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A simple numerical algorithm is developed for the implementation of the harmonic balance method to analyse periodic responses of a general dynamic system having geometrical non‐linearities of the quadratic and cubic types. The resulting non‐linear algebraic equations which are not explicitly determined are solved by non‐linear equation routines available in most mathematical libraries. Various non‐linear responses, such as the combinational resonances of a hinged‐clamped beam, the non‐linear effect on degenerate vibration modes of a square plate and the non‐linear oscillation of thin rings, are presented to demonstrate the versatility of the algorithm.

The non‐linear damping mechanisms are expressed in two general forms: velocity dependent and displacement dependent. The non‐linear damping phenomena are expressed by an array of ‘local constants’, whose value depends upon excitation frequency, excitation amplitude, and type of non‐linearity. Thus, the non‐linear system is replaced by several localized linear systems corresponding to every discrete frequency and amplitude of excitation. Each of the localized linear systems, thus formulated, characterizes the response behaviour of the original non‐linear system, quite accurately in the vicinity of the specific frequency and amplitude of excitation. An algorithm is developed, which expresses the non‐linear damping by an array of ‘local constants’. The algorithm then employs the usual linear design tools to generate the response characteristics almost identical to the response behaviour of the non‐linear system.

A local equivalent linearization methodology is proposed to simulate non‐linear shock absorbers and dual‐phase dampers in the convenient frequency domain. The methodology based on principle of energy similarity, characterizes the non‐linear dual‐phase dampers via an array of local damping constants as function of local excitation frequency and amplitude, response, and type of non‐linearity. The non‐linear behaviour of the dual‐phase dampers can thus be predicted quite accurately in the entire frequency range. The frequency response characteristics of a vehicle model employing non‐linear dual‐phase dampers, evaluated using local linearization algorithm, are compared to those of the non‐linear system, established via numerical integration, to demonstrate the effectiveness of the algorithm. An error analysis is performed to quantify the maximum error between the damping forces generated by non‐linear and locally linear simulations. The influence of damper parameters on the ride improvement potentials of dual‐phase dampers is further evaluated using the proposed methodology and discussed.

The purpose of this paper is to examine the price–volume relationship in the bitcoin market to validate near-stock properties of bitcoin.

Daily data of bitcoin returns, returns volatility and trading volume (TV) are utilized for the period August 17, 2010–April 16, 2017. Linear and non-linear causality tests are employed to examine price–volume relationship in the bitcoin market.

The linear causality analysis indicates that the bitcoin TV cannot be used to predict return; however, the reverse causality is significant. In contrast, the non-linear causality analysis shows that there are non-linear feedbacks between the bitcoin TV and returns. The bitcoin TV, which represents new information, leads to price changes, and large positive price changes lead to increased trading activity. Similarly, in recent periods (post-break period), the results of the non-linear causality test show a unidirectional causality from TV to the volatility of returns.

This study uses the average index value of major bitcoin exchanges. But further research on this relationship using data from different bitcoin exchanges may provide further insights into the price–volume relationship of bitcoin and its near-stock properties.

These findings from the non-linear causality analysis, therefore, suggest that investors cannot simply base their decisions on the linear dynamics of the bitcoin market. This is because new information in terms of the TV is neither linearly related to the price nor it is a one-to-one kind of relationship as most investors commonly understand it to be. Rather, investors’ decisions should be based on non-linear models, in general, and the best-fitting non-linear model, in particular.

The study examines bitcoin’s near-stock properties in a price–volume relationship framework with the help of both linear and non-linear causality tests, which to the best of the authors’ knowledge remains unexplored.

In this paper an outline of the development of methods for the analysis of non‐linear circuits is presented. Non‐inert and inert elements have been discerned and an inertia measure has been proposed. For this purpose, an exponential function with time constant T equal to T_R, T_L, or T_C is formulated for an element of type R, L, or C, respectively. In circuits with the imposed cause, which varies sinusoidally in time with period T_e, the following situations are distinguished and considered: T_e ≫ T; T_e ≪ T; and T_e ≈ T. In the first case, the effect changes in time according to the non‐linear characteristic of the element. In the second case, the respective circuit is referred to as the “quasi‐linear”, because for the sinusoidal cause the effect is also sinusoidal. In the third case, the hysteresis occurs and the effect is a two‐valued function. The hysteresis effect occurs also in resistive elements.

In this paper, six forms of non-linear Taylor rule have been applied to compare the fitting and prediction of response function of monetary policy of China, in an attempt to figure out a form of non-linear Taylor rule that accords with Chinese practices. The paper aims to discuss this issue.

In this paper, the authors will conduct in-sample fitting and out-of-sample prediction on the response function of monetary policy of China by introducing the factor of exchange rate and by applying forward-looking, backward-looking and within-quarters non-linear Taylor rule with data from the first quarter of 1994 to the second quarter of 2011, with a view to provide reference for formulation and implementation of monetary policies of China.

By analyzing the experimental data, the authors find that first, after introducing the factor of exchange rate, both the implementation effect and prediction ability of the monetary policies improve. Exchange rate has a relatively greater influence on the effect of the monetary policies during low inflation period. Introduction of exchange rate can improve the prediction accuracy of our monetary policies significantly. Second, as the implementation effect of monetary policy under different macro-background varies greatly, the situation should be correctly appraised when formulating and implementing monetary policies. According to the empirical results, the monetary policies have obvious non-linear characteristics, and transit smoothly with the change of inflation rate. On the two sides of inflation rate of 2.174 percent, there is an asymmetry response.

Surely, the conclusions are reached on the basis of quarterly data and one-step prediction method. It is no doubt that use of frequency mixing data including quarterly and monthly data will provide more sample information for studying relevant issues. And the use of multiple-step prediction method may cause a dynamic change of prediction indicators of models, which will help choose more appropriate prediction models. That is what the authors will study next.

First, by introducing exchange rate, this paper will extend non-linear Taylor rules and test its applicability and fitting effect in China. Second, figure out a non-linear Taylor rule that conforms to Chinese practices with data. In this paper, multiple forms of non-linear Taylor rules and actual macro date will be adopted for fitting and finding out a non-linear Taylor rule that fits Chinese practices. Third, empirical basis will be provided for further perfecting monetary policies prediction models. As there are few studies in connection with the prediction accuracy of non-linear Taylor rules so far, this paper will compare and study the prediction accuracy of non-linear Taylor rules by utilizing multiple advanced prediction techniques, so as to offer a beneficial thinking for predicting and formulating monetary policies by the central bank.

Gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1992‐1995.

This paper deals with theory and computation of fluid flow in fractured rock. Non‐Darcian flow behavior was observed in pumping tests at the geothermal research site at Soultz‐sous‐Forêts (France). Examples are examined to demonstrate the influence of fracture roughness and pressure‐gradient dependent permeability on pressure build‐up. A number of test examples based on classical models are investigated, which may be suited as benchmarks for non‐linear flow. This is a prelude of application of the non‐linear flow model to real pumping test data. Frequently, conceptual models based on simplified geometric approaches are used. Here, a realistic fracture network model based on borehole data is applied for the numerical simulations. The obtained data fit of the pumping test shows the capability of fracture network models to explain observed hydraulic behavior of fractured rock systems.

The purpose of this paper is to show how non-linear causal modelling knowledge, already accumulated by other disciplines, is central to unravelling wicked problem scoping and definition in social marketing.

The paper is an illustrative case study approach, highlighting three real-world exemplars of causal modelling for wicked problem definition.

The findings show how the traditional linear research methods of social marketing are not sensitive enough to the dynamics and complexities of wicked problems. A shift to non-linear causal modelling techniques and methods, using interaction as the unit of analysis, provides insight and understanding into the chains of causal dependencies underlying social marketing problems.

This research extends the application of systems thinking in social marketing through the illustration of three non-linear causal modelling techniques, namely, collective intelligence, fuzzy cognitive mapping and system dynamics modelling. Each technique has the capacity to visualise structural and behavioural properties of complex systems and identify the central interactions driving behaviour.

Non-linear causal modelling methods provide a robust platform for practical manifestations of collaborative-based strategic projects in social marketing, when used with participatory research, suitable for micro, meso, macro or systems wide interventions.

The paper identifies non-linear causality as central to wicked problem scoping identification, documentation and analysis in social marketing. This paper advances multi-causal knowledge in the social marketing paradigm by using fuzzy, collective and interpretative methods as a bridge between linear and non-linear causality in wicked problem research.