The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one ... more The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one of the basic routes to chaos. It is rarely mentioned that there are virtually always infinitely many cascades whenever there is one. We report that for one- and two-dimensional phase space, in the transition from no chaos to chaos -- as a parameter is varied -- there must be infinitely many cascades under some mild hypotheses. Our meaning of chaos includes the case of chaotic sets which are not attractors. Numerical studies indicate that this result applies to the forced-damped pendulum and the forced Duffing equations, viewing the solutions once each period of the forcing. We further show that in many cases cascades appear in pairs connected (in joint parameter-state space) by an unstable periodic orbit. Paired cascades can be destroyed or created by perturbations, whereas unpaired cascades are conserved under even significant perturbations.
The occurrence of quasiperiodic motions in nonconservative dynami cal systems is of great-fundame... more The occurrence of quasiperiodic motions in nonconservative dynami cal systems is of great-fundamental importance. However, current understanding concerning the question of how prevalent such m otions should be is incomplete. With this in mind, the types of attractors which can exist for flows on an N•torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N•frequeney quasiperiodic attractors. These perturbations can cause the original N-frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N-frequency quasiperiodic attractors are the most common, followed by (N-Ij-Irequcncy quasiperiodic attractors , ... , followed by periodic attractors. However, as the nonlinearity is further increased. N•frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to mod erate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaot ic attractors which apparently fill the ent ire N-torus (i.e., limit sets of orbits on these attractors are the entire torus); Iurthermore, these are the most. common types of chaotic attractors at moderate nonlinearit ies.
One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimens... more One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. In this paper, in trying to characterize such systems we define a property called "multichaos". A set X is multi-chaotic if X has a dense trajectory and for at least 2 values of k, the k-dimensionally unstable periodic points are dense in X. All proofs that such a behavior holds have been based on hyperbolicity in the sense that (i) there is a chaotic set X with a dense trajectory and (ii) in X there are two or more hyperbolic sets with different unstable dimensions. We present a simple 2-dimensional paradigm for multi-chaos in which a quasiperiodic orbit plays the key role, replacing the large hyperbolic set.
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are un... more Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos). While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.
The appearance of numerous period-doubling cascades is among the most prominent features of param... more The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R × M → M, where M is a smooth locally compact manifold without boundary, typically R N. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe-such as in all the examples we investigate here-that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic F. We illustrate this theory with several examples. We show that there is a close connection between the transition through infinitely many cascades and the creation of a horseshoe.
The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some ... more The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena. Prediction and simulation for chaotic systems occur throughout science. Predictability is more difficult when the "chaotic attractor" is heterogeneous, i.e. if different regions of the chaotic attractor are unstable in more directions than in others. More precisely, when arbitrarily close to each point of the attractor there are different periodic points with different unstable dimensions, we say the chaos is heterogeneous and we call it hetero-chaos. Simple illustrative models of hetero-chaos have been lacking in the literature, and here we present the simplest examples we have found.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017
Transient chaos is a characteristic behaviour in nonlinear dynamics where trajectories in a certa... more Transient chaos is a characteristic behaviour in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations, the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper, we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. We also show, for the first time, the application of this method in three dimensions, which is the major step forward in this work. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyse three quite different ways to implement the method. First, we apply this method by building an one-dimensional map using the successive maxima of one of the var...
Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability.... more Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we say that the set of basins has theWada property and initial conditions near that boundary are even more unpredictable. Many physical systems of interest with this topological property appear in the literature. However, so far the only approach to study Wada basins has been restricted to two-dimensional phase spaces. Here we report a simple algorithm whose purpose is to look for the Wada property in a given dynamical system. Another benefit of this procedure is the possibility to classify and study intermediate situations known as partially Wada boundaries.
International Journal of Bifurcation and Chaos, 2015
There are many ways that a person can encounter chaos, such as through a time series from a lab e... more There are many ways that a person can encounter chaos, such as through a time series from a lab experiment, a basin of attraction with fractal boundaries, a map with a crossing of stable and unstable manifolds, a fractal attractor, or in a system for which uncertainty doubles after some time period. These encounters appear so diverse, but the chaos is the same in all of the underlying systems; it is just observed in different ways. We describe these different types of chaos. We then give two conjectures about the types of dynamical behavior that is observable if one randomly picks out a dynamical system without searching for a specific property. In particular, we conjecture that from picking a system at random, one observes (1) only three types of basic invariant sets: periodic orbits, quasiperiodic orbits, and chaotic sets; and (2) that all the definitions of chaos are in agreement.
Recently physical and computer experiments involving systems describable by continuous maps that ... more Recently physical and computer experiments involving systems describable by continuous maps that are nondi8'erentiable on some surface in phase space have revealed novel bifurcation phenomena. These phenomena are part of a rich new class of bifurcations which we call border collision bifurcations A. general criterion for the occurrence of border-collision bifurcations is given. Illustrative numerical results, including transitions to chaotic attractors, are presented. These border-collision bifurcations are found in a variety of physical experiments.
We consider simple Lyapunov-exponent-based conditions under which the response of a system to a c... more We consider simple Lyapunov-exponent-based conditions under which the response of a system to a chaotic drive is a smooth function of the drive state. We call this differentiable generalized synchronization ͑DGS͒. When DGS does not hold, we quantify the degree of nondifferentiability using the Hölder exponent. We also discuss the consequences of DGS and give an illustrative numerical example. ͓S1063-651X͑97͒02704-9͔
The average lifetime of a chaotic transient versus a system parameter is studied for the case whe... more The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the a~erage lifetime T depends upon the system parameter p via T-~pp,~& , where p, denotes the value of p at the crisis and we caII y the critical exponent of the chaotic transient. A theory determining y for two-dimensional maps is developed and com-pared~ith numerical experiments. The theory also applies to critical behavior at interior crises.
The second sentence in Ref. 8 should read as follows: "Three exceptions (which are quite differen... more The second sentence in Ref. 8 should read as follows: "Three exceptions (which are quite different from our approach) are the papers of Huberman and Lumer, of Hubler (who typically requires large controlling signals), and of
This paper addresses the question of how chaotic scattering arises and evolves as a system parame... more This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e. , not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of "fully developed" chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed. Recently, some progress has been made on the general problem we address in this paper: understanding how and why scattering can become chaotic as a parameter is varied. ' In Ref.
Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologica... more Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologically equivalent, but in fact to be related by a nearly linear change of parameter coordinates. This effect has been observed numerically for one-parameter families of maps, and we offer an analytical explanation for this phenomenon. We further present numerical evidence of the same phenomenon for two-parameter families, and give a mathematical explanation like that for the one-parameter case.
The system of equations introduced by Lorenz to model turbulent convective flow is studied here f... more The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbers r somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence upon r is studied both numerically and (very close to the critical r) analytically.
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1998
Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter... more Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter. We demonstrate that most of these bifurcations can be categorized as "border-collision bifurcations." A method of predicting the local bifurcation structure through the construction of a normal form is applied. This method applies to many power electronic circuits as well as other piecewise smooth systems.
The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one ... more The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one of the basic routes to chaos. It is rarely mentioned that there are virtually always infinitely many cascades whenever there is one. We report that for one- and two-dimensional phase space, in the transition from no chaos to chaos -- as a parameter is varied -- there must be infinitely many cascades under some mild hypotheses. Our meaning of chaos includes the case of chaotic sets which are not attractors. Numerical studies indicate that this result applies to the forced-damped pendulum and the forced Duffing equations, viewing the solutions once each period of the forcing. We further show that in many cases cascades appear in pairs connected (in joint parameter-state space) by an unstable periodic orbit. Paired cascades can be destroyed or created by perturbations, whereas unpaired cascades are conserved under even significant perturbations.
The occurrence of quasiperiodic motions in nonconservative dynami cal systems is of great-fundame... more The occurrence of quasiperiodic motions in nonconservative dynami cal systems is of great-fundamental importance. However, current understanding concerning the question of how prevalent such m otions should be is incomplete. With this in mind, the types of attractors which can exist for flows on an N•torus are studied numerically for N = 3 and 4. Specifically, nonlinear perturbations are applied to maps representing N•frequeney quasiperiodic attractors. These perturbations can cause the original N-frequency quasiperiodic attractors to bifurcate to other types of attractors. Our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows: N-frequency quasiperiodic attractors are the most common, followed by (N-Ij-Irequcncy quasiperiodic attractors , ... , followed by periodic attractors. However, as the nonlinearity is further increased. N•frequency quasiperiodicity becomes less common, ceasing to occur when the map becomes noninvertible. Chaotic attractors are very rare for N = 3 for small to mod erate nonlinearity, but are somewhat more common for N = 4. Examination of the types of chaotic attractors that occur for N = 3 reveals a rich variety of structure and dynamics. In particular, we see that there are chaot ic attractors which apparently fill the ent ire N-torus (i.e., limit sets of orbits on these attractors are the entire torus); Iurthermore, these are the most. common types of chaotic attractors at moderate nonlinearit ies.
One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimens... more One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. In this paper, in trying to characterize such systems we define a property called "multichaos". A set X is multi-chaotic if X has a dense trajectory and for at least 2 values of k, the k-dimensionally unstable periodic points are dense in X. All proofs that such a behavior holds have been based on hyperbolicity in the sense that (i) there is a chaotic set X with a dense trajectory and (ii) in X there are two or more hyperbolic sets with different unstable dimensions. We present a simple 2-dimensional paradigm for multi-chaos in which a quasiperiodic orbit plays the key role, replacing the large hyperbolic set.
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are un... more Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos). While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.
The appearance of numerous period-doubling cascades is among the most prominent features of param... more The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R × M → M, where M is a smooth locally compact manifold without boundary, typically R N. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe-such as in all the examples we investigate here-that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic F. We illustrate this theory with several examples. We show that there is a close connection between the transition through infinitely many cascades and the creation of a horseshoe.
The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some ... more The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena. Prediction and simulation for chaotic systems occur throughout science. Predictability is more difficult when the "chaotic attractor" is heterogeneous, i.e. if different regions of the chaotic attractor are unstable in more directions than in others. More precisely, when arbitrarily close to each point of the attractor there are different periodic points with different unstable dimensions, we say the chaos is heterogeneous and we call it hetero-chaos. Simple illustrative models of hetero-chaos have been lacking in the literature, and here we present the simplest examples we have found.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017
Transient chaos is a characteristic behaviour in nonlinear dynamics where trajectories in a certa... more Transient chaos is a characteristic behaviour in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations, the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper, we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. We also show, for the first time, the application of this method in three dimensions, which is the major step forward in this work. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyse three quite different ways to implement the method. First, we apply this method by building an one-dimensional map using the successive maxima of one of the var...
Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability.... more Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we say that the set of basins has theWada property and initial conditions near that boundary are even more unpredictable. Many physical systems of interest with this topological property appear in the literature. However, so far the only approach to study Wada basins has been restricted to two-dimensional phase spaces. Here we report a simple algorithm whose purpose is to look for the Wada property in a given dynamical system. Another benefit of this procedure is the possibility to classify and study intermediate situations known as partially Wada boundaries.
International Journal of Bifurcation and Chaos, 2015
There are many ways that a person can encounter chaos, such as through a time series from a lab e... more There are many ways that a person can encounter chaos, such as through a time series from a lab experiment, a basin of attraction with fractal boundaries, a map with a crossing of stable and unstable manifolds, a fractal attractor, or in a system for which uncertainty doubles after some time period. These encounters appear so diverse, but the chaos is the same in all of the underlying systems; it is just observed in different ways. We describe these different types of chaos. We then give two conjectures about the types of dynamical behavior that is observable if one randomly picks out a dynamical system without searching for a specific property. In particular, we conjecture that from picking a system at random, one observes (1) only three types of basic invariant sets: periodic orbits, quasiperiodic orbits, and chaotic sets; and (2) that all the definitions of chaos are in agreement.
Recently physical and computer experiments involving systems describable by continuous maps that ... more Recently physical and computer experiments involving systems describable by continuous maps that are nondi8'erentiable on some surface in phase space have revealed novel bifurcation phenomena. These phenomena are part of a rich new class of bifurcations which we call border collision bifurcations A. general criterion for the occurrence of border-collision bifurcations is given. Illustrative numerical results, including transitions to chaotic attractors, are presented. These border-collision bifurcations are found in a variety of physical experiments.
We consider simple Lyapunov-exponent-based conditions under which the response of a system to a c... more We consider simple Lyapunov-exponent-based conditions under which the response of a system to a chaotic drive is a smooth function of the drive state. We call this differentiable generalized synchronization ͑DGS͒. When DGS does not hold, we quantify the degree of nondifferentiability using the Hölder exponent. We also discuss the consequences of DGS and give an illustrative numerical example. ͓S1063-651X͑97͒02704-9͔
The average lifetime of a chaotic transient versus a system parameter is studied for the case whe... more The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the a~erage lifetime T depends upon the system parameter p via T-~pp,~& , where p, denotes the value of p at the crisis and we caII y the critical exponent of the chaotic transient. A theory determining y for two-dimensional maps is developed and com-pared~ith numerical experiments. The theory also applies to critical behavior at interior crises.
The second sentence in Ref. 8 should read as follows: "Three exceptions (which are quite differen... more The second sentence in Ref. 8 should read as follows: "Three exceptions (which are quite different from our approach) are the papers of Huberman and Lumer, of Hubler (who typically requires large controlling signals), and of
This paper addresses the question of how chaotic scattering arises and evolves as a system parame... more This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e. , not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of "fully developed" chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed. Recently, some progress has been made on the general problem we address in this paper: understanding how and why scattering can become chaotic as a parameter is varied. ' In Ref.
Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologica... more Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologically equivalent, but in fact to be related by a nearly linear change of parameter coordinates. This effect has been observed numerically for one-parameter families of maps, and we offer an analytical explanation for this phenomenon. We further present numerical evidence of the same phenomenon for two-parameter families, and give a mathematical explanation like that for the one-parameter case.
The system of equations introduced by Lorenz to model turbulent convective flow is studied here f... more The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbers r somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence upon r is studied both numerically and (very close to the critical r) analytically.
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1998
Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter... more Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter. We demonstrate that most of these bifurcations can be categorized as "border-collision bifurcations." A method of predicting the local bifurcation structure through the construction of a normal form is applied. This method applies to many power electronic circuits as well as other piecewise smooth systems.
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Papers by James Yorke