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An RF phase-quadrature oscillator is analyzed. The designer "ensured" the phase-quadrature working condition through a suitable cross-coupled electrical connection between the two parts composing the complete oscillator. Through a detailed analysis of the dynamics of this oscillator, we show that device mismatch can lead to an unexpected stable working mode where the oscillator does not operate in a phase-quadrature condition. The proposed analysis has been first performed on a simplified model of the oscillator to allow the use of software packages for numerical continuation. The results of the analysis of the simplified oscillator dynamics have been then checked by means of accurate device models from a 65 nm RF technology library.

We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a boundary point. The boundary point is found using an expression for the boundary in terms of the vertices of a set of finite, convex polyhedra. The resulting algorithm is quite simple, allows adaptive spacing of the computed points, and deals with the problems of local and global overlap in a natural way. The algorithm is robust (the new points need only be near the boundary), and is well suited to problems with large embedding dimension, and small to moderate dimension.

We present a robust method to locate and continue period-doubling, saddle-node and symmetry-breaking bifurcations of periodically driven experimental systems. The method is illustrated from results obtained for an electronic implementation of a Duffing oscillator.

A nonlinear state feedback controller is proposed for the generation of chaos in third-order linear autonomous continuous time systems. With a simple feedback controller of the form g(x)=βx|x|, chaotic dynamics can be observed. Bifurcation analysis and the verification of the existence of chaos are given based on numerical continuation techniques and the Shil'nikov theorem. Both computer simulation and circuit implementation demonstrate the chaotification effect of this simple feedback controller.

We present the set of bifurcation tracking algorithms which have been developed in the LOCA software library to work with large scale application codes that use fully coupled Newton's method with iterative linear solvers. Turning point (fold), pitchfork, and Hopf bifurcation tracking algorithms based on Newton's method have been implemented, with particular attention to the scalability to large problem sizes on parallel computers and to the ease of implementation with new application codes. The ease of implementation is accomplished by using block elimination algorithms to solve the Newton iterations of the augmented bifurcation tracking systems. The applicability of such algorithms for large applications is in doubt since the main computational kernel of these routines is the iterative linear solve of the same matrix that is being driven singular by the algorithm. To test the robustness and scalability of these algorithms, the LOCA library has been interfaced with the MPSalsa massively parallel finite element reacting flows code. A bifurcation analysis of an 1.6 Million unknown model of 3D Rayleigh–Bénard convection in a 5 × 5 × 1 box is successfully undertaken, showing that the algorithms can indeed scale to problems of this size while producing solutions of reasonable accuracy.

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of nonlinear systems of delay differential equations (DDEs). Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at a period-doubling bifurcation is made simple by the present scheme as a parameter is simply increased from zero to a small positive value so that a solution on the new branch is obtained. Subsequent continuation of an emanating branch is also discussed. The advantage of the PI method lies in its simplicity and ease of implementation.

Nonlinearities are an important feature of high Reynolds numbers flows about aircraft. Standard time stepping schemes, used in computational fluid dynamics simulations, are unable to capture the whole solution space, breaking down in the region of bifurcations. The extension of continuation techniques to such flows is therefore attractive. CFD schemes yield large systems of equations and the associated difficulties of applying continuation methods to such large systems need to be overcome. Whilst previous studies of fluids using continuation have been published, these are mainly limited to much lower Reynolds numbers. In high Reynolds number flows, inertial forces dominate and turbulence must be modeled. This study has shown that continuation can be used effectively for high Reynolds number flows demonstrated through the presentation of a number of test cases.

High Reynolds number flows are typical for many applications including those found in aerospace. In these conditions nonlinearities arise which can, under certain conditions, result in instabilities of the flow. The accurate prediction of these instabilities is vital to enhance understanding and aid in the design process. The stability boundary can be traced by following the path of a bifurcation as two parameters are varied using a direct bifurcation tracking method. Historically, these methods have been applied to small-scale systems and only more recently have been used for large systems as found in Computational Fluid Dynamics. However, these have all been concerned with flows that are inviscid. We show how direct bifurcation tracking methods can be applied efficiently to high Reynolds number flows around an airfoil. This has been demonstrated through the presentation of a number of test cases using both flow and geometrical parameters.

We consider complex networks where the dynamics of each interacting agent is given by a nonlinear vector field and the connections between the agents are defined according to the topology of undirected simple graphs. The aim of the work is to explore whether the asymptotic dynamic behavior of the entire network can be fully determined from the knowledge of the dynamic properties of the underlying constituent agents. While the complexity that arises by connecting many nonlinear systems hinders us to analytically determine general solutions, we show that there are conditions under which the dynamical properties of the constituent agents are equivalent to the dynamical properties of the entire network. This feature, which depends on the nature and structure of both the agents and connections, leads us to define the concept of regions of reduced dynamics, which are subsets of the parameter space where the asymptotic solutions of a network behave equivalently to the limit sets of the constituent agents. On one hand, we discuss the existence of regions of reduced dynamics, which can be proven in the case of diffusive networks of identical agents with all-to-all topologies and conjectured for other topologies. On the other hand, using three examples, we show how to locate regions of reduced dynamics in parameter space. In simple cases, this can be done analytically through bifurcation analysis and in other cases we exploit numerical continuation methods.

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

We present the set of bifurcation tracking algorithms which have been developed in the LOCA software library to work with large scale application codes that use fully coupled Newton's method with iterative linear solvers. Turning point (fold), pitchfork, and Hopf bifurcation tracking algorithms based on Newton's method have been implemented, with particular attention to the scalability to large problem sizes on parallel computers and to the ease of implementation with new application codes. The ease of implementation is accomplished by using block elimination algorithms to solve the Newton iterations of the augmented bifurcation tracking systems. The applicability of such algorithms for large applications is in doubt since the main computational kernel of these routines is the iterative linear solve of the same matrix that is being driven singular by the algorithm. To test the robustness and scalability of these algorithms, the LOCA library has been interfaced with the MPSalsa massively parallel finite element reacting flows code. A bifurcation analysis of an 1.6 Million unknown model of 3D Rayleigh–Bénard convection in a 5 × 5 × 1 box is successfully undertaken, showing that the algorithms can indeed scale to problems of this size while producing solutions of reasonable accuracy.

In this note we show how very general continuation results can be used to describe (1) families of periodic orbits in conservative systems, and (2) families of symmetric and doubly-symmetric orbits in reversible conservative systems. We describe a general approach which repairs the lack of submersivity for such problems due to the presence of first integrals. Our results can in particular be applied in the Hamiltonian context; we very briefly describe how they can be used for the continuation of choreographies in the N-body problem.

Parametric roll excitations of a containership are investigated by a "Mathieu-type" roll model. The results of analytical closed-form formulas for the investigated ship are compared with the results obtained by the numerical technique of continuation. The aim of the study is twofold; firstly to extract the parametric roll boundaries of the investigated containership and secondly to validate the analytical expressions developed by international organisations.^1,2 The continuation technique implemented in our study is able to unveil in detail the dynamic portrait of the investigated dynamical system and to identify all associated bifurcations. The method of continuation in nonlinear roll dynamics not only predicts the instability roll boundaries and the steady amplitudes of roll oscillation by following the branches of stable solutions, but is also able to track the branches of unstable solutions and trace the system's bifurcations.