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The aim of this paper is to develop a new method for finding multiple bifurcation points in structures.

A brief review of nonlinear analysis is presented. A powerful method (called arc‐length) for tracing nonlinear equilibrium path is described. Techniques for monitoring critical points are discussed to find the rank deficiency of the stiffness matrix. Finally, by using eigenvalue perturbation of tangent stiffness matrix, load parameter associated with multiple bifurcation points is obtained.

Since other methods of finding simple bifurcation points diverge in the neighborhood of critical points, this paper introduces a new method to find multiple bifurcation points. It should be remembered that a simple bifurcation point is a multiple bifurcation point with rank deficiency equal to one. Therefore, the method is applicable to simple critical points as well.

Global buckling of the structures should be considered in design. Many structures (specially symmetric space structures) have multiple bifurcation points, therefore, analyst and designer should be aware of these points and should control them (for example, by changing the geometry or other related factors) for obtaining a safe and optimum design.

In this paper a robust method to find multiple bifurcation points is introduced. By using this method, engineers can be aware of critical load of multiple bifurcation points to control global buckling of related structures.

The practical behaviour of problems exhibiting bifurcation with secondary branches cannot be studied in general by using standard path‐following methods such as arc‐length schemes. Special algorithms have to be employed for the detection of bifurcation and limit points and furthermore for branch‐switching. Simple methods for this purpose are given by inspection of the determinant of the tangent stiffness matrix or the calculation of the current stiffness parameter. Near stability points, the associated eigenvalue problem has to be solved in order to calculate the number of existing branches. The associated eigenvectors are used for a perturbation of the solution at bifurcation points. This perturbation is performed by adding the scaled eigenvector to the deformed configuration in an appropriate way. Several examples of beam and shell problems show the performance of the method.

In the inelastic stability analysis of plated structures, incremental‐iterative finite element methods sometimes encounter prohibitive solution difficulties in the vicinity of sharp limit points, branch points and other regions of abrupt non‐linearity. Presents an analysis system that attempts to trace the non‐linear response associated with these types of problems at minor computational cost. Proposes a semi‐heuristic method for automatic load incrementation, termed the adaptive arc‐length procedure. This procedure is capable of detecting abrupt non‐linearities and reducing the increment size prior to encountering iterative convergence difficulties. The adaptive arc‐length method is also capable of increasing the increment size rapidly in regions of near linear response. This strategy, combined with consistent linearization to obtain the updated tangent stiffness matrix in all iterative steps, and with the use of a “minimum residual displacement” constraint on the iterations, is found to be effective in avoiding solution difficulties in many types of severe non‐linear problems. However, additional procedures are necessary to negotiate branch points within the solution path, as well as to ameliorate convergence difficulties in certain situations. Presents a special algorithm, termed the bifurcation processor, which is effective for solving many of these types of problems. Discusses several example solutions to illustrate the performance of the resulting analysis system.

This paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs.

The parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory.

Periodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur.

By treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

This paper is concerned with rank analysis of rectangular matrix of a homogeneous set of incremental equations regarded as an element of continuation method. The rank analysis is based on a known feature that every rectangular matrix can be transformed into the matrix of echelon form. By inspection of the rank, correct control parameters are chosen and this allows not only for rounding limit and turning points but also for branch‐switching near bifurcation points.

The purpose of the chapter is to describe the concept of conflicts of socio-economic systems on the basis of the theory and methodology of the systemic approach.

The authors use the provisions of the systemic approach.

It is determined that dynamic socio-economic system is susceptible to conflicts, which, according to the Systemic approach, are bifurcation points. The features of conflict that allow defining it as a bifurcation point are its short duration, violation of sustainability of socio-economic system, uncertainty of consequences, influence on the system on the whole (cascade effect), and multiplicity of scenarios of development of socio-economic system after the conflict.

Studying conflict with the help of the theory and methodology of the systemic approach allowed determining the fact that conflict is a violation of order in a socio-economic system (which specifies definition of conflict and its essence) and could be evaluated through the measure of the system’s order (which specifies methodology of conflict evaluation). Moreover, conflict in a socio-economic system emerges not at once – it is accumulated under joint negative influences of internal and external factors. Further, the performed research allows specifying the classification of conflicts, which is offered in previous chapters – according to scenarios of development of socio-economic system after the conflict, normal (leading to normal change of system), abrupt (leading to abrupt change of the system), and attraction (all following states of the system are sustainable) conflicts are possible.

Cascaded DC-DC converters system is the main structure of distributed power system, and it has complex nonlinear phenomena during operation, which affect the power quality. Therefore, the dynamic behavior of the cascaded buck converter and boost converter system, as one of the typical cascaded DC-DC converters systems is analyzed.

Firstly, the studied cascaded system of the buck converter with peak current control and the boost converter with PI current control is introduced and its discrete modeling is built. Then, the Jacobian matrix of the cascaded system is calculated to research the stability when the parameter change. Finally, simulation by PSIM and experiments are carried out to verify the theoretical analysis.

The coexistence of fast and slow time scale bifurcations with the changes of reference current and input voltage are studied in the cascaded system, and using simulation analysis to further study the sensitivity of the inductor current of the front-stage converter and back-stage converter to different parameters.

A discrete model of the cascaded buck converter and boost converter system is established, and its dynamic behavior is analyzed in detail for the first time.

The information age has highlighted the complex nature of our organizing systems. Complex systems move between ordered and disordered states. Ordered states condense near attractor points which include bifurcation points where order is redefined for the system. Such points have occurred repeatedly in our economy over the past decade, causing an unpredictable environment. As our economy moves from a stable state, a different set of organizing rules is needed. When this new set of organizing rules are overlaid with the conditions and processes facilitated by the use of advanced communication and computing technology, we find that virtual or network organizations are an emerging logical form for organizing.

The purpose of this paper is to offer the theory of a parametrical regulation of market economy development, and the results of the theory development and usage.

Theoretical results of the abstract have been obtained by way of applying the theory of ordinary differential equations, geometrical methods in variation tasks and the theory of dynamic systems. These results have been used for solving a number of practical tasks.

The market economy development parametrical regulation theory structure has been offered. The approach to parametrical regulation of a nonlinear dynamic system's development has been suggested. An assumption about the existence of solution to the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms has been set forward. An assumption about the conditions sufficient for the existence of an extremal's bifurcation point of the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms is presented, formulated and proved. Theory application samples have been provided.

Future papers would be focused on studies of rigidness of other mathematic models of economic systems.

The research findings could be applied to the choice and realization of an effective budget and tax as well as monetary and loan state policy.

The market economy development parametrical regulation theory has been offered for consideration for the first time.