random perturbations of heteroclinic attractors estimates are derived for the mean recurrence time of orbits in the neighborhood of an attracting homoclinic orbit or heteroclinic cycle in an ordinary differential equation subject to small additive random noise the theory presented is illustrated with numerical simulations of several systems including ones invariant under symmetry groups for which such heteroclinic attractors are structurally stable the physical implications of the work presented are briefly discussed

near optimal control of nonstandard singularly perturbed systems abstract near optimal control of nonstandard multiparameter multitime scale singularly perturbed systems is studied using the descriptor variable approach in particular the slow models of the considered systems are allowed to exhibit impulses in the case of multiparameter singular perturbations the near optimal control is a cascade connection of separately designed slow and fast subregulators in the case of multitime scale singular perturbations the near optimal control is hierarchically composed of n 1 subregulators a parallel algorithm is provided for designing the subregulators separately the knowledge of the small singular perturbation parameters is not required in the overall design furthermore the frequency responses of the near optimal systems are analyzed and several design examples are given

most probable dynamics of the tumor growth model with immune surveillance under cross correlated noises abstract the noise is inherent and indispensable in the tumor growth system therefore the behavior of the tumor system will display randomness different from the traditional analysis method such as the mean value variance etc the literature presents another deterministic tool i e the most probable trajectories which are defined by computing the spatial maximizers of the probability density function here we will investigate the tumor cell growth system with immune surveillance under correlated white noises from a deterministic point of view then the most probable extinction time is defined by the time when the most probable trajectories first escape to the extinction state from the tumor state afterward the probability ratio of extinction state versus tumor state characterizes treatment effects from the numerical simulation we derive that for the increasing cross correlation intensity of noises the most probable extinction time is enhanced and the therapeutic effect is weaken and conversely for the intensity of multiplicative noise in contrast there exists a critical intensity of additive noise at which the most probable extinction time is the smallest meanwhile treatment effects would be improved with the shrinking intensity of additive noise

a deterministic and stochastic model for the system dynamics of tumor immune responses to chemotherapy abstract modern medical studies show that chemotherapy can help most cancer patients especially for those diagnosed early to stabilize their disease conditions from months to years which means the population of tumor cells remained nearly unchanged in quite a long time after fighting against immune system and drugs in order to better understand the dynamics of tumor immune responses under chemotherapy deterministic and stochastic differential equation models are constructed to characterize the dynamical change of tumor cells and immune cells in this paper the basic dynamical properties such as boundedness existence and stability of equilibrium points are investigated in the deterministic model extended stochastic models include stochastic differential equations sdes model and continuous time markov chain ctmc model which accounts for the variability in cellular reproduction growth and death interspecific competitions and immune response to chemotherapy the ctmc model is harnessed to estimate the extinction probability of tumor cells numerical simulations are performed which confirms the obtained theoretical results

a note on optimal control of generalized state space descriptor systems linear time invariant systems of the form e dx dt ax bu are considered where e is a square matrix which may be singular and b is a rectangular matrix having full column rank it is assumed that for any admissible initial state x 0 any control vector u t yields one and only one state vector x t the problem is this find a control vector u t that will drive the system from an admissible initial state x 0 to a fixed final state xf in a fixed time tf while minimizing some cost functional only elementary matrix and variational techniques are used necessary conditions are derived for the existence of minima of j the problem of finding sufficient conditions of the existence of minima of j is not considered it is shown that in many cases the necessary conditions for the existence of minima of j yield a two point boundary value problem consisting of a system of ordinary differential equations containing only elements of x t and the boundary conditions prescribed at 0 and tf if some

numerical algorithms and existence results on lq control of descriptor systems with conditions on x 0 and stability we present a numerical algorithm based on using orthogonal matrices and sufficient existence conditions for an lq control problem of descriptor systems which generalise the existing results the associated hamiltonian pencil may be singular and may have infinite generalised eigenvalues iges of multiplicity 1 besides on x 0 we present optimality conditions on x 0 we present sufficient conditions for stability of the optimal closed loop descriptor system as well as a stabilisation algorithm

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

h control for descriptor systems a matrix inequalities approach abstract this paper considers the h control problem for descriptor systems that possibly have impulsive modes and or j axis zeros first we propose matrix inequalities that give a generalized stability condition and an h norm condition for descriptor systems using these matrix inequalities we show that the solvability of a set of matrix inequalities is necessary and sufficient to the existence of a proper controller that satisfies a prescribed h norm condition as well as stabilizing the closed loop system and eliminating all impulsive modes these inequalities are equivalent to certain linear matrix inequalities to which we can get solutions whenever they exist using efficient polynomial time algorithms

an algebraic model for performance evaluation of timed event multigraphs this note demonstrated that a certain class of petri nets called timed event multigraphs can be represented as a set of linear equations by using a novel dioid with this new algebra some properties are presented in the text an example is discussed in the end and an interesting result is computed

modelling routing phenomenon with bounds estimation in dioids abstract in job shop systems each product is routed according to its own production cycle however routings or conflicts cannot be modelled in scalar dioid algebraic structures such as z min or z max also denoted z min and z max the main reason is that choices cannot be represented in such modellings in this article the input output behaviour of a whole system with several sub systems in conflict is bounded by those of two linear systems in dioid z min by doing so we model behaviours as intervals an interval contains all the possible system behaviours in terms of number of pallets coming and going delays and production rates when the routing policy therein is periodic as a consequence even though the input output behaviour of such a system is not linear in a scalar dioid we can nevertheless use an imprecise modelling over a dioid of intervals this allows for using dioid theory contributions as for control problem synthesis issues

the linear quadratic optimal regulator for descriptor systems discrete time case abstract in this paper the linear quadratic optimal regulator problem for the discrete time descriptor system ex k 1 ax k bu k where e is in general a singular matrix is investigated the continuous time version of this problem was treated elsewhere the discrete time theory is largely parallel to the continuous time theory but there are several important differences one is that in the discrete time case a fairly general problem can be solved with singular cost matrices the key to the solution as for the continuous time problem is a coordinate system which can be computed by performing a singular value decomposition of e this is called an svd coordinate system the optimization problem is solved by applying hamiltonian minimization to compute the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors in the general singular cost case this trajectory is computed by recursively solving a discrete time riccati equation from the optimal trajectory the optimal descriptor variable feedback gain relating the control to the descriptor variable can be computed several riccati difference equations are derived all of which have the same solution this solution can be used to compute the optimal trajectory and cost for non singular problems even in the finite horizon case the optimal feedback gain can be computed non recursively by solving an eigenvalue eigenvector problem formulated from the system parameters in the svd coordinate system the optimal gain for the infinite horizon steady state problem on the other hand can be computed from the solution of an eigenvector problem involving the untransformed system parameters

new sufficient conditions for linear feedback closed loop stackelberg strategy of descriptor systems this paper is concerned with the derivation of linear feedback closed loop stackelberg lfcls strategies for a class of continuous time two person nonzero sum differential games characterized by linear descriptor systems and the quadratic cost functionals compared with existing results new sufficient conditions for the existence of a lfcls strategy are obtained under less restrictive conditions furthermore from the results of this paper the authors also arrive at some conclusions about the linear quadratic closed loop nash game for continuous time descriptor systems these conclusions are distinguished from those of the linear quadratic closed loop nash game for state space systems

stochastic bifurcation for a tumor immune system with symmetric levy noise abstract in this paper we investigate stochastic bifurcation for a tumor immune system in the presence of a symmetric non gaussian levy noise stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index the effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation we find that stochastic dynamics induced by gaussian and non gaussian levy noises are quite different

asymptotic stability of heteroclinic cycles in systems with symmetry ii systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry in previous work we began a systematic investigation into the asymptotic stability of such cycles in particular we found a sufficient condition for asymptotic stability and we gave algebraic criteria for deciding when this condition is also necessary these criteria are satisfied for cycles in r 3 field and swift and hofbauer considered examples in r 4 for which our sufficient condition for stability is not optimal they obtained necessary and sufficient conditions for asymptotic stability using a transition matrix technique in this paper we combine our previous methods with the transition matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles in particular we obtain a complete theory for simple heteroclinic cycles in r 4 thereby proving and extending results for homoclinic cycles that were stated without proof by chossat krupa melbourne and scheel a partial classification of simple heteroclinic cycles in r 4 is also given finally our stability results generalize naturally to higher dimensions and many of the higher dimensional examples in the literature are covered by this theory

descriptor variable systems and optimal state regulation linear systems of the form e dot x ax bu with e singular are treated it is desired to find a control which drives the system asymptotically to the origin minimizing a quadratic cost functional no restrictions are placed on initial conditions the cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order riccati equation

optimal control of drug therapy in cancer treatment abstract we present an optimal control strategy for nonlinear systems with application to the drug therapy of cancer the tumour growth model is represented by a system of equations from population dynamics which is based on the competition between normal cells and tumour cells the effect of the immune system in the presence of cancer is also included in the model a linear time varying approximation technique is proposed to construct an optimal control strategy for the nonlinear system which is valid not only within small perturbations around the equilibrium point but also for global dynamics of the system the optimal control method is applied to eliminate the tumour cells whilst minimizing the amount of drug in the simulated model simulation results show that cancer cells can be eradicated 1 in a very short time with a small amount of drug using an optimal administration of the therapy

a descriptor system approach to singular perturbation of linear regulators the relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system is investigated meanwhile a decomposition approach is provided for solution of the linear quadratic regulator problem for singularly perturbed systems this approach is a significant advance over the two stage method

stability and diversity in collective adaptation we derive a class of macroscopic differential equations that describe collective adaptation starting from a discrete time stochastic microscopic model the behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior we show that although individual agents interact with their environment and other agents in a purely self interested way macroscopic behavior can be interpreted as game dynamics application to several familiar explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors the simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation we also analyze the adaptation dynamics from an information theoretic viewpoint and discuss self organization induced by the dynamics of uncertainty giving a novel view of collective adaptation

lq control of descriptor systems by cancelling structure at infinity this paper presents results concerning lq control of arbitrary descriptor systems and a numerical algorithm for achieving it the algorithm consists of two steps the first step is using the silverman algorithm orthogonal version transformation of the problem of lq control of descriptor system into a problem of lq control of a strictly proper system with singular or non singular new matrix 22 if new 22 is singular we perform the second step i e solving the lq singular control problem using again the silverman algorithm regarding the existing results on the lq singular control problem we have new results a counter example that this problem may not have a stable solution most general solvability conditions with or without stability and a new numerical algorithm based on using orthogonal matrices when stabilizability conditions are satisfied the algorithm places simultaneously stable poles in the closed loop system we show that the traditional riccati equation associated to this proble

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

mathematical modeling of tumor immune competitive system considering the role of time delay abstract in this paper we consider a three dimensional nonlinear delay differential system tumor cells cytotoxic t lymphocytes t helper cells with single interaction delay we perform linear stability of the equilibria and the existence of hopf bifurcation in which the discrete time delay is used as a bifurcation parameter we estimate the length of delay to preserve the stability of period 1 limit cycle we also investigate the direction period and the stability of bifurcated periodic solutions by applying normal form method and center manifold theory we observe that the discrete time delay plays an important role in stability switching numerical simulations are presented to illustrate the rich dynamical behavior of the model with different values for the time delay including the existence of periodic oscillations which demonstrate the phenomena of long term tumor relapse

dynamics in a tumor immune system with time delays in this paper we study the dynamical behavior of a tumor immune system t is interaction model with two discrete delays namely the immune activation delay for effector cells ecs and activation delay for helper t cells htcs by analyzing the characteristic equations we establish the stability of two equilibria tumor free equilibrium and immune control equilibrium and the existence of hopf bifurcations when two delays are used as the bifurcation parameter our results exhibit that both delays do not affect the stability of tumor free equilibrium however they are able to destabilize the immune control equilibrium and cause periodic solutions we numerically illustrate how these two delays can change the stability region of the immune control equilibrium and display the different impacts to the control of tumors the numerical simulation results show that the immune activation delay for htcs can induce heteroclinic cycles to connect the tumor free equilibrium and immune control equilibrium furthermore we observe that the immune activation delay for htcs can even stabilize the unstable immune control equilibrium

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

variational calculus for descriptor problems the first order necessary condition for optimality is derived from a variational argument that involves an ad hoc modification of the bliss method resulting in a hamiltonian characterization in terms of edx dt rather than dx dt the former being smoother than the latter this approach sidesteps the regularity conditions of the lagrange multiplier theory under some mild assumptions the necessary condition for optimality is also sufficient and the optimal control exists the numerically relevant result is a generalized eigenvector inverse free characterization of optimality

efficient numerical procedures for solving closed loop stackelberg strategies with small singular perturbation parameter in this paper the computation of the linear closed loop stackelberg strategies with small singular perturbation parameter that characterizes singularly perturbed systems sps are studied the attention is focused on a new numerical algorithm for solving a set of cross coupled algebraic lyapunov and riccati equations calre it is proven that the new algorithm guarantees the local quadratic convergence a numerical example is solved to show a reduction of the average cpu time compared with the existing algorithm

stability and bifurcation analysis of delay induced tumor immune interaction model a modified mathematical model of kuznetsov et al 1994 representing tumor immune interaction with discrete time delay is proposed in this paper by choosing the discrete time delay as the bifurcation parameter we establish the sufficient condition for local stability of interior steady state hopf bifurcation occurs when the time delay passes a critical value the estimation of the length of delay to preserve stability has been derived moreover based on the normal form theory and center manifold theorem we establish explicit expressions to determine the direction of hopf bifurcation and the stability of bifurcating periodic solutions numerical simulations are obtained to validate our analytical findings by varying the system parameter a two parameters bifurcation diagrams are also shown by varying the system parameters which bring out the rich biological dynamics of the system

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

modeling and control of weight balanced timed event graphs in dioids the class of timed event graphs tegs has widely been studied thanks to an approach known as the theory of max plus linear systems in particular the modeling of tegs via formal power series in a dioid called minax has led to input output representations on which some model matching control problems have been solved our work attempts to extend the class of systems for which a similar control synthesis is possible to this end a subclass of timed petri nets that we call weight balanced timed event graphs wbtegs will be first defined they can model synchronization and delays wbtegs contain tegs and can also describe dynamic phenomena such as batching and event duplications unbatching their behavior is described by rational compositions sum product and kleene star of four elementary operators n t m and b on a dioid of formal power series denoted e the main feature is that the transfer series of wbtegs have a property of ultimate periodicity such as rational series in minax finally the existing results on control synthesis for max plus linear systems find a natural application in this framework

hopf bifurcation in differential equations with delay for tumor immune system competition model this paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor the model which describes this competition is governed by a system of differential equations with one delay it is shown that the dynamics depends crucially on the time delay parameter by using the time delay as a parameter of bifurcation the analysis is focused on the hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state the obtained results depict the oscillations given by simulations see m galach int j appl math comput sci 13 2003 pp 395 406 which are observed in reality see d kirschner and j c panetta j math biol 37 1998 pp 235 252 it is suggested to examine by laboratory experiments how to employ these results for control of tumor growth

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

h2 output feedback control for descriptor systems this paper solves the h 2 output feedback control problem for continuous time descriptor systems an existence condition of the h 2 optimal controller is derived based on the standard model matching technique and a characterization of the optimal controller is given in terms of the solutions of two generalized algebraic riccati equations it turns out that the optimal controller has an observer based structure well known in the state space case so that the results in this paper are natural extensions of the state space results to the descriptor systems a numerical example is also included to show the applicability of the proposed results

solutions of differential algebraic equations as outputs of lti systems application to lq control problems abstract in this paper we propose a unified geometric framework for representing all solutions of a linear time invariant differential algebraic equation dae lti as outputs of classical linear time invariant ordinary differential equations ode lti the proposed framework is then used to solve an lq optimal control problem for dae ltis with rectangular matrices

timed event graphs with multipliers and homogeneous min plus systems the authors study fluid analogues of a subclass of petri nets called fluid timed event graphs with multipliers which are a timed extension of weighted t systems studied in the petri net literature these event graphs can be studied naturally with a new algebra analogous to the min plus algebra but defined on piecewise linear concave increasing functions endowed with the pointwise minimum as addition and the composition of functions as multiplication a subclass of dynamical systems in this algebra which have a property of homogeneity can be reduced to standard min plus linear systems after a change of counting units the authors give a necessary and sufficient condition under which a fluid timed event graph with multipliers can be reduced to a fluid timed event graph without multipliers in the fluid case this class corresponds to the so called expansible timed event graphs with multipliers of munier 1993 or to conservative weighted t systems the change of variable is called here a potential its restriction to the transition nodes of the event graph is a t semiflow

a time delay model of tumour immune system interactions global dynamics parameter estimation sensitivity analysis abstract recently a large number of mathematical models that are described by delay differential equations ddes have appeared in the life sciences in this paper we present a delay differential model to describe the interactions between the effector and tumour cells the existence of the possible steady states and their local stability and change of stability via hopf bifurcation are theoretically and numerically investigated parameter estimation problem for given real observations using least squares approach is studied the global stability and sensitivity analysis are also numerically proved for the model the stability and periodicity of the solutions may depend on the time lag parameter the model is qualitatively consistent with the experimental observations of immune induced tumour dormancy the model also predicts dormancy as a transient period of growth which necessarily results in either tumour elimination or tumour escape

a new test for chaos in deterministic systems we describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic in contrast to the usual method of computing the maximal lyapunov exponent our metho

numerical solution of differential algebraic riccati equations abstract we consider riccati matrix differential algebraic equations arising from singular or descriptor control problems we discuss the solvability of such equations under different conditions in order to apply numerical methods for differential algebraic systems one has to transform the equation unfortunately these equations then have a linear part which is described by a singular pencil and thus the usual integration methods do not apply under some conditions which we discuss these singularities can be removed by a preprocessing algorithm and the equation can then be solved by well known methods for differential algebraic systems like dassl of l petzold or limex of deuflhard hairer and zugck we discuss the numerical procedures and give some numerical examples

bifurcations from an attracting heteroclinic cycle under periodic forcing abstract there are few examples of non autonomous vector fields exhibiting complex dynamics that may be proven analytically we analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two sphere we derive the first return map near the heteroclinic cycle for small amplitude of the perturbing term and we reduce the analysis of the non autonomous system to that of a two dimensional map on a cylinder interesting dynamical features arise from a discrete time bogdanov takens bifurcation when the perturbation strength is small the first return map has an attracting invariant closed curve that is not contractible on the cylinder near the centre of frequency locking there are parameter values with bistability the invariant curve coexists with an attracting fixed point increasing the perturbation strength there are periodic solutions that bifurcate into a closed contractible invariant curve and into a region where the dynamics is conjugate to a full shift on two symbols

stability and bifurcations for the chronic state in marchuk s model of an immune system in the paper we present known and new results concerning stability and the hopf bifurcation for the positive steady state describing a chronic disease in marchuk s model of an immune system we describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system we compare these results with the results known in the literature we show that the positive steady state can be stable only for very specific parameter values stability analysis is illustrated by mikhailov s hodographs and numerical simulations conditions for the hopf bifurcation and stability of arising periodic orbit are also studied these conditions are checked for arbitrary chosen realistic parameter values numerical examples of arising due to the hopf bifurcation periodic solutions are presented

frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle received 18 july 2006 published 6 november 2006robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invarianthyperplanes in particular they occur frequently in models for ecological dynamics and uid mechanicalinstabilities we consider the effect of small amplitude time periodic forcing and describe how to reduce thedynamics to a two dimensional map in the limit where the heteroclinic cycle loses asymptotic stability intervals of frequency locking appear in the opposite limit where the heteroclinic cycle becomes stronglystable the dynamics remains chaotic and no frequency locking is observed doi 10 1103 physreve 74 055201 pacs number s 05 45 a 82 40 bj

eigenvalue placement for generalized linear systems this paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by e ax bu where e is a singular map it is shown that controllability of the infinite eigenvalues of the pencil se a is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre specified complex numbers a procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed

filtering and lqg problems for discrete time stochastic singular systems discrete time stochastic singular systems are discussed filtering and linear quadratic gaussian lqg problems are considered the problems are first transformed into the normal form via state augmentation and then solved by utilizing standard results for nonsingular systems the linear unbiased least squares state estimation algorithm and the optimal control law for the lqg problem are given the order of the filtering algorithm obtained in this way is not much increased moreover this algorithm allows the presence of some kinds of control inputs

sensitivity analysis for dynamic systems with time lags many problems in bioscience for which observations are reported in the literature can be modelled by suitable functional differential equations incorporating time lags other terminology delays or memory effects parameterized by scientifically meaningful constant parameters p or and variable parameters for example control functions u t it is often desirable to have information about the effect on the solution of the dynamic system of perturbing the initial data control functions time lags and other parameters appearing in the model the main purpose of this paper is to derive a general theory for sensitivity analysis of mathematical models that contain time lags in this paper we use adjoint equations and direct methods to estimate the sensitivity functions when the parameters appearing in the model are not only constants but also variables of time to illustrate the results the methodology is applied numerically to an example of a delay differential model

observers for singular systems the following three major aspects of the design of observers for linear time invariant singular systems are resolved the necessary and sufficient conditions for the problem to have a solution the order of the minimal observer and the properties of the closed loop system separation principle an algorithm is presented for the construction of the minimal order observer

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

feedback strategies for discrete time linear quadratic two player descriptor games this paper deals with discrete time linear quadratic descriptor two player games the two frameworks of nash and stackelberg strategies with feedback information structure are considered the aim is to provide sufficient conditions ensuring for any initial state the existence and the uniqueness of a tri trajectory gathering the state and the controls of the two players which reaches the equilibrium the provided results are mainly based on a discrete time matrix block formulation the provided algorithms consist in an iterative backward in time procedure issued from dynamic programming numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

dynamics near a periodically perturbed robust heteroclinic cycle abstract robust heteroclinic cycles rhcs arise naturally in collections of symmetric differential equations derived as dynamical models in many fields including fluid mechanics game theory and population dynamics in this paper we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle rhc we give a detailed derivation of the poincare map for trajectories near the rhc asymptotically correct in the limit of small amplitude perturbations this reduces the nonautonomous system in r 3 to a 2d map we identify three distinct dynamical regimes the distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem the first regime corresponds to the rhc being only weakly attracting here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque in the second regime the periodically perturbed rhc is more strongly attracting and the system dynamics corresponds to that of a non invertible or invertible circle map in the third regime of yet stronger attraction the dynamics of the return map is chaotic and no longer reducible to a one dimensional map this third regime has been noted previously our analysis in this paper focusses on providing quantitative results in the first two regimes

stochastic model for tumor growth with immunization we analyze a stochastic model for tumor cell growth with both multiplicative and additive colored noises as well as nonzero cross correlations in between whereas the death rate within the logistic model is altered by a deterministic term characterizing immunization the birth rate is assumed to be stochastically changed due to biological motivated growth processes leading to a multiplicative internal noise moreover the system is subjected to an external additive noise which mimics the influence of the environment of the tumor the stationary probability distribution p s is derived depending on the finite correlation time the immunization rate and the strength of the cross correlation p s offers a maximum which becomes more pronounced for increasing immunization rate the mean first passage time is also calculated in order to find out under which conditions the tumor can suffer extinction its characteristics are again controlled by the degree of immunization and the strength of the cross correlation the behavior observed can be interpreted in terms of a biological model of tumor evolution

controlling bifurcation in a delayed fractional predator prey system with incommensurate orders this paper investigates an issue of bifurcation control for a novel incommensurate fractional order predator prey system with time delay firstly the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter and the conditions of creation for hopf bifurcation are established it is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously then a linear delayed feedback controller is introduced to successfully control the hopf bifurcation for such system it is shown that the control effort is markedly influenced by feedback gain it is further found that the onset of the bifurcation can be delayed as feedback gain decreases finally two illustrative examples are exploited to verify the validity of the obtained newly results

on equivalence of pencils from discrete time and continuous time control we study two matrix pencils that arise respectively in discrete time and continuous time optimal and robust control we introduce a one to one transformation between these two pencils we show that for the pencils under the transformation their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related the eigen structures of the pencils under consideration have strong connections with the associated control problems our result may be applied to connect the discrete time and continuous time control problems and eventually lead to a unified treatment of these two types of control problems