Methods and Models for Accelerating Dynamic Simulation of Fluid Power Circuits

The objective of this dissertation is to improve the dynamic simulation of fluid power circuits. A fluid power circuit is a typical way to implement power transmission in mobile working machines, e.g. cranes, excavators etc. Dynamic simulation is an essential tool in developing controllability and energy-efficient solutions for mobile machines. Efficient dynamic simulation is the basic requirement for the real-time simulation. In the real-time simulation of fluid power circuits there exist numerical problems due to the software and methods used for modelling and integration. A simulation model of a fluid power circuit is typically created using differential and algebraic equations. Efficient numerical methods are required since differential equations must be solved in real time. Unfortunately, simulation software packages offer only a limited selection of numerical solvers. Numerical problems cause noise to the results, which in many cases leads the simulation run to fail. Mathematically the fluid power circuit models are stiff systems of ordinary differential equations. Numerical solution of the stiff systems can be improved by two alternative approaches. The first is to develop numerical solvers suitable for solving stiff systems. The second is to decrease the model stiffness itself by introducing models and algorithms that either decrease the highest eigenvalues or neglect them by introducing steady-state solutions of the stiff parts of the models. The thesis proposes novel methods using the latter approach. The study aims to develop practical methods usable in dynamic simulation of fluid power circuits using explicit fixed-step integration algorithms. In this thesis, twomechanisms whichmake the systemstiff are studied. These are the pressure drop approaching zero in the turbulent orifice model and the volume approaching zero in the equation of pressure build-up. These are the critical areas to which alternative methods for modelling and numerical simulation are proposed. Generally, in hydraulic power transmission systems the orifice flow is clearly in the turbulent area. The flow becomes laminar as the pressure drop over the orifice approaches zero only in rare situations. These are e.g. when a valve is closed, or an actuator is driven against an end stopper, or external force makes actuator to switch its direction during operation. This means that in terms of accuracy, the description of laminar flow is not necessary. But, unfortunately, when a purely turbulent description of the orifice is used, numerical problems occur when the pressure drop comes close to zero since the first derivative of flow with respect to the pressure drop approaches infinity when the pressure drop approaches zero. Furthermore, the second derivative becomes discontinuous, which causes numerical noise and an infinitely small integration step when a variable step integrator is used. A numerically efficient model for the orifice flow is proposed using a cubic spline function to describe the flow in the laminar and transition areas. Parameters for the cubic spline function are selected such that its first derivative is equal to the first derivative of the pure turbulent orifice flow model in the boundary condition. In the dynamic simulation of fluid power circuits, a tradeoff exists between accuracy and calculation speed. This investigation is made for the two-regime flow orifice model. Especially inside of many types of valves, as well as between them, there exist very small volumes. The integration of pressures in small fluid volumes causes numerical problems in fluid power circuit simulation. Particularly in realtime simulation, these numerical problems are a great weakness. The system stiffness approaches infinity as the fluid volume approaches zero. If fixed step explicit algorithms for solving ordinary differential equations (ODE) are used, the system stability would easily be lost when integrating pressures in small volumes. To solve the problem caused by small fluid volumes, a pseudo-dynamic solver is proposed. Instead of integration of the pressure in a small volume, the pressure is solved as a steady-state pressure created in a separate cascade loop by numerical integration. The hydraulic capacitance V/Be of the parts of the circuit whose pressures are solved by the pseudo-dynamic method should be orders of magnitude smaller than that of those partswhose pressures are integrated. The key advantage of this novel method is that the numerical problems caused by the small volumes are completely avoided. Also, the method is freely applicable regardless of the integration routine applied. The superiority of both above-mentioned methods is that they are suited for use together with the semi-empirical modelling method which necessarily does not require any geometrical data of the valves and actuators to be modelled. In this modelling method, most of the needed component information can be taken from the manufacturer’s nominal graphs. This thesis introduces the methods and shows several numerical examples to demonstrate how the proposed methods improve the dynamic simulation of various hydraulic circuits.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Mekanismien dynamiikan simuloinnissa sovellettavia numeerisia- ja mallinnusmenetelmiä

Koneet voidaan usein jakaa osajärjestelmiin, joita ovat ohjaus- ja säätöjärjestelmät, voimaa tuottavat toimilaitteet ja voiman välittävät mekanismit. Eri osajärjestelmiä on simuloitu tietokoneavusteisesti jo usean vuosikymmenen ajan. Osajärjestelmien yhdistäminen on kuitenkin uudempi ilmiö. Usein esimerkiksi mekanismien mallinnuksessa toimilaitteen tuottama voimaon kuvattu vakiona, tai ajan funktiona muuttuvana voimana. Vastaavasti toimilaitteiden analysoinnissa mekanismin toimilaitteeseen välittämä kuormitus on kuvattu vakiovoimana, tai ajan funktiona työkiertoa kuvaavana kuormituksena. Kun osajärjestelmät on erotettu toisistaan, on niiden välistenvuorovaikutuksien tarkastelu erittäin epätarkkaa. Samoin osajärjestelmän vaikutuksen huomioiminen koko järjestelmän käyttäytymissä on hankalaa. Mekanismien dynamiikan mallinnukseen on kehitetty erityisesti tietokoneille soveltuvia numeerisia mallinnusmenetelmiä. Useimmat menetelmistä perustuvat Lagrangen menetelmään, joka mahdollistaa vapaasti valittaviin koordinaattimuuttujiin perustuvan mallinnuksen. Numeerista ratkaisun mahdollistamiseksi menetelmän avulla muodostettua differentiaali-algebraaliyhtälöryhmää joudutaan muokkaamaan esim. derivoimalla rajoiteyhtälöitä kahteen kertaan. Menetelmän alkuperäisessä numeerisissa ratkaisuissa kaikki mekanismia kuvaavat yleistetyt koordinaatit integroidaan jokaisella aika-askeleella. Tästä perusmenetelmästä johdetuissa menetelmissä riippumattomat yleistetyt koordinaatit joko integroidaan ja riippuvat koordinaatit ratkaistaan rajoiteyhtälöiden perusteella tai yhtälöryhmän kokoa pienennetään esim. käyttämällä nopeus- ja kiihtyvyysanalyyseissä eri kiertymäkoordinaatteja kuin asema-analyysissä. Useimmat integrointimenetelmät on alun perin tarkoitettu differentiaaliyhtälöiden (ODE) ratkaisuunjolloin yhtälöryhmään liitetyt niveliä kuvaavat algebraaliset rajoiteyhtälöt saattavat aiheuttaa ongelmia. Nivelrajoitteiden virheiden korjaus, stabilointi, on erittäin tärkeää mekanismien dynamiikan simuloinnin onnistumisen ja tulosten oikeellisuuden kannalta. Mallinnusmenetelmien johtamisessa käytetyn virtuaalisen työn periaatteen oletuksena nimittäin on, etteivät rajoitevoimat tee työtä, eli rajoitteiden vastaista siirtymää ei tapahdu. Varsinkaan monimutkaisten järjestelmien pidemmissä analyyseissä nivelrajoitteet eivät toteudu tarkasti. Tällöin järjestelmän energiatasapainoei toteudu ja järjestelmään muodostuu virtuaalista energiaa, joka rikkoo virtuaalisen työn periaatetta, Tästä syystä tulokset eivät enää pidäpaikkaansa. Tässä raportissa tarkastellaan erityyppisiä mallinnus- ja ratkaisumenetelmiä, ja vertaillaan niiden toimivuutta yksinkertaisten mekanismien numeerisessa ratkaisussa. Menetelmien toimivuutta tarkastellaan ratkaisun tehokkuuden, nivelrajoitteiden toteutumisen ja energiatasapainon säilymisen kannalta.

Selective catalytic reduction of nitrogen oxides with ammonia in forced unsteady state reactors - Case based reasoning and mathematical model simulation reasoning

The application of forced unsteady-state reactors in case of selective catalytic reduction of nitrogen oxides (NOx) with ammonia (NH3) is sustained by the fact that favorable temperature and composition distributions which cannot be achieved in any steady-state regime can be obtained by means of unsteady-state operations. In a normal way of operation the low exothermicity of the selective catalytic reduction (SCR) reaction (usually carried out in the range of 280-350°C) is not enough to maintain by itself the chemical reaction. A normal mode of operation usually requires supply of supplementary heat increasing in this way the overall process operation cost. Through forced unsteady-state operation, the main advantage that can be obtained when exothermic reactions take place is the possibility of trapping, beside the ammonia, the moving heat wave inside the catalytic bed. The unsteady state-operation enables the exploitation of the thermal storage capacity of the catalyticbed. The catalytic bed acts as a regenerative heat exchanger allowing auto-thermal behaviour when the adiabatic temperature rise is low. Finding the optimum reactor configuration, employing the most suitable operation model and identifying the reactor behavior are highly important steps in order to configure a proper device for industrial applications. The Reverse Flow Reactor (RFR) - a forced unsteady state reactor - corresponds to the above mentioned characteristics and may be employed as an efficient device for the treatment of dilute pollutant mixtures. As a main disadvantage, beside its advantages, the RFR presents the 'wash out' phenomena. This phenomenon represents emissions of unconverted reactants at every switch of the flow direction. As a consequence our attention was focused on finding an alternative reactor configuration for RFR which is not affected by the incontrollable emissions of unconverted reactants. In this respect the Reactor Network (RN) was investigated. Its configuration consists of several reactors connected in a closed sequence, simulating a moving bed by changing the reactants feeding position. In the RN the flow direction is maintained in the same way ensuring uniformcatalyst exploitation and in the same time the 'wash out' phenomena is annulated. The simulated moving bed (SMB) can operate in transient mode giving practically constant exit concentration and high conversion levels. The main advantage of the reactor network operation is emphasizedby the possibility to obtain auto-thermal behavior with nearly uniformcatalyst utilization. However, the reactor network presents only a small range of switching times which allow to reach and to maintain an ignited state. Even so a proper study of the complex behavior of the RN may give the necessary information to overcome all the difficulties that can appear in the RN operation. The unsteady-state reactors complexity arises from the fact that these reactor types are characterized by short contact times and complex interaction between heat and mass transportphenomena. Such complex interactions can give rise to a remarkable complex dynamic behavior characterized by a set of spatial-temporal patterns, chaotic changes in concentration and traveling waves of heat or chemical reactivity. The main efforts of the current research studies concern the improvement of contact modalities between reactants, the possibility of thermal wave storage inside the reactor and the improvement of the kinetic activity of the catalyst used. Paying attention to the above mentioned aspects is important when higher activity even at low feeding temperatures and low emissions of unconverted reactants are the main operation concerns. Also, the prediction of the reactor pseudo or steady-state performance (regarding the conversion, selectivity and thermal behavior) and the dynamicreactor response during exploitation are important aspects in finding the optimal control strategy for the forced unsteady state catalytic tubular reactors. The design of an adapted reactor requires knowledge about the influence of its operating conditions on the overall process performance and a precise evaluation of the operating parameters rage for which a sustained dynamic behavior is obtained. An apriori estimation of the system parameters result in diminution of the computational efforts. Usually the convergence of unsteady state reactor systems requires integration over hundreds of cycles depending on the initial guess of the parameter values. The investigation of various operation models and thermal transfer strategies give reliable means to obtain recuperative and regenerative devices which are capable to maintain an auto-thermal behavior in case of low exothermic reactions. In the present research work a gradual analysis of the SCR of NOx with ammonia process in forced unsteady-state reactors was realized. The investigation covers the presentationof the general problematic related to the effect of noxious emissions in the environment, the analysis of the suitable catalysts types for the process, the mathematical analysis approach for modeling and finding the system solutions and the experimental investigation of the device found to be more suitable for the present process. In order to gain information about the forced unsteady state reactor design, operation, important system parameters and their values, mathematical description, mathematicalmethod for solving systems of partial differential equations and other specific aspects, in a fast and easy way, and a case based reasoning (CBR) approach has been used. This approach, using the experience of past similarproblems and their adapted solutions, may provide a method for gaining informations and solutions for new problems related to the forced unsteady state reactors technology. As a consequence a CBR system was implemented and a corresponding tool was developed. Further on, grooving up the hypothesis of isothermal operation, the investigation by means of numerical simulation of the feasibility of the SCR of NOx with ammonia in the RFRand in the RN with variable feeding position was realized. The hypothesis of non-isothermal operation was taken into account because in our opinion ifa commercial catalyst is considered, is not possible to modify the chemical activity and its adsorptive capacity to improve the operation butis possible to change the operation regime. In order to identify the most suitable device for the unsteady state reduction of NOx with ammonia, considering the perspective of recuperative and regenerative devices, a comparative analysis of the above mentioned two devices performance was realized. The assumption of isothermal conditions in the beginningof the forced unsteadystate investigation allowed the simplification of the analysis enabling to focus on the impact of the conditions and mode of operation on the dynamic features caused by the trapping of one reactant in the reactor, without considering the impact of thermal effect on overall reactor performance. The non-isothermal system approach has been investigated in order to point out the important influence of the thermal effect on overall reactor performance, studying the possibility of RFR and RN utilization as recuperative and regenerative devices and the possibility of achieving a sustained auto-thermal behavior in case of lowexothermic reaction of SCR of NOx with ammonia and low temperature gasfeeding. Beside the influence of the thermal effect, the influence of the principal operating parameters, as switching time, inlet flow rate and initial catalyst temperature have been stressed. This analysis is important not only because it allows a comparison between the two devices and optimisation of the operation, but also the switching time is the main operating parameter. An appropriate choice of this parameter enables the fulfilment of the process constraints. The level of the conversions achieved, the more uniform temperature profiles, the uniformity ofcatalyst exploitation and the much simpler mode of operation imposed the RN as a much more suitable device for SCR of NOx with ammonia, in usual operation and also in the perspective of control strategy implementation. Theoretical simplified models have also been proposed in order to describe the forced unsteady state reactors performance and to estimate their internal temperature and concentration profiles. The general idea was to extend the study of catalytic reactor dynamics taking into account the perspectives that haven't been analyzed yet. The experimental investigation ofRN revealed a good agreement between the data obtained by model simulation and the ones obtained experimentally.

On state and parameter estimation in chaotic systems

State-of-the-art predictions of atmospheric states rely on large-scale numerical models of chaotic systems. This dissertation studies numerical methods for state and parameter estimation in such systems. The motivation comes from weather and climate models and a methodological perspective is adopted. The dissertation comprises three sections: state estimation, parameter estimation and chemical data assimilation with real atmospheric satellite data. In the state estimation part of this dissertation, a new filtering technique based on a combination of ensemble and variational Kalman filtering approaches, is presented, experimented and discussed. This new filter is developed for large-scale Kalman filtering applications. In the parameter estimation part, three different techniques for parameter estimation in chaotic systems are considered. The methods are studied using the parameterized Lorenz 95 system, which is a benchmark model for data assimilation. In addition, a dilemma related to the uniqueness of weather and climate model closure parameters is discussed. In the data-oriented part of this dissertation, data from the Global Ozone Monitoring by Occultation of Stars (GOMOS) satellite instrument are considered and an alternative algorithm to retrieve atmospheric parameters from the measurements is presented. The validation study presents first global comparisons between two unique satellite-borne datasets of vertical profiles of nitrogen trioxide (NO3), retrieved using GOMOS and Stratospheric Aerosol and Gas Experiment III (SAGE III) satellite instruments. The GOMOS NO3 observations are also considered in a chemical state estimation study in order to retrieve stratospheric temperature profiles. The main result of this dissertation is the consideration of likelihood calculations via Kalman filtering outputs. The concept has previously been used together with stochastic differential equations and in time series analysis. In this work, the concept is applied to chaotic dynamical systems and used together with Markov chain Monte Carlo (MCMC) methods for statistical analysis. In particular, this methodology is advocated for use in numerical weather prediction (NWP) and climate model applications. In addition, the concept is shown to be useful in estimating the filter-specific parameters related, e.g., to model error covariance matrix parameters.

Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.

Dispersal and metacommunity dynamics in a soft-sediment benthic system : how well is the seafloor connected?

Connectivity depends on rates of dispersal between communities. For marine soft-sediment communities continued small-scale dispersal as post-larvae and as adults can be equally important in maintaining community composition, as initial recruitment of substrate by pelagic larvae. In this thesis post-larval dispersal strategies of benthic invertebrates, as well as mechanisms by which communities are connected were investigated. Such knowledge on dispersal is scarce, due to the difficulties in actually measuring dispersal directly in nature, and dispersal has not previously been quantified in the Baltic Sea. Different trap-types were used underwater to capture dispersing invertebrates at different sites, while in parallel measuring waves and currents. Local community composition was found to change predictably under varying rates of dispersal and physical connectivity (waves and currents). This response was, however, dependent on dispersal-related traits of taxa. Actively dispersing taxa will be relatively better at maintaining their position, as they are not as dependent on hydrodynamic conditions for dispersal and will be less prone to be passively transported by currents. Taxa also dispersed in relative proportions that were distinctly different from resident community composition and a significant proportion (40 %) of taxa were found to lack a planktonic larval life-stage. Community assembly was re-started in a large-scale manipulative field experiment over one year across several sites, which revealed how patterns of community composition (α-, β- and λ-diversity) change depending on rates of dispersal. Results also demonstrated that in response to small-scale disturbance, initial recruitment was by nearby-dominant species after which other species arrived from successively further away. At later assembly time, the number of coexisting species increased beyond what was expected purely by local niche requirements (species sorting), transferring regional differences in community composition (β-diversity) to the local scale (α-diversity, mass effect). Findings of this thesis complement more theoretical studies in metacommunity ecology by demonstrating that understanding how and when individuals disperse relative to underlying environmental heterogeneity is key to interpreting how patterns of diversity change across different spatial scales. Such information from nature is critical when predicting responses to, for example, different types of disturbances or management actions in conservation.

JNK, a versatile regulator of kinesin-1 transport and cytoskeleton dynamics

C-Jun N-terminal kinase (JNK) is traditionally recognized as a crucial factor in stress response and inducer of apoptosis upon various stimulations. Three isoforms build the JNK subfamily of MAPK; generally expressed JNK1 and JNK2 and brain specific JNK3. Degenerative potency placed JNK in the spotlight as potential pharmacological option for intervention. Unfortunately, adverse effects of potential drugs and observation that expression of only JNK2 and JNK3 are induced upon stress, restrained initial enthusiasm. Notably, JNK1 demonstrated atypical high constitutive activity in neurons that is not responsive to cellular stresses and indicated existence of physiological activity. This thesis aimed at revealing the physiological functions of JNK1 in actin homeostasis through novel effector MARCKS-Like 1 (MARCKSL1) protein, neuronal trafficking mediated by major kinesin-1 motor protein and microtubule (MT) dynamics via STMN2/SCG10. The screen for novel physiological JNK substrates revealed specific phosphorylation of C-terminal end of MARCKSL1 at S120, T148 and T183 both ex vivo and in vitro. By utilizing site-specific mutagenesis, various actin dynamics and migrations assays we were able to demonstrate that JNK1 phosphorylation specifically facilitates F-actin bundling and thus filament stabilisation. Consecutively, this molecular mechanism was proved to enhance formation of filopodia; cell surface projections that allow cell sensing surrounding environment and migrate efficiently. Our results visualize JNK dependent and MARCKSL1 executed induction of filopodia in neurons and fibroblast indicating general mechanism. Subsequently, inactivation of JNK action on MARCKSL1 shifts cellular actin machinery into lamellipodial dynamic arrangement. Tuning of actin cytoskeleton inevitably melds with cell migration. We observed that both active JNK and JNK pseudo-phosphorylated form of MARCKSL1 reduce actin turnover in intact cells leading to overall diminished cell motility. We demonstrate that tumour transformed cells from breast, prostate, lung and muscle-derived cancers upregulate MARCKSL1. We showed on the example of prostate cancer PC-3 cell line that JNK phosphorylation negatively controls MARCKSL1 ability to induce migration, which precedes cancer cell metastasis. The second round of identification of JNK physiological substrates resulted in detection of predominant motor protein kinesin-1 (Kif5). Mass spectrometry detailed analysis showed evident endogenous phosphorylation of kinesin-1 on S176 within motor domain that interacts with MT. In vitro phosphorylation of bacterially expressed kinesin heavy chain by JNK isoforms displayed higher specificity of JNK1 when compared to JNK3. Since, JNK1 is constitutively active in neurons it signified physiological aspect of kinesin-1 regulation. Subsequent biochemical examination revealed that kinesin-1, when not phosphorylated on JNK site, exhibits much higher affinity toward MTs. Expression of the JNK non-phosphorable kinesin-1 mutant in intact cells as well as in vitro single molecule imaging using total internal reflection fluorescence microscopy indicated that the mutant loses normal speed and is not able to move processively into proper cellular compartments. We identify novel kinesin-1 cargo protein STMN2/SCG10, which along with known kinesin-1 cargo BDNF is showing impaired trafficking when JNK activity is inhibited. Our data postulates that constitutive JNK activity in neurons is crucial for unperturbed physiologically relevant transport of kinesin-1 dependant cargo. Additionally, my work helps to validate another novel physiological JNK1 effector STMN2/SCG10 as determinant of axodendritic neurites dynamics in the developing brain through regulation of MT turnover. We show successively that this increased MT dynamics is crucial during developmental radial migration when brain layering occurs. Successively, we are able to show that introduction of JNK phosphorylation mimicking STMN2/SCG10 S62/73D mutant rescues completely JNK1 genetic deletion migration phenotype. We prove that STMN2/SCG10 is predominant JNK effector responsible for MT depolymerising activity and neurite length during brain development. Summarizing, this work describes identification of three novel JNK substrates MARCKSL1, kinesin-1 and STMN2/SCG10 and investigation of their roles in cytoskeleton dynamics and cargo transport. This data is of high importance to understand physiological meaning of JNK activity, which might have an adverse effect during pharmaceutical intervention aiming at blocking pathological JNK action.

Market integration, return and volatility dynamics; empirical evidence from African stock markets

Frontier and Emerging economies have implemented policies with the objective of liberalizing their equity markets. Equity market liberalization opens the domestic equity market to foreign investors and as well paves the way for domestic investors to invest in foreign equity securities. Among other things, equity market liberalization results in diversification benefits. Moreover, equity market liberalization leads to low cost of equity capital resulting from the lower rate of return by investors. Additionally, foreign and local investors share any potential risks. Liberalized equity markets also become liquid considering that there are more investors to trade. Equity market liberalization results in financial integration which explains the movement of two markets. In crisis period, increased volatility and co-movement between two markets may result in what is termed contagion effects. In Africa, major moves toward financial liberalization generally started in the late 1980s with South Africa as the pioneer. Over the years, researchers have studied the impact of financial liberalization on Africa’s economic development with diverse results; some being positive, others negative and still others being mixed. The objective of this study is to establish whether African stock-markets are integrated into the United States (US) and World market. Furthermore, the study helps to see if there are international linkages between the Africa, US and the world markets. A Bivariate- VAR- GARCH- BEKK model is employed in the study. In the study, the effect of thin trading is removed through series of econometric data purification. This is because thin trading, also known as non-trading or inconsistency of trading, is a main feature of African markets and may trigger inconsistency and biased results. The study confirmed the widely established results that the South Africa and Egypt stock markets are highly integrated with the US and World market. Interestingly, the study adds to knowledge in this research area by establishing the fact that Kenya is very integrated with the US and World markets and that it receives and exports past innovations as well as shocks to and from the US and World market.