Robert A. Van Gorder

CLASSIFICATION OF CHAOTIC REGIMES IN THE T SYSTEM BY USE OF COMPETITIVE MODES

We study chaotic behavior of the T system, a three-dimensional autonomous nonlinear system introduced by G. Tigan [Analysis of a dynamical system derived from the Lorenz system, Sci. Bull. Politehnica Univ Timisoara50 (2005) 61–72] which has potential application in secure communications. The recently-developed technique of competitive modes analysis is applied to determine parameter regimes for which the system may exhibit chaotic behavior. We verify that the T system exhibits interesting behaviors in the many parameter regimes thus obtained, thereby demonstrating the great utility of the competitive modes approach in delineating chaotic regimes in multiparemeter systems, where their identification can otherwise involve tedious numerical searches. An additional, novel finding is that one may use competitive modes at infinity in order to identify parameter regimes admitting stable equilibria in dynamical models such as the T system.

The variational iteration method is a special case of the homotopy analysis method

Abstract In the present paper, we demonstrate that the variational iteration method (and all of its optimal analogues) are specific cases of the more general homotopy analysis method. To do so, we derive the variational iteration method starting with the homotopy analysis method. The optimal variational iteration method, which also appears in the literature, can be described completely within the context of the optimal homotopy analysis method. Alternately, the optimal homotopy analysis method can be used to construct more general iterative methods of the same ilk, which is potentially useful for solving partial differential equations.

Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

We study the chaotic behavior of the T system, a three dimensional autonomous nonlinear system introduced by Tigan (2005, Analysis of a Dynamical System Derived From the Lorenz System, Scientific Bulletin Politehnica University of Timisoara, Tomul, 50, pp. 61-72), which has potential application in secure communications. Here, we first recount the heteroclinic orbits of Tigan and Dumitru (2008, Analysis of a 3D Chaotic System, Chaos, Solitons Fractals, 36, pp. 1315-1319), and then we analytically construct homoclinic orbits describing the observed Smale horseshoe chaos. In the parameter regimes identified by this rigorous Shilnikov analysis, the occurrence of interesting behaviors thus predicted in the T system is verified by the use of numerical diagnostics.

Nonlinear Flow Phenomena and Homotopy Analysis

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Role of seasonality on predator-prey-subsidy population dynamics.

The role of seasonality on predator-prey interactions in the presence of a resource subsidy is examined using a system of non-autonomous ordinary differential equations (ODEs). The problem is motivated by the Arctic, inhabited by the ecological system of arctic foxes (predator), lemmings (prey), and seal carrion (subsidy). We construct two nonlinear, nonautonomous systems of ODEs named the Primary Model, and the n-Patch Model. The Primary Model considers spatial factors implicitly, and the n-Patch Model considers space explicitly as a Stepping Stone system. We establish the boundedness of the dynamics, as well as the necessity of sufficiently nutritional food for the survival of the predator. We investigate the importance of including the resource subsidy explicitly in the model, and the importance of accounting for predator mortality during migration. We find a variety of non-equilibrium dynamics for both systems, obtaining both limit cycles and chaotic oscillations. We were then able to discuss relevant implications for biologically interesting predator-prey systems including subsidy under seasonal effects. Notably, we can observe the extinction or persistence of a species when the corresponding autonomous system might predict the opposite.

CHAOTIC REGIMES, POST-BIFURCATION DYNAMICS, AND COMPETITIVE MODES FOR A GENERALIZED DOUBLE HOPF NORMAL FORM

We discuss the post-bifurcation dynamics of the general double Hopf normal form, which allows us to study two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including the classical Ruelle–Takens and quasiperiodic routes to chaos. However, it has not, to the best of our knowledge, been considered in the context of the double Hopf normal form, although it has been numerically observed and tracked in the post-double-Hopf regime. We then apply the method of competitive modes to verify parameter regimes for which the double Hopf normal form exhibits chaotic behavior. Such an analysis is useful, as it allows us to potentially identify specific parameter regimes for which the system may exhibit strange or irregular behavior, something which would be extremely difficult otherwise in a system with so many parameters. Indeed, it is conjectured that for parameter sets with two of the square-mode frequencies competitive or nearly competitive, chaotic behavior is likely to be observed in the system. We apply the method of competitive modes to two representative cases where intermittent chaos is found, and the competitive mode analysis there seems to verify the occurrence of chaos in each of the two types of regimes.

Several Types of Similarity Solutions for the Hunter–Saxton Equation*

We study separable and self-similar solutions to the Hunter–Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director Geld of a nematic liquid crystal (among other applications). Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the Hunter–Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact solutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter–Saxton equation.

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic and is implemented in symbolic software such as MAPLE or SageMath. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.

Predator–prey–subsidy population dynamics on stepping-stone domains

Predator-prey-subsidy dynamics on stepping-stone domains are examined using a variety of network configurations. Our problem is motivated by the interactions between arctic foxes (predator) and lemmings (prey) in the presence of seal carrion (subsidy) provided by polar bears. We use the n-Patch Model, which considers space explicitly as a Stepping Stone system. We consider the role that the carrying capacity, predator migration rate, input subsidy rate, predator mortality rate, and proportion of predators surviving migration play in the predator-prey-subsidy population dynamics. We find that for certain types of networks, added mobility will help predator populations, allowing them to survive or coexist when they would otherwise go extinct if confined to one location, while in other situations (such as when sparsely distributed nodes in the network have few resources available) the added mobility will hurt the predator population. We also find that a combination of favourable conditions for the prey and subsidy can lead to the formation of limit cycles (boom and bust dynamic) from stable equilibrium states. These modifications to the dynamics vary depending on the specific network structure employed, highlighting the fact that network structure can strongly influence the predator-prey-subsidy dynamics in stepping-stone domains.