within the 7th European Community Framework Programme
Hotel PIRAMIDA, Maribor, Slovenia
23 August 2013 - 24 August 2013
Organizer
C ENTER FOR A PPLIED M ATHEMATICS AND T HEORETICAL P HYSICS
U NIVERZA V M ARIBORU • U NIVERSITY OF M ARIBOR
K REKOVA 2 • SI-2000 M ARIBOR • S LOVENIA
Phone +(386) (2) 2355 350 and 2355 351 • Fax +(386) (2) 2355 360
www.camtp.uni-mb.si
Co-organizer
FACULTY OF N ATURAL S CIENCES AND M ATHEMATICS
U NIVERSITY OF M ARIBOR , K ORO Sˇ KA CESTA 160, SI-2000 M ARIBOR , S LOVENIA
Phone +(386) (2) 2293 844 • Fax +(386) (2) 2518 180
www.fnm.uni-mb.si
Sponsor
Slovenian Research Agency
CIP - Kataloˇzni zapis o publikaciji Univerzitetna knjiˇznica Maribor
001.891:061.1EU:517.93(082)(048.3)
WORKSHOP dynamical systems and applications in the framework of the project
FP7-PEOPLE-2012-IRSES-316338 within the 7th European Community Framework Programme, 23 August - 24 August 2013 : book of abstracts / [urednik
Valery Romanovski ; organizer Center for applied mathematics and theoretical
physics]. - Maribor : Center za uporabno matematiko in teoretiˇcno fiziko, 2013
ISBN 978-961-281-121-1
1. Center for applied mathematics and theoretical physics (Maribor)
COBISS.SI-ID 75254273
1
Organizing Committee
Prof. Dr. Marko Robnik
CAMTP, University of Maribor, Slovenia
Prof. Dr. Valery Romanovski
CAMTP, University of Maribor, Slovenia
Prof. Dr. Yonghui Xia
Zhejiang Normal University, China
2
Invited Speakers
Prof. Dr. Vladimir Amel’kin, Belarusian State University, Belarus
Prof. Dr. Ilknur Kusbeyzi Aybar, Yeditepe University, Turkey
Mr. Orhan Ozgur Aybar, Yeditepe University, Turkey
Prof. Dr. Xingwu Chen, Sichuan University, China
Prof. Dr. Wei Ding, Shanghai Normal University, China
Prof. Dr. Maoan Han, Shanghai Normal University, China
Dr. Athanasios Manos, CAMTP, University of Maribor
and University of Nova Gorica, Slovenia
Prof. Dr. Matej Mencinger, University of Maribor, Slovenia
Prof. Dr. Alexander Prokopenya, Warsaw University of Life Sciences, Poland
Prof. Dr. Marko Robnik, CAMTP, University of Maribor, Slovenia
Prof. Dr. Valery Romanovski, CAMTP, University of Maribor, Slovenia
Prof. Dr. Andreas Ruffing, Technical University of Munchen, Germany
Prof. Dr. Ekaterina Vasil’eva, Saint Petersburg State University, Russia
Prof. Dr. Andreas Weber, University of Bonn, Germany
Prof. Dr. Yepeng Xing, Shanghai Normal University, China
Prof. Dr. Weinian Zhang, Sichuan University, China
ˇ
Prof. Dr. Vesna Zupanovi´
c, University of Zagreb, Croatia
3
SCHEDULE
Friday 23 August
9:30-9:40
Opening
Chairman
Weinian Zhang
09:40-10:20 Maoan Han
10:20-11:00 Andreas Weber
11:00-11:30 Tea & Coffee
11:30-12:10 Ekaterina Vasileva
12:10-12:50 Athanasios Manos
12:50-13:30 Wei Ding
13:30-15:00 Lunch
Chairman
Andreas Weber
15:00-15:40 Vladimir Amel’kin
15:40-16:20 Xingwu Chen
16:20-16:50 Tea & Coffee
16:50-17:30 Andreas Ruffing
17:30-18:10 Ilknur Kusbeyzi Aybar
18:10-18:30 Lingling Liu
19:00
Welcome Dinner
4
Saturday 24 August
Chairman
Maoan Han
09:00-09:40 Marko Robnik
09:40-10:20 Weinian Zhang
10:20-11:10 Tea & Coffee
11:10-11:50 Alexander Prokopenya
11:50-12:30 Yepeng Xing
ˇ
12:30-13:10 Vesna Zupanovi´
c
13:10-15:00 Lunch
Chairman
Alexander Prokopenya
15:00-15:40 Valery Romanovski
15:40-16:20 Matej Mencinger
16:20-16:50 Tea & Cofee
16:50-17:30 Orhan Ozgur Aybar
17:30-17:50 Yuanyuan Liu
17:50-18:10 Maˇsa Dukari´c
18:30
Dinner
5
ABSTRACTS
6
Strongly isochronous two-dimensional holomorphic
systems of the ordinary differential equations
VLADIMIR V. AMEL’KIN
Belarusian State University
Minsk, Nezavisimosti avenue 4
220030 Minsk, Republic of Belarus
[email protected]
http://www.bsu.by
We consider the real autonomous holomorphic systems on the plane
with pure imaginary roots of the characteristic equation of the first
approximation system.
We give an overview of practically all known results concerning the
strong isochronicity of polynomial centers with the indication of the
maximum order of strong isochronicity and the initial polar angle.
In particular, quadratic systems, systems with incomplete polynomials of the third, fourth, fifth degrees are considered.
7
Stability, bifurcation and normal form analysis of
predator-prey systems
ILKNUR KUSBEYZI AYBAR
Yeditepe University
34755 Istanbul, Turkey
[email protected]
http://www.yeditepe.edu.tr
The predator-prey system attempts to model the relationship in the
populations of different species that share the same environment
where some of the species (predators) prey on the others. The wellknown problem the Lotka-Volterra model which stands as a starting point for more complex generalizations of predator-prey system
lacks to represent realistic models with its unrealistic stability characteristics. Therefore generalizations including higher nonlinear interactions are studied. Changes in the number and stability of equilibrium points in the generalized predator-prey system that lead to
qualitative changes in the behavior of the system as a function of
some of its parameters are studied by bifurcation analysis. Numerical methods and the approach provided by approximate techniques
near equilibrium points are used in the analysis.
References
[1] I. Kusbeyzi, O.O. Aybar, A. Hacinliyan, Stability and bifurcation in two species predator-prey models, Nonlinear Analysis:
Real World Applications 12 (2011) 377-387.
[2] I. Kusbeyzi, A. Hacinliyan, Bifurcation scenarios of some modified predator-prey nonlinear systems, Journal of Applied Functional Analysis 4(3) (2009) 519-527.
8
[3] Y. Nutku, Hamiltonian structure of the Lotka-Volterra equations, Physics Letters A I145 (1990) 27-28.
9
Regular or chaotic behavior
in a class of hamiltonian systems
ORHAN OZGUR AYBAR
Gebze Institute of Technology
Department of Mathematics, Gebze-Kocaeli, Turkey
Yeditepe University
Department of Information Systems and Technologies, Istanbul, Turkey
[email protected]
In this paper we study possible regular and chaotic behavior and
approximately conserved quantities in a class of generalized H´enonHeiles and classical Yang-Mills Hamiltonians. We present a method
for identifying chaotic and regular behavior by considering the evolution of its tangent space. The method is applied to a class of
Hamiltonian systems to demonstrate the idea. In order to distinguish a small positive characteristic exponent from a zero characteristic exponent, a method for identifying the presence of zero characteristic exponents is suggested. Approximately conserved quantities are investigated via Poisson brackets of the Hamiltonian and
possible invariants.
This work is based on my PhD Thesis supervised by Prof. A. Hacinliyan.
References
[1] O. Ozgur Aybar, I. Birol, A. Hacinliyan, I. Kusbeyzi Aybar, Regular or Chaotic Behavior in a Class of Hamiltonian Systems, in
preparation
10
Local bifurcation of critical periods for isochronous
centers
XINGWU CHEN
Department of Mathematics, Sichuan University
Chengdu, Sichuan 610064, P. R. China
[email protected]
In this talk we discuss the local bifurcation of critical periods for
planar differential systems. From 1989 when Chicone and Jacobs
founded the basic theory of the critical period bifurcation, many
works have been done for several families of systems, specially for
polynomial systems. Here we introduce some interesting results
and mainly discuss the case that critical periods bifurcating from
isochronous centers. For the isochronous centers of an n-th degree
homogeneous vector field perturbing in m-th degree time-reversible
systems, period-bifurcation functions are presented as integrals of
analytic functions which depend on perturbation coefficients and
the problem of critical periods is reduced to finding zeros of a judging function.
References
[1] C. Chicone, M. Jacobs, Bifurcation of critical periods for plane
vector fields, Trans. Amer. Math. Soc. 312(1989) 433-486.
[2] X. Chen, V. G. Romanovski, W. Zhang, Critical periods of perturbations of reversible rigidly isochronous centers, J. Differential Equations 251(2011) 1505-1525.
11
Traveling wave solutions
for some reaction-diffusion models
WEI DING
Department of Applied Mathematics
Shanghai Normal University
Shanghai, China, 200234
[email protected]
http://www.shnu.edu.cn
We study the traveling wave solutions of two kinds of reactiondiffusion models. We discuss the existence of traveling wave solutions with the techniques of qualitative analysis, upper-lower solutions, and monotone iteration method. We also find the precise
value of the minimum speeds that are significant to study the spreading speed.
References
[1] L.J.S. Allen, B.M. Boller, Y. Lou and A.L. Nevai, Asymptotic
profiles of the steady states for an SIS epidemic reactiondiffusion model, Discrete Contin. Dyn. Syst. A, 21(2008), 1-20.
[2] J. Yang, S. Liang and Y. Zhang, Travelling Waves of a
Delayed SIR Epidemic Model with Nonlinear Incidence
Rate and Spatial Diffusion.
PLoS ONE 6(6): e21128.
doi:10.1371/journal.pone.0021128 (2011).
[3] W.Huang, Problem on Minimum Wave Speed foe a LotkaVolterra Reaction- Diffusion Competition Model. J Dyn Diff
Equat. 22(2010),285-297.
12
[4] W.Huang, W.Wu, Minimum wave speed for a diffusive competition model with time delay. Journal of Applied Analysis and
Computation. 1(2)(2011),205-218.
[5] R. Peng and S. Liu, Global stability of the steady states of an SIS
epidemic reaction-diffusion model, Non. Analysis, 71(2009),
239-247.
13
The solution of the 1 : -3 resonant center problem in
the quadratic case
ˇ DUKARIC,
´ BRIGITA FERCEC
ˇ
MASA
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor, Krekova 2
SI-2000 Maribor, Slovenia
[email protected]; [email protected]
JAUME GINE´
Departamento de Matematica
Universitat de Lleida, Avda. Jaume II, 69
25001 Lleida, Spain
[email protected]
The 1 : −3 resonant center problem in the quadratic case is to find
necessary and sufficient conditions for existence of local analytic
first integrals of the differential system
x˙ = x − a10 x2 − a01 xy − a12 y 2 ,
y˙ = −3y + b21 x2 + b10 xy + b01 y 2 .
Dividing this problem into two smaller problems, where a01 = 0
and a01 = 1, we gain 25 center cases for a01 = 1 and 11 cases for
a01 = 0. The necessary conditions are obtained using modular arithmetics. The main tool to prove the sufficiency of the obtained conditions is the method of Darboux, however some other technics were
used as well.
14
References
[1] X. Chen, J. Gine, V.G. Romanovski, D.S. Shafer, The 1 : −q resonant center problem for certain cubic Lotka-Volterra systems,
Appl. Math. Comput. 218 (2012), 11620–11633.
[2] V.G. Romanovski, M. Preˇsern, An approach to solving systems
of polynomials via modular arithmetics with applications, J.
Comput. Appl. Math. 236 (2011), no. 2, 196–208.
[3] M. Dukari´c, B. Ferˇcec, J. Gin´e, The solution of the 1:-3 resonant
center problem in the quadratic case, submitted to J. Comput.
Appl. Math. (2013).
15
The stability of some kinds of generalized homoclinic
loops in planar piecewise smooth systems
FENG LIANG, MAOAN HAN†
Department of Mathematical, Shanghai Normal University
Shanghai, China, 200234
†
[email protected]
In this paper we present some kinds of generalized homoclinic loops
in planar piecewise smooth systems. By establishing Poincar´
e map,
we study the stability of these homoclinic loops. For each case, a
criteria for stability is given. As applications, several concrete piecewise systems are considered.
References
[1] A. F. Filippov, Differential Equations with Discontinuous
Righthand Sides. Kluwer Academic Netherlands (1988) .
[2] M. Han, Asymptotic expansions of Melnikov function and limit
cycle bifurcations. International Journal of Bifurcation and Chaos
12 (2012) 1-20
[3] M. Han, Bifurcation theory of limit cycles. Science Press Beijing
(2013) .
[4] M. Han and W. Zhang, On the Hopf bifurcation in nonsmooth
planar systems. Journal of Differential Equations 248 (2010) 2399416.
[5] M. Kunze, Non-Smooth Dynamical Systems. Springer-Verlag
Berlin (2000).
[6] F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with
a homoclinic loop. Nonlinear Analysis: Theory, Methods & Applications 75 (2012) 4355-4374.
16
[7] F. Liang and M. Han, Limit cycles near generalized homoclinic
and double homoclinic loops in piecewise smooth systems.
Chaos, Solitons & Fractals 45 (2012) 454-464.
[8] J. Yang, M. Han and W. Huang, On Hopf bifurcations of piecewise planar Hamiltonian systems. Journal of Differential Equations 250 (2011) 1026-1051.
17
Identification of focus and center in a 3-dim system
LINGLING LIU
Department of Mathematics, Sichuan University
Chengdu, Sichuan 610064, P.R.China
[email protected]
We identify focus and center for a generalized Lorenz system, a 3dimensional quadratic polynomial differential system. It is a wellknown routine to restrict the system to its a center manifold, but the
approximation of the center manifold brings a great complexity in
the computation of Lyapunov quantities and their dependence even
if the original system is not of high degree. On the other hand, in
order to determine if an center-focus equilibrium is a center, those
criteria for planar systems such as time-reversibility and integrability are not available on an approximated center manifold. We prove
its Darboux integrability by finding an invariant surface, showing
that the equilibrium of center-focus type is actually a rough center
on a center manifold.
References
[1] Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang, Identification of Focus and Center in a 3-Dimensional System, submitted to Discr. Contin. Dyn. Syst. B (2013)
18
Limit cycle bifurcations of a class of piecewise smooth
Li´enard systems
YUANYUAN LIU, MAOAN HAN
Department of Mathematical, Shanghai Normal University
Shanghai, China, 200234
[email protected]; [email protected]
VALERY ROMANOVSKI
Center for Applied Mathematics and Theoretical Physics
University of Maribor
Krekova 2, Maribor 2000, Slovenia
[email protected]
In this paper, we consider a class of piecewise smooth Li´enard systems as follows
x˙ = y,
y˙ = −g(x) − f (x)y,
where g(x) and f (x) are piecewise polynomials. We classify the unperturbed system into three types and study the bifurcation of limit
cycles for the above system. By studying the expansions of the first
order Melnikov function, we give some new results on the number
of limit cycles in Hopf and homoclinic bifurcations.
References
[1] M. Han, Bifurcation theory of limit cycles. Science Press Beijing
(2013) .
[2] M. Han, Asymptotic expansions of Melnikov function and limit
cycle bifurcations. International Journal of Bifurcation and Chaos
12 (2012) 1-20
19
[3] M. Han and W. Zhang, On the Hopf bifurcation in nonsmooth
planar systems. Journal of Differential Equations 248 (2010) 2399416.
[4] X. Liu and M. Han, Bifurcation of limit cycles by perturbing
piecewise Hamiltionian systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 5 (2010) 1-12.
[5] L. Wei, F. Liang and S. Lu, Limit cycle bifurcations near a generalized homoclinic loop in piecewise smooth systems with a hyperbolic saddle on a switch line , submitted to Applied Mathematics and Computation. (2013)
[6] F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with
a homoclinic loop. Nonlinear Analysis: Theory, Methods & Applications 75 (2012) 4355-4374.
[7] F. Liang and M. Han, Limit cycles near generalized homoclinic
and double homoclinic loops in piecewise smooth systems.
Chaos, Solitons & Fractals 45 (2012) 454-464.
[8] J. Yang, M. Han and W. Huang, On Hopf bifurcations of piecewise planar Hamiltonian systems. Journal of Differential Equations 250 (2011) 1026-1051.
20
Dynamical localization, spectral statistics and
localization measure in the kicked rotator
THANOS MANOS1,2 , MARKO ROBNIK1
1
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
and
2
School of applied sciences
University of Nova Gorica, Vipavska 11c, SI-5270, Ajdovˇscˇ ina, Slovenia
[email protected] • www.camtp.uni-mb.si/camtp/manos
We study the kicked rotator in the classically fully chaotic regime
using Izrailev’s N -dimensional model for various N ≤ 4000, which
in the limit N → ∞ tends to the quantized kicked rotator. We do not
treat only the case K = 5 as studied previously, but many different
values of the classical kick parameter 5 ≤ K ≤ 35, and also many
different values of the quantum parameter k ∈ [5, 60]. We describe
the features of dynamical localization of chaotic eigenstates as a
paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed type Hamilton systems. We
generalize the scaling variable Λ = l∞ /N to the case of anomalous
diffusion in the classical phase space, by deriving the localization
length l∞ for the case of generalized classical diffusion. We greatly
improve the accuracy and statistical significance of the numerical
calculations, giving rise to the following conclusions: (C1) The level
spacing distribution of the eigenphases (or quasienergies) is very
well described by the Brody distribution, systematically better than
by other proposed models, for various Brody exponents βBR . (C2)
We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by βloc in the interval [0, 1].
The level repulsion parameters βBR and βloc are almost linearly related, close to the identity line. (C3) We show the existence of a
21
scaling law between βloc and the relative localization length Λ, now
including the regimes of anomalous diffusion. The above findings
are important also for chaotic eigenstates in time-independent systems (Batisti´c and Robnik 2010,2013), where the Brody distribution
is c onfirmed to a very high degree of precision for dynamically
localized chaotic eigenstates even in the mixed-type systems (after
separation of regular and chaotic eigenstates).
References
[1]
[2]
[3]
[4]
[5]
[6]
Izrailev F M 1988 Phys. Lett. A 134 13
Izrailev F M 1990 Phys. Rep. 196 299
Batisti´c B and Robnik M 2010 J. Phys. A: Math. Gen. 43
Manos T and Robnik M 2013 Phys. Rev. E 87 062905
Batisti´c B, Manos T and Robnik M 2013 EPL 102 50008
Batisti´c B and Robnik M 2013 J. Phys. A (in press)
22
The study of isochronicity and critical period
bifurcations on center manifolds of some 3-dim
polynomial systems
MATEJ MENCINGER
FG - Faculty of Civil Engineering
University of Maribor, Slomˇskov trg 15
SI-2000 Maribor, Slovenia
[email protected]
http://www.fg.uni-mb.si/
and
IMFM - Institute of Mathematics, Physics and Mechanics
SI-1000 Ljubljana, Slovenia
The main topic of this talk is the investigation of a quadratic 3D
system of ODEs
u˙ = −v + au2 + av 2 + cuw + dvw
v˙ = u + bu2 + bv 2 + euw + f vw
w˙ = −w + Su2 + Sv 2 + T uw + U vw
(1)
with real coefficients a, b, c, d, e, f, S, T and U . System (1) was studied already in [1], and further in [2,3], where planar polynomial systems of ODEs appearing on the center manifold of (1) were studied.
Using the solutions of the center-focus problem from [1], confirmed
in [3] by the so called modular approach [4], we present the investigation of four (at most three parameter-) families of (at most third
degree) polynomial systems corresponding to the center varieties
of (1). Thus, all systems under consideration correspond to a center
manifold filled with closed trajectories (corresponding to periodic
solutions of (1)).
23
In particular, in the talk I shall present the criteria on the coefficients of the system to distinguish between the cases of isochronous
and non-isochronous oscillations, considered in [2,3]. Bifurcations
of critical periods of the system will be presented as well.
References
[1] V. F. Edneral, A. Mahdi, V. G. Romanovski and D. S. Shafer, The
center problem on a center manifold in R3 , Nonlinear Anal. 75
(2012) 2614-2622.
[2] B. Ferˇcec, M. Mencinger , Isochronicity of centers at a center
manifold. AIP conference proceedings, 1468. Melville, N.Y.: American Institute of Physics, 2012, pp. 148-157.
[3] V. G. Romanovski, M. Mencinger and, B. Ferˇcec, Investigation
of center manifolds of 3-dim systems using computer algebra,
Program. Comput. Softw. 39 (2013) 67-73.
[4] V. G. Romanovski, M. Preˇsern, An approach to solving systems
of polynomials via modular arithmetic with applications. Journal of Computational and Applied Mathematics 236 (2011) 196-208.
24
Adiabatic invariants and some statistical properties of
the time-dependent linear and nonlinear oscillators
MARKO ROBNIK
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor, Krekova 2
SI-2000 Maribor, Slovenia
[email protected]
http://www.camtp.uni-mb.si
We consider 1D time dependent Hamiltonian systems and their statistical properties, namely the time evolution of microcanonical distributions, whose properties are very closely related to the existence and preservation of the adiabatic invariants. We review the
elements of the recent developments by Robnik and Romanovski
(2006-2008) for the entirely general 1D time dependent linear oscillator and try to generalize the results to the 1D nonlinear Hamilton
oscillators, in particular the power-law potentials like e.g. quartic
oscillator. Furthermore we consider the limit opposite to the adiabatic limit, namely parametrically kicked 1D Hamiltonian systems.
Even for the linear oscillator interesting properties are revealed: an
initial kick disperses the microcanonical distribution to a spread
one, but an anti-kick at an appropriate moment of time can annihilate it, kicking it back to the microcanonical distribution. The case of
periodic parametric kicking is also interesting. Finally we propose
that in the parametric kicking of a general 1D Hamilton system the
average value of the adiabatic invariant always increases, which we
prove for the power-law potentials. We find that the approximation
of kicking is good for quite long times of the parameter variation,
up to the order of not much less than one period of the oscillator.
We also look at the behaviour of the quartic oscillator for the case of
the kick and anti-kick, and also the periodic kicking.
25
References
[1] G. Papamikos, M. Robnik, J.Phys.A: Math.Theor.
315102.
44 (2011)
[2] M. Robnik, V.G. Romanovski, J.Phys.A: Math.Gen.
L35-L41.
39 (2006)
[3] M. Robnik, V.G. Romanovski, Open Sys. and Inf. Dynamics 13
(2006) 197-222.
[4] M.
Robnik,
V.G.
Romanovski,
J.Phys.A:Math.Gen. 39 (2006) L551-L554.
H.-J.
¨
Stockmann,
[5] A.V. Kuzmin, M. Robnik, Rep. Math. Phys. 60 (2007) 69-84.
[6] M. Robnik, V.G. Romanovski, ”Let’s Face Chaos through Nonlinear Dynamics”, Proceedings of the 7th International summer
school/conference, Maribor-Slovenia, 2008, in AIP Conf. Proc.
1076, Eds. M.Robnik and V.G. Romanovski.
[7] M. Robnik, V.G. Romanovski, J.Phys.A: Math. Theor.
5093.
33 (2000)
[8] G. Papamikos, B.C. Sowden, M. Robnik, Nonl.Phen.Compl.Sys.
15, No 3, (2012) 227-240.
26
Integrability of polynomial systems of ODEs
VALERY G. ROMANOVSKI
CAMTP - Center for Applied Mathematics and Theoretical Physics
University of Maribor, Krekova 2
SI-2000 Maribor, Slovenia
[email protected]
http://www.camtp.uni-mb.si
Consider polynomial vector fields on Rn having a singularity at the
origin with non-degenerate linear part. We discuss the problems
of existence of analytic first integrals, center manifolds, periodic solutions and the problem of stability, and their interconnection. We
also explicitly compute center manifolds and first integrals for several families of systems with quadratic and cubic higher order terms
and describe computational methods for the study. We also discuss
main mechanisms for integrability: Darboux integrability and timereversibility.
Bifurcations of periodic solutions for some families of polynomial
systems are considered as well.
References
[1] A. Mahdi, V. G. Romanovski, D. S. Shafer, Stability and periodic
oscillations in the Moon-Rand systems, Nonlinear Analysis: Real
World Applications 14 (2012) 294-313.
[2] V. G. Romanovski, M. Preˇsern, An approach to solving systems
of polynomials via modular arithmetic with applications. Journal of Computational and Applied Mathematics 236 (2011) 196-208.
27
[3] Z. Hu, M. Aldazharova, T. M. Aldibekov, V. G. Romanovski,
Integrability of 3-dim polynomial systems with three invariant
planes, submitted to Nonlinear Dynamics (2013)
[4] Z. Hu, H. Maoan, V. G. Romanovski, Local integrability of a
family of three-dimensional quadratic systems, submitted to
Physica D (2013)
28
Quantum difference–differential equations
ANDREAS L. RUFFING
Technische Universit¨at Munchen
¨
Department of Mathematics
Boltzmannstraße 3
D-85747 Garching, Germany
[email protected]
http://www-m6.ma.tum.de/∼ruffing/
Differential equations which contain the parameter of a scaling process are usually referred to by the name Quantum Difference–Differential
Equations. Some of their applications to discrete models of the
¨
Schrodinger
equation are presented and some of their rich, filigrane
und sometimes unexpected analytic structures are revealed.
A Lie-algebraic concept for obtaining basic adaptive discretizations
is explored, generalizing the concept of deformed Heisenberg algebras by Julius Wess. They are also related to algebraic foundations
of quantum groups in the spirit of Ludwig Pittner.
Some of the moment problems of the underlying basic difference
¨
equations are investigated. Applications to discrete Schrodinger
theory are worked out and some spectral properties of the aris¨
ing operators are presented, also in the case of Schrodinger
operators with basic shift–potentials and in the case of ground state
difference–differential operators.
For the arising orthogonal function systems, the concept of inherited orthogonality is explained. The results in this talk are mainly
related to a recent joint work with Sophia Roßkopf and Lucia Birk.
29
References
¨
¨
[1] A. Ruffing: Habilitationsschrift TU Munchen,
Fakult¨at fur
Mathematik: Contributions to Discrete Schr¨odinger Theory (2006).
[2] M. Meiler, A. Ruffing: Constructing Similarity Solutions to Discrete Basic Diffusion Equations, Advances in Dynamical Systems
and Applications, Vol. 3, Nr. 1, (2008) 41–51.
[3] A. Ruffing, M. Simon: Difference Equations in Context of a qFourier Transform, New Progress in Difference Equations, CRC
press (2004), 523–530.
[4] A. Ruffing, M. Simon: Analytic Aspects of q-Delayed Exponentials:
Minimal Growth, Negative Zeros and Basis Ghost States, Journal of
Difference Equations and Applications, Vol. 14, Nr. 4 (2008),
347–366.
[5] L. Birk, S. Roßkopf, A. Ruffing: New Potentials in Discrete
Schr¨odinger Theory, American Institute of Physics Conference
Proceedings 1468, 47 (2012).
[6] L. Birk, S. Roßkopf, A. Ruffing: Difference-Differential Operators
for Basic Adaptive Discretizations and Their Central Function Systems, Advances in Difference Equations (2012), 2012 : 151.
30
Stable periodic points of n-dimentional
diffeomorphisms
EKATERINA V. VASILEVA
St. Petersburg State University
Universitetski pr. 28, St. Petersburg, Russia
[email protected]
We consider a self-diffeomorphism of (n+m)-space with a fixed hyperbolic point at the origin. The existence of nontransversal homoclinic point is assumed; i.e., intersection of the stable and unstable
manifolds contains a point, referred to as a homoclinic point, and
this point is a point of tangency of these manifolds.
It follows from results of Sh. Newhouse, B.F. Ivanov, L.P.Shilnikov,
S. V. Gonchenko and other authors that, when the stable and unstable manifolds are tangent in a certain way, a neighborhood of the
homoclinic point may contain stable periodic points, but at least one
of the characteristic exponents for such points tends to zero with increasing the period.
The goal of the talk is to show that under certain conditions imposed mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the homoclinic point contains an
infinite set of stable periodic points whose characteristic exponents
are negative and bounded away from zero. An example of such a
diffeomorphism was considered in the monograph of V.A. Pliss.
31
References
[1] Newhouse Sh., Diffeomorphisms with finitely many sinks,
Topology v.12 p. 9-12 (1974).
[2] Ivanov B.F., Stability of Trajectories That Do Not Leave a Neighborhood of a Homoclinic Curve , Differ. Uravn., 1979, v. 15,
no.8, pp. 1411-1419.
[3] Gonchenko S.V., Turaev D. V., Shilnikov L.P., Dynamical Phenomena in Multidimensinal Systems with a Structurally Unstable Poincare Curve, Dokl. RAN, 1993, v. 330, no. 2, pp. 144-147.
[4] Pliss V. A. Integralnye mnozhestva perodicheskikh system differentsialnykh uravnenii (Integral Sets of Periodic Systems of
Differential Equations), Moscow; Nauka, 1977.
[5] Vasileva E.V. Multidimensional Diffeomorphisms with Stable
Periodic Points, Dokl. RAN, 2011, v. 441, no. 3, pp. 299-301.
[6] Vasileva E.V. Diffeomorphisms of the Plane with Stable Perodic
Points, Differ. Uravn., 2012, v. 48, no.3, pp. 307-315.
32
Efficient symbolic method to detect
Hopf bifurcations in chemical reaction networks
ANDREAS WEBER1 , HASSAN ERRAMI1 ,
MARKUS EISWIRTH1 ,DIMA GRIGORIEV2 ,
WERNER M. SEILER3 , THOMAS STURM4
1
Institut fur
¨ Informatik II, Universit¨at Bonn
Friedrich-Ebert-Allee 144, 53113 Bonn, Germany
[email protected]
2
CNRS, Math´ematiques, Universit´e de Lille
Villeneuve d’Ascq, 59655, France
3
Institut fur
¨ Mathematik, Universit¨at Kassel
Heinrich-Plett-Strasse 40 34132 Kassel, Germany
4
Max-Planck-Institut fur
¨ Informatik , RG 1: Automation of Logic
66123 Saarbrucken,
¨
Germany
Determining potential oscillatory behavior of (bio-)chemical reaction networks is closely related to determining the existence of Hopf
bifurcation fixed points for positive values of variables and parameters in the corresponding dynamical systems.
On the basis of a classical theorem of Orlando’s on pairs of purely
imaginary eigenvalues of a matrix and using combinations of ideas
of stoichiometric network analysis with ideas of tropical geometry
we have developed (and implemented) a symbolic method for detecting Hopf bifurcation fixed points in chemical reaction networks
(for some positive reaction rate parameters). With our fully algorithmic method we could successfully examine several networks of
dimension up to 20, and and we have found Hopf bifurcation fixed
points for several of them.
33
Normal forms and versal unfoldings in GLV
WEINIAN ZHANG
Yangtze Center of Mathematics and Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China
[email protected]
In the case of the full zero degeneracy, the classic Poincare-Birhoff
normal form theory is not available. In this talk a predator-prey system with such a degeneracy is considered. A new normal form is
given for generalized Lotka-Volterra systems (abbreviated by GLV),
which is employed to unfold the degenerate predator-prey system
in GLV class. The universality of unfolding is proved by verifying
the transversality. Further, all possible bifurcations at the degenerate equilibrium including transcritical bifurcation, Hopf bifurcation
and heteroclinic bifurcation are discussed.
References
[1] Shigui Ruan, Yilei Tang, Weinian Zhang, Versal unfoldings
of predator-prey systems with ratio-dependent functional response, J. Differential Equations 249 (2010) 1410-1435.
[2] S.-N. Chow, C. Li, D. Wang, Normal Forms and Bifurcation of
Planar Vector Fields., Cambridge University Press, Cambridge,
1994.
34
Stability and bifurcation analysis on a kind of hybrid
system
YEPENG XING
Department of Mathematics
Shanghai Normal University
Rd. Guilin, 100, Shanghai, P.R. China
[email protected]
The mathematical descriptions of many hybrid dynamical systems
can be characterized by impulsive differential equations. Impulsive
dynamical systems can be viewed as a subclass of hybrid systems
and consist of three elementsłnamely, a continuous time differential equation, which governs the motion of the dynamical system
between impulsive or resetting events; a difference equation, which
governs the way the system states are instantaneously changed when
a resetting event occurs; and a criterion for determining when the
states of the system are to be reset. Hybrid and impulsive dynamical systems exhibit a very rich dynamical behavior. In this talk, we
are concerned with the stability of the origin and the existence of
disconnected limit cycles. Also, we give examples to illustrate how
the continuous subsystem and the discontinuous subsystem influence each other, i.e. how the impulses affect the behavior of the
orbits to the continuous system.
References
[1] Yefeng He and Yepeng Xing, Poincar´
e Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe, Abstract and Applied Analysis, Vol 2013.
35
[2] Z. Hu and M. Han, ”Periodic solutions and bifurcations of
firstorder periodic impulsive differential equations,” International Journal of Bifurcation and Chaos in Applied Sciences and
Engineering, vol. 19, no. 8, pp. 2515-2530, 2009.
[3] D.D. Bainov and P. S. Simenov, Impulsive Differential Equations:Periodic Solutions and Applications, Longman Scientific
and Technical, Harlow, UK, 1993.
36
Multiplicity of fixed points and growth of
ε-neighborhoods of orbits
ˇ C
´ 1 , MAJA RESMAN2 , VESNA ZUPANOVI
´2
ˇ
PAVAO MARDESI
C
1
Universit´e de Bourgogne, Department de Math´ematiques
Institut de Math´ematiques de Bourgogne
B.P. 47 870-21078-Dijon Cedex, France
2
University of Zagreb, Department of Applied Mathematics
Faculty of Electrical Engineering and Computing
Unska 3, 10000 Zagreb, Croatia
[email protected]
We study the relationship between the multiplicity of a fixed point
of a function g, and the dependence on ε of the length of ε-neighborhood of any orbit of g, tending to the fixed point. The relationship
ˇ
between these two notions was discovered in Elezovi´c, Zubrini´
c,
ˇ
Zupanovi´
c in the differentiable case, and related to the box dimension of the orbit.
Here, we generalize these results to non-differentiable cases. We
study the space of functions having a development in a Chebyshev
scale and use multiplicity with respect to this space of functions.
With these new definitions, we recover the relationship between
multiplicity of fixed points and the dependence on ε of the length of
ε-neighborhoods of orbits in non-differentiable cases where results
ˇ
ˇ
from Elezovi´c, Zubrini´
c, Zupanovi´
c do not apply.
Applications include in particular Poincar´e map near homoclinic
loop and Abelian integrals.
37
References
ˇ
ˇ
[1] N. Elezovi´c, D. Zubrini´
c, V. Zupanovi´
c, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons
Fractals 34(2) (2007) 244-252.
ˇ
[2] P. Mardeˇsi´c, M. Resman, V. Zupanovi´
c, Multiplicity of fixed
points and ε-neighborhoods of orbits. Journal of Differential
Equations 253 (2012) 2493-2514.
38
Contents
Organizing Committee . . . . . . . . . . . . . . . . . . . . .
1
Invited Speakers . . . . . . . . . . . . . . . . . . . . . . . . .
2
Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Index
40
39
Index
Amel’kin V., 6
Aybar I., 7
Aybar O., 9
Chen X., 10
Ding W., 11
Dukari´c M., 13
Han M., 15
Liu L., 17
Liu Y., 18
Manos T., 20
Mencinger M., 22
Robnik M., 24
Romanovski V., 26
Ruffing A., 28
Vasileva E., 30
Weber A., 32
Xing Y., 34
Zhang W., 33
Zupanovi´c V., 36
40